Pohl's Introduction to Physics

Home , Lichtenberg figure, Robert Wichard Pohl, Wimshurst machine

Klaus Lüders Robert O. Pohl Editors Pohl’s Introduction to Physics Volume 2: Electrodynamics and Optics

With videos: Demonstrations and biography of Robert W.Pohl Pohl’s Introduction to Physics Klaus Lüders Robert Otto Pohl Editors

Pohl’s Introduction to Physics

Volume 2: Electrodynamics and Optics Editors Klaus Lüders Robert Otto Pohl Fachbereich Physik Department of Physics Freie Universität Berlin Cornell University Berlin, Germany Ithaca, NY, USA

Translated by Prof. William D. Brewer, PhD, Fachbereich Physik, Freie Universität Berlin, Berlin, Germany

Additional material to this book can be downloaded from http://extras.springer.com.

ISBN 978-3-319-50267-0 ISBN 978-3-319-50269-4 (eBook) https://doi.org/10.1007/978-3-319-50269-4

Library of Congress Control Number: 2017944730

© Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a war- ranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface to the 24th German Edition (2018) and the 2nd English Edition (2018)

Accompanying Volume 1, this second volume of “Pohl” will be published in an up-to-date format, with a modern system of numbering of the chapters, equations and figures and with exercises at the end of each chapter. Once again, we have taken the opportunity to carry out a critical reading of the whole text. Along with numerous clarifications and new formulations, we have revised many figures and comments to conform with modern notation and symbols, in order to make reviewing the material and reference to other literature as straightforward as possible. As in Volume 1, we have carried out major revisions to the accompanying videos. In the e-book format, they will be readily accessible and can be opened directly by clicking on the links provided. This is true also of supplementary references from the Internet, and of the historical documentary “Simplicity is the Mark of Truth – the Life and Work of Robert Wichard Pohl” by the science journalist Ekkehard Sieker (see the Preface to the 22nd edition, sidenote). He has dealt extensively with the biography of Robert Wichard Pohl in Göttingen. Once again, it is our pleasure and duty to expressly thank the Erstes Physikalisches Institut at the Georg-August-Universität Göttingen and the Fachbereich Physik at the Freie Universität Berlin for generous support. Special thanks go to Prof. K. Samwer, Prof. G. Beuermann, Mr. J. Feist and Mr. C. Mahn from Göttingen, as well as most especially Dr. J. Kirstein from the Freie Universität Berlin, without whose help the preparation of the videos for this new edition would not have been possible. The first English edition of Pohl’s “Physical Principles of Electricity and Magnetism” appeared in 1930 (translated, as was his “Physical Principles of Mechanics and Acoustics” two years later, by Winifred M. Deans). It was published by Blackie & Son, Ltd., London and Glasgow. The translation was based on the second edition of Pohl’s “Einführung in die Physik, Elektrizitätslehre” (Julius Springer, 1929). The present new, second English edition is based on the 24th edition of “Pohls Einführung in die Physik”,Vol.2(Elektrizitätslehre und Optik, Springer Spektrum, Berlin Heidelberg 2018). Again, we gratefully acknowledge the help of Professor W.D. Brewer from the Fachbereich Physik of the Freie Universität Berlin, not only for carrying out the translation of the text with great quality and speed, but also, and this is probably even more important, for his help with the identification and clarification of unclear parts in the text and in our comments. The English-language readers will appreciate the numerous links he added for further information. We owe thanks also to E. Sieker for his translation and production of the English version of his video biography of R.W. Pohl, which is included in the links for this edition. We also wish to thank Dr. T. Schneider and the Springer-Verlag for making this edition possible, and for their generous help in carrying out its preparation and production.

Berlin, Göttingen, Ithaca, March 2017 K. Lüders R.O. Pohl

v From the Preface to the 23rd German Edition (2009)

After the publication of Vol. 1 of R.W. POHL’s Introduction to Physics, covering the topics of Mechanics, Acoustics and Thermodynamics in a new, revised edition in 2008, we now follow it with this Vol. 2, dealing with Electromagnetism and Optics, likewise as a newly-revised edition. We have again taken the opportunity to add supplemental information where it seemed appropriate to us. In addition to new or revised comments and a number of clariﬁcations in the text, the novel features include in particular a series of videos showing demonstration experiments as well as a collection of exercises for the readers. The chapter on Ferromagnetism from earlier editions was againincludedinthesectiononElectromagnetism. The additional videos for this edition were recorded under our own direction in the new physics lecture hall at the University of Göttingen, or else in cooperation with the Physics Didactics Group at the Free University in Berlin. The main part of the exercises on electromagnetism comes from the original English-language edition of 1930; however, we have again added new exercises to both the Electromagnetism and Optics sections. They deal in particular with questions which arise in the text, the ﬁgures, or the videos, and so provide additional information and an aid to understanding the concepts introduced; they are thus intended to make it easier for the reader to review and digest the material in the book.

Berlin and Göttingen, June 2009 K. Lüders R.O. Pohl

vii From the Preface to the 22nd German Edition

ROBERT WICHARD POHL completed his three-volume “Introduction to Physics” in 1940 with the publication of the first edition of his “Optics”. After we had edited the new, revised first volume covering the fields of Mechanics, Acoustics and Thermodynamics in 2004, we decided to combine the chapters on the fundamentals of Electromagnetism and Optics into a second volume. As with the first volume, we wanted to make an appropriate selection of the material from the many previous editions. The present Vol. 2 is based on the 20th edition of the “Elektrizitätslehre” and the 12th edition of “Optik und Atomphysik”, both of which appeared in 1967. For this volume, as for Vol. 1, the IWF Wissen und Medien (Institute for Scientific Films) in Göttingen prepared short videos showing original demonstration experiments. They are made available with the book as a DVD. In addition, the DVD contains the historical documentary film1 “Simplicity is the Mark of Truth” (Original title: “Einfachheit ist das Zeichen des Wahren”, Pohl’s scientific motto; see the Comment), which was planned, researched and, together with the Düsseldorf production studio ‘Kiosque’, filmed by the scientific journalist Ekkehard Sieker. The film offers a detailed view of the life and work of Robert Wichard Pohl in Göttingen. It describes how R.W. Pohl, together with his famous colleagues Max Born and James Franck, made essential contributions to research and the teaching of physics in Germany in the 1920’s. The Physics Institutes of the University of Göttingen in those days became one of the internationally most important centers of physics. Max Born engaged early on in research into Einstein’s theory of relativity and made important contributions to the theoretical foundation of modern quantum theory. James Franck’s research interests likewise lay in the field of quantum mechanics, in particular in the areas of atomic and molecular physics. R.W. Pohl was a pioneer of solid-state physics, and through his research and as a brilliant teacher, he influenced generations of physicists from all over the world. The excellent standing of physics in Göttingen was brought to an abrupt end by the political takeover of the National Socialist (Nazi) Party at the end of January, 1933. Max Born and James Franck were forced to emigrate from Germany; R.W. Pohl remained as the sole physics professor in Göttingen. He was not a publicly political person – he was a scientist, whose area of commitment was in his Institute.

1 Video: “Simplicity is the Mark of Truth” – the Life and Work of Robert Wichard Pohl – http://tiny.cc/xucxny The title of this ﬁlm is the translation of Pohl’s scientiﬁc motto, “Simplex sigillum veri”, under which he held his famous lectures over several decades. It was written on the front wall of the physics lecture hall in Göttingen, and was occasionally mistranslated by jokesters as “Sealing wax is the only truth”. Pohl’s successor, R. Hilsch, felt that the motto was no longer timely and had it removed during a renovation of the lecture hall some time later. “Summer Festival 1952” “Color Centers” “R.W. Pohl’s Farewell Lecture” (Audio only) “Dr. h.c. for Ernest Rutherford” (Audio only)

ix x From the Preface to the 22nd German Edition

However, in the course of his research for the film, E. Sieker came upon the little-known fact that R.W. Pohl had contacts to the civil resistance group around Carl Friedrich Goerdeler, which opposed the Nazi regime. Following the failed assassination attempt on Hitler of July 20th, 1944, his friend and contact to the Goerdeler Group, teacher and lecturer Hermann Kaiser, was sentenced to death and was executed on the 23rd of January 1945 in Berlin-Plötzensee. After the German capitulation, the British Occupying Forces made R.W. Pohl, among others, a member of the De- nazification Commission for the University of Göttingen. Pohl himself regarded it as an obligation of the University to make major restitutions to the scientists who were forced to leave in 1933. The film contains many historical documents and interviews with relatives, friends and other contem- poraries, giving an exceptional insight into the life of the physicist Robert Wichard Pohl.

Berlin and Göttingen, August 2005 K. Lüders R.O. Pohl R.W. Pohl (1884–1976)

R.W. POHL (1884–1976) discussing color centers (F-centers), elementary crystal lattice defects which were discovered at his institute and investigated there for many years. He is shown during a visit to the Ansco Research Laboratory in Binghamton, NY, in the year 1951. Details of POHL’s life and work can be found on the website http://rwpohl.mpiwg-berlin.mpg.de of the Max Planck Institute for the History of Science (MPIWG). There, one can ﬁnd links to other literature, scientiﬁc institutions and websites which offer information and documents on the teaching and research of the famous physicist in Göttingen. In addition, the documentary video “Simplicity is the Mark of Truth” by EKKEHARD SIEKER (Video 1) can be found on the MPIWG web site, together with all the other videos from both volumes and other audiovisual materials, available both for videostreaming or as downloads. Contents

Part I Electricity and Magnetism

1 The Measurement of Electric Current and Voltage ...... 3 1.1 Preliminary Remarks ...... 3 1.2 Electric Current ...... 4 1.3 The Technical Design of Current Meters or Ammeters ...... 11 1.4 The Calibration of Current Meters or Ammeters ...... 12 1.5 The Electric Voltage or Potential Difference ...... 13 1.6 The Technical Design of Static Voltmeters ...... 14 1.7 The Calibration of Voltmeters ...... 15 1.8 Current-Carrying Voltmeters. Electrical Resistance ...... 16 1.9 Some Examples of Currents and Voltages of Varying Strengths ... 17 1.10 Current Impulses and Their Measurement ...... 20 1.11 Current and Voltage Instruments with a Short Response Time. Cathode-Ray Tubes ...... 22 1.12 Measurements of Electrical Energy ...... 23 Exercises ...... 25

2 The Electric Field ...... 27 2.1 Preliminary Remarks ...... 27 2.2 Basic Observations. Different Forms of Electric Fields ...... 27 2.3 Electric Fields in Vacuum ...... 33 2.4 The Electric Charge ...... 34 2.5 Fields in Matter ...... 36 2.6 Charges Can Move Within Conductors ...... 37 2.7 Inﬂuence and Its Explanation ...... 38 2.8 The Free Charges on the Surface of a Conductor ...... 41 2.9 Current Sources Delivering High Voltages ...... 44

xiii xiv Contents

2.10 Currents During Field Decay ...... 45 2.11 Measurements of Electric Charge from Current Impulses. The Relation Between Charge and Current ...... 46 2.12 The Electric Field ...... 49 2.13 The Proportionality of Charge Density and Electric Field Strength . 51 2.14 The Electric Field of the Earth. Space Charge and Field Gradients. MAXWELL’s First Law ...... 53 2.15 The Capacitance of Condensers and Its Calculation ...... 55 2.16 Charging and Discharging a Condenser ...... 59 2.17 Various Types of Condensers. Dielectrics and Their Polarization ... 60 Exercises ...... 64

3 Forces and Energy in Electric Fields ...... 67 3.1 Three Preliminary Remarks ...... 67 3.2 The Fundamental Experiment ...... 67 3.3 The General Deﬁnition of the Electric Field E ...... 70 3.4 First Applications of the Equation F = QE ...... 71 3.5 Pressure on the Surfaces of Charged Bodies. Reduction of Their Surface Tension ...... 74

3.6 GUERICKE’s Levitation Experiment (1672). The Elementary Charge, e = 1:602 1019 As...... 75 3.7 The Energy of the Electric Field ...... 78 3.8 The Electric Potential. Equipotential Surfaces ...... 79 3.9 The Electric Dipole. Electric Dipole Moments ...... 81 3.10 Induced and Permanent Electric Dipole Moments. Pyro- and Piezo- electric Crystals ...... 84 Exercises ...... 86

4 The Magnetic Field ...... 87 4.1 The Production of Magnetic Fields by Electric Currents ...... 87 4.2 The Magnetic Field H ...... 91 4.3 The Motion of Electric Charges Produces a Magnetic Field. ROWLAND’s Experiment (1878) ...... 94 4.4 The Magnetic Fields of Permanent Magnets are also Produced by the Motion of Electric Charges ...... 96 Contents xv

5 Induction Phenomena ...... 101 5.1 Preliminary Remark ...... 101

5.2 Induction Phenomena (M. FARADAY, 1832) ...... 101 5.3 Induction in Conductors at Rest ...... 104 5.4 The Deﬁnition and Measurement of the Magnetic Flux ˚ and the Magnetic Flux Density B ...... 106 5.5 Induction in Moving Conductors ...... 108 5.6 The Most General Form of the Law of Induction ...... 110 Exercises ...... 111

6 The Relation Between Electric and Magnetic Fields ...... 113

6.1 Detailed Treatment of Induction. MAXWELL’s Second Law ...... 113 6.2 Measuring Magnetic Potential Differences ...... 115 6.3 The Magnetic Potential Difference of a Current. Applications .... 117

6.4 The Displacement Current and MAXWELL’s Third Law ...... 122

6.5 MAXWELL’s Fourth Law ...... 125 Exercises ...... 126

7 How the Fields Depend on the Frame of Reference ...... 129 7.1 Preliminary Remarks ...... 129

7.2 A Quantitative Evaluation of ROWLAND’s Experiment ...... 129 7.3 Induction in Moving Conductors ...... 131 7.4 Fields and the Principle of Relativity ...... 132 7.5 Summary: The Electromagnetic Field ...... 135

8 Forces in Magnetic Fields ...... 137 8.1 Demonstration of the Forces on Moving Charges ...... 137 8.2 Forces Between Two Parallel Currents ...... 139

8.3 LENZ’s Law and Eddy Currents ...... 142 8.4 Damping of Rotating-Coil Instruments. The Creeping Galvanometer. Magnetic Flux Circuits ...... 145 8.5 The Magnetic Dipole Moment m ...... 147 8.6 The Localization of Magnetic Flux ...... 152 Exercises ...... 157 xvi Contents

9 Applications of Induction, in Particular to Generators and Motors ..... 159 9.1 Preliminary Note. General Remarks on Current Sources ...... 159 9.2 Inductive Current Sources, Generators ...... 160 9.3 Electric Motors ...... 166 9.4 Three-Phase Motors for Alternating Current ...... 170

10 The Inertia of the Magnetic Field. Alternating Current ...... 173 10.1 Self-Inductance and the Inductance L ...... 173 10.2 The Inertia of the Magnetic Field as a Result of Self-Inductance ... 175 10.3 Alternating Current: A Quantitative Discussion ...... 180 10.4 Coils in Alternating-Current Circuits ...... 182 10.5 Condensers in Alternating-Current Circuits ...... 184 10.6 Coils and Condensers in Series in Alternating-Current Circuits .... 185 10.7 Coils and Condensers in Parallel in Alternating-Current Circuits ... 188 10.8 Power in Alternating-Current Circuits ...... 189 10.9 Transformers and Inductances (Chokes) ...... 190 Exercises ...... 192

11 Electrical Oscillations ...... 195 11.1 Preliminary Remark ...... 195 11.2 Free Electrical Oscillations ...... 195 11.3 High-Frequency Alternating Currents for Demonstration Experiments ...... 198 11.4 The Production of Undamped Electrical Oscillations Using Feedback with Triodes ...... 202 11.5 Feedback with Diodes ...... 204 11.6 Forced Electrical Oscillations ...... 205 11.7 A Quantitative Treatment of Forced Oscillations in a “Tank Circuit” ...... 207 Exercises ...... 210

12 Electromagnetic Waves ...... 211 12.1 Preliminary Remarks ...... 211 12.2 A Simple Electrical Harmonic Oscillator Circuit ...... 212 12.3 A Rod-Shaped Electric Dipole ...... 213

12.4 Standing Waves Between Two Parallel Wires: The LECHER Line .... 219 Contents xvii

12.5 Travelling Electromagnetic Waves Between Two Parallel Wires: Their Velocity ...... 221 12.6 The Electric Field of a Dipole. The Emission of Free Electromagnetic Waves ...... 223 12.7 Wave Resistance ...... 231 12.8 The Essential Similarity of Electromagnetic Waves and Light Waves ...... 233 12.9 The Technical Importance of Electromagnetic Waves ...... 234 12.10 The Production of Undamped Microwaves. Demonstration Experiments for Wave Optics ...... 234 12.11 Waveguides for Short-Wavelength Electromagnetic Waves (Microwaves) ...... 237 Exercises ...... 241

13 Matter in an Electric Field ...... 243 13.1 Introduction. The Dielectric Constant " ...... 243 13.2 Measuring the Dielectric Constant " ...... 243 13.3 Two Quantities Which Can Be Derived from the Dielectric Constant " ...... 245 13.4 Distinguishing Dielectric, Paraelectric and Ferroelectric Materials ...... 245 13.5 Deﬁnitions of the Electric Field Quantities E and D Within Matter ...... 248 13.6 Depolarization ...... 249 13.7 The Electric Field Within a Cavity ...... 251 13.8 Paraelectric and Dielectric Materials in an Inhomogeneous Electric Field ...... 252 13.9 The Molecular Electric Polarizability. The CLAUSIUS-MOSSOTTI Equation ...... 253 13.10 The Permanent Electric Dipole Moments of Polar Molecules ..... 256 13.11 The Frequency Dependence of the Dielectric Constant " ...... 257 Exercises ...... 260

14 Matter in a Magnetic Field ...... 261 14.1 Introduction. The Permeability ...... 261 14.2 Two Quantities Derived from the Permeability ...... 262 14.3 Measuring the Permeability ...... 263 14.4 Distinguishing Diamagnetic, Paramagnetic and Ferromagnetic Materials ...... 263 xviii Contents

14.5 Deﬁnition of the Magnetic Field Quantities H and B Within Matter ...... 269 14.6 Demagnetization ...... 271 14.7 The Molecular Magnetizability ...... 273

14.8 The Permanent Magnetic Moments mp of Paramagnetic Molecules 274 14.9 The Elementary Magnetic Moment or Magneton. The Gyromagnetic Ratio and the Electronic Spin ...... 276 14.10 The Atomic Interpretation of Diamagnetic Polarization. LARMOR Precession ...... 279 14.11 Ferromagnetism, Antiferromagnetism, and Ferrimagnetism ..... 280 Exercises ...... 288

Part II Optics

15 Introduction. Measuring the Optical Radiant Power ...... 293 15.1 Introduction ...... 293

15.2 The Eye as a Radiation Detector. MACH’s Stripes ...... 293 15.3 Physical Radiation Detectors. Direct Measurements of Radiant Power ...... 296 15.4 Indirect Measurements of Radiant Power ...... 297

16 The Simplest Optical Observations ...... 299 16.1 Light Beams and Light Rays ...... 299 16.2 Light Sources of Small Diameters ...... 301 16.3 The Fundamental Facts of Reflection and Refraction ...... 302 16.4 The Law of Reflection as a Limiting Case. Scattered Light ...... 305 16.5 Total Reflection ...... 306 16.6 Prisms ...... 309 16.7 Lenses and Concave Mirrors. The Focal Length ...... 310 16.8 The Separation of Parallel Light Rays by Image Formation ...... 316 16.9 Description of Light Propagation by Travelling Waves. Diffraction . 317 16.10 Radiation of Different Wavelengths. Dispersion ...... 320 16.11 Some Technical Resources: Angled Mirrors and Mirror Prisms .... 324 Exercises ...... 325 Contents xix

17 Image Formation and Light-Beam Boundaries ...... 327 17.1 Fundamentals of Image Formation ...... 327 17.2 Image Points as Diffraction Patterns from the Lens Mount ...... 328 17.3 The Resolving Power of Lenses, Particularly in the Eye and in Astronomical Telescopes ...... 331 Exercises ...... 334

18 Fundamental and Technical Details of Image Formation and Beam Limitation ...... 335 18.1 Preliminary Remark ...... 335 18.2 Principal Planes, Nodes ...... 335 18.3 Pupils and the Boundaries of Light Beams ...... 340 18.4 Spherical Aberration ...... 344 18.5 Astigmatism and Arching of the Image Plane ...... 345 18.6 Coma and the Sine Condition ...... 348 18.7 Geometric Distortion ...... 349 18.8 Chromatic Aberration ...... 350

18.9 Achievements of Optical Technology. The SCHMIDT Mirror ...... 352 18.10 Increasing the Angle of Vision with a Magnifying Glass or a Telescope ...... 353 18.11 Increasing the Angle of Vision with a Projector or a Microscope ...... 356 18.12 Resolving Power of the Microscope. The Numerical Aperture .... 356 18.13 Telescope Systems ...... 360 18.14 The Field of View of Optical Instruments ...... 362 18.15 Imaging of Three-Dimensional Objects and Depth of Field ...... 365 18.16 Perspective ...... 366 Exercises ...... 370

19 Radiation Energy and Beam Limitation ...... 371 19.1 Preliminary Remark ...... 371

19.2 Radiation and Opening Angle: Deﬁnitions. LAMBERT’s Cosine Law . 371 19.3 Radiation from the Surface of the Sun ...... 376

19.4 The Radiance Le and the Irradiance Ee in Image Formation ...... 376 19.5 Emitters with Direction-Independent Radiant Flux ...... 379 19.6 Parallel Light Beams as an Unattainable Limiting Case ...... 381 xx Contents

20 Interference ...... 383 20.1 Preliminary Remark ...... 383 20.2 The Interference of Wave Groups from Pointlike Wave Centers ... 383 20.3 Replacing Pointlike Wave Centers by Extended Centers. The Coherence Condition ...... 385 20.4 General Remarks on the Interference of Light Waves ...... 387 20.5 A Three-Dimensional Interference Field with Two Openings as Wave Centers ...... 388 20.6 The Spatial Interference Field in Front of a Flat Plate with Two Mirror Images ...... 390 20.7 The Spatial Interference Field in Front of a Wedge Plate ...... 392 20.8 Interference in the Image Plane of a Pinhole Camera ...... 394 20.9 Interference in the Focal Plane of a Lens: Longitudinal Observation 395 20.10 Sharpening the Interference Fringes. Interference Microscopy, MÜLLER’s Stripes ...... 397 20.11 The Lengths of Wave Groups ...... 399 20.12 Redirection of the Radiant Power by Interference ...... 401 20.13 Interference Filters ...... 402 20.14 Standing Light Waves ...... 403 20.15 Interference due to Particles which Redirect the Light ...... 404

20.16 YOUNG’s Interference Experiment in the FRAUNHOFER Limit ...... 406 20.17 Optical Interferometers ...... 408 20.18 Coherence and Fluctuations in the Wave Field ...... 410 Exercises ...... 411

21 Diffraction ...... 413 21.1 Casting Shadows ...... 413

21.2 BABINET’s Theorem ...... 416 21.3 Diffraction by Many Equal-Sized and Randomly-Arranged Openings or Particles ...... 418 21.4 The Rainbow ...... 419 21.5 Diffraction by a Step ...... 422 21.6 Diffracting Objects with an Amplitude Structure ...... 422 21.7 Gratings with a Phase Structure ...... 425 21.8 The Pinhole Camera and the Ring Grating ...... 427 21.9 A Ring Grating with Only One Focal Length ...... 429 Contents xxi

21.10 Holography ...... 431 21.11 Visible Imaging of Invisible Objects. The Schlieren Methods ..... 432

21.12 ERNST ABBE’s Description of Microscopic Image Formation ...... 434 21.13 Making Invisible Structures Visible Under the Microscope ...... 436 21.14 X-ray Diffraction ...... 438 Exercises ...... 441

22 Optical Spectrometers ...... 443 22.1 Prism Spectrometers and Their Resolving Power ...... 443 22.2 Grating Spectrometers and Their Resolving Power ...... 445 22.3 The Line Shapes and Halfwidths of Spectral Lines ...... 448 22.4 Spectrometers and Incandescent Light ...... 449 22.5 Comparison of Prisms and Gratings ...... 451 22.6 Various Forms of Line Gratings ...... 452 22.7 Interference Spectrometers ...... 454 Exercises ...... 456

23 The Velocity of Light, and Light in Moving Frames of Reference ...... 457 23.1 Preliminary Remark ...... 457 23.2 Example of a Measurement of the Velocity of Light ...... 458 23.3 The Group Velocity of Light ...... 459 23.4 Light in Moving Frames of Reference ...... 460

23.5 The DOPPLER Effect with Light ...... 463

24 Polarized Light ...... 467 24.1 Distinguishing Transverse and Longitudinal Waves ...... 467 24.2 Light as a Transverse Wave ...... 468 24.3 Polarizers of Various Types ...... 470 24.4 Birefringence, in Particular in Calcite and Quartz ...... 472 24.5 Elliptically-Polarized Light ...... 476 24.6 The Interference of Parallel Beams of Polarized Light ...... 481 24.7 Interference with Diverging Beams of Polarized Light ...... 483 24.8 Optically-Active Materials – Rotation of the Oscillation Plane. The FARADAY Effect ...... 486 24.9 Strain Birefringence. Conclusions ...... 488 xxii Contents

25 The Relation Between Absorption, Reﬂection and Refraction of Light ... 491 25.1 Preliminary Remark ...... 491 25.2 The Extinction and Absorption Constants ...... 491 25.3 The Mean Penetration Depth w of the Radiation. The Extinction and Absorption Coefﬁcient k ...... 493

25.4 BEER’s Law. The Interaction Cross-Section of a Single Molecule ... 494 25.5 Distinguishing Between Weakly- and Strongly-Absorbing Materials 496 25.6 Specular Light Reﬂection by Planar Surfaces ...... 497 25.7 Phase Changes on Reﬂection of Light ...... 500

25.8 FRESNEL’s Formulas for Weakly-Absorbing Materials. Applications . 501 25.9 Detailed Description of Total Reﬂection ...... 504 25.10 The Mathematical Representation of Damped Travelling Waves .. 506

25.11 BEER’s Formula for Perpendicular Reﬂection by Strongly-Absorbing Materials ...... 508 25.12 Light Absorption by Strongly-Absorbing Materials at Oblique Incidence ...... 511 25.13 Conclusion: Pictures Used in Physical Descriptions ...... 513

26 Scattering ...... 517 26.1 Preliminary Remark ...... 517 26.2 The Basic Ideas Underlying the Quantitative Treatment of Scattering ...... 517

26.3 The Radiation from Oscillating Dipoles. PURCELL’s Experiment .... 518 26.4 Quantitative Treatment of Dipole Radiation ...... 521

26.5 The Wavelength Dependence of RAYLEIGH Scattering ...... 522 26.6 The Extinction of X-rays by Scattering ...... 526 26.7 The Number of Scattering Electrons in Light Atoms ...... 526 26.8 Scattering as a Means of Producing and Detecting Polarized X-rays 528 26.9 The Scattering of Visible Light by Large, Weakly-Absorbing Particles ...... 530 26.10 Diffuse Reﬂection from Matte Surfaces ...... 532 Exercises ...... 535 Contents xxiii

27 Dispersion and Absorption ...... 537 27.1 Preliminary Remark and Overview of the Chapter ...... 537 27.2 The Wavelength Dependence of Refraction and Extinction ...... 537 27.3 The Special Status of the Metals ...... 541 27.4 ‘Metallic’ Reflectivity ...... 542 27.5 The Penetration Depth of X-rays ...... 544 27.6 The Explanation of Refraction in Terms of Scattering ...... 545 27.7 The Qualitative Interpretation of Dispersion ...... 546 27.8 A Quantitative Treatment of Dispersion ...... 548 27.9 Refractive Indices for X-rays ...... 551 27.10 The Refractive Index and the Number Density. Light Drag ...... 551 27.11 Curved Light Beams ...... 553 27.12 The Qualitative Interpretation of Absorption ...... 556 27.13 A Quantitative Treatment of Absorption ...... 557 27.14 The Shapes of Absorption Bands ...... 559 27.15 Quantitative Analysis Using Absorption Spectra ...... 560 27.16 The Properties of Optically-Effective Resonators ...... 562 27.17 The Mechanism of Light Absorption in Metals ...... 565 27.18 Dispersion by Free Electrons with Weak Absorption (Plasma Oscillations) ...... 566 27.19 The Total Reflection of Electromagnetic Waves by Free Electrons in the Atmosphere ...... 568 27.20 Extinction by Small Particles of Strongly-Absorbing Materials .... 569 27.21 Extinction by Large Metal Colloids. Artificial Dichroism, Artificial Birefringence ...... 572 Exercises ...... 576

28 Thermal Radiation ...... 577 28.1 Preliminary Remark ...... 577 28.2 Basic Empirical Results ...... 577

28.3 KIRCHHOFF’s Law ...... 578 28.4 The Black Body, and the Laws of Black-Body Radiation ...... 580 28.5 Selective Thermal Radiation ...... 583 28.6 Thermal Light Sources ...... 584 28.7 Optical Thermometry. The Black-Body Temperature and the Color Temperature ...... 586 xxiv Contents

29 Visual Perception and Photometry ...... 591 29.1 Preliminary Remark. The Need for Photometry ...... 591 29.2 Experimental Aids for Varying the Irradiance ...... 592 29.3 The Principles of Photometry ...... 593 29.4 Deﬁnition of the Equality of Two Illuminances ...... 595 29.5 The Spectral Distribution of the Sensitivity of the Eye: The Luminosity Function ...... 597 29.6 The Rise Time and Accumulation Time of the Eye ...... 600 29.7 Brightness ...... 600 29.8 Achromatic Colors. The Conditions for Their Formation ...... 603 29.9 Chromatic Colors, Their Hues, and Their Tinting and Shading .... 604 29.10 Color Filters for Producing Pure Hues ...... 606 29.11 Dyes and Pigments ...... 610 29.12 Brilliance and Shine ...... 611 29.13 Shimmering Colors ...... 611 Color Plates ...... 613

Table of Physical Constants ...... 615

Solutions to the Exercises ...... 617

Index ...... 625 List of Videos

All of the videos for this volume, and also those for Vol. 1, may be downloaded from http://extras.springer.com.

1 “Simplicity is the Mark of Truth” http://tiny.cc/z8fgoy ...... ix 1a “Summer Festival 1952” http://tiny.cc/38fgoy ...... ix 1b “Color Centers” http://tiny.cc/d9fgoy ...... ix 1c “R.W. Pohl’s Farewell Lecture” (Audio only) http://tiny.cc/j9fgoy ...... ix 1d “Dr. h.c. for Ernest Rutherford” (Audio only) http://tiny.cc/q9fgoy ...... ix

Part I Electricity and Magnetism 2.1 “Matter in electric ﬁelds” http://tiny.cc/s9fgoy ...... 28, 244

2.2 “LICHTENBERG’s ﬁgures” http://tiny.cc/99fgoy ...... 38 2.3 “Measuring the electric ﬁeld of the earth” http://tiny.cc/69fgoy ...... 54 2.1 “The capacitance of a sphere” http://tiny.cc/jaggoy ...... 58 2.5 “The Electric Wind” http://tiny.cc/maggoy ...... 59 3.1 “Forces in an Electric Field” http://tiny.cc/xaggoy ...... 73 3.2 “Soap Bubbles in an Electric Field” http://tiny.cc/9aggoy ...... 76 5.1 “Induction in Conductors at Rest” http://tiny.cc/ebggoy ...... 105, 108

xxv xxvi List of Videos

5.2 “Induction in Moving Conductors” http://tiny.cc/hbggoy ...... 109 6.1 “The Magnetic potentiometer” http://tiny.cc/obggoy ...... 117, 118, 121, 122

8.1 “LENZ’s Law” http://tiny.cc/mcggoy ...... 142 8.2 “The Eddy-Current Brake” http://tiny.cc/1bggoy ...... 143 8.3 “Induction Rotor” http://tiny.cc/5bggoy ...... 143 8.4 “Electromagnet” http://tiny.cc/xbggoy ...... 157, 273 9.1 “Circular Vibrations” http://tiny.cc/qcggoy ...... 170, 476 10.1 “The Inertia of a Magnetic Field” http://tiny.cc/wcggoy ...... 177 10.2 “Rotating Magnetic Field” http://tiny.cc/8cggoy ...... 184

11.1 “TESLA Coil” http://tiny.cc/adggoy ...... 199

12.1 “LECHER line” http://tiny.cc/gdggoy ...... 221 14.1 “Paramagnetic materials” http://tiny.cc/kdggoy ...... 266

Part II Optics 16.1 “Polarized light” http://tiny.cc/5dggoy ...... 299, 470, 523 16.2 “Reﬂection cones” http://tiny.cc/qdggoy ...... 305 16.3 “Diffraction and coherence” http://tiny.cc/xdggoy ...... 319, 386, 388, 406, 415, 416 17.1 “Resolving power” http://tiny.cc/9dggoy ...... 333, 418 18.1 “Spherical aberration” http://tiny.cc/1eggoy ...... 345 18.2 “Astigmatism” http://tiny.cc/reggoy ...... 346 18.3 “Chromatic aberration” http://tiny.cc/seggoy ...... 351 List of Videos xxvii

18.4 “Perspective” http://tiny.cc/5eggoy ...... 368 20.1 “Interference” http://tiny.cc/9eggoy ...... 388, 406, 447

20.2 “POHL’s interference experiment” http://tiny.cc/kfggoy ...... 391 25.1 “Absorption” http://tiny.cc/sfggoy ...... 496 27.1 “Curved light beams” http://tiny.cc/wfggoy ...... 300, 553 29.1 “The color white” http://tiny.cc/dgggoy ...... 603 29.2 “Tinting and shading of chromatic colors” http://tiny.cc/sgggoy ...... 605 29.3 “Complementary colors” http://tiny.cc/4gggoy ...... 607 29.4 “How brilliance comes about” http://tiny.cc/zgggoy ...... 611 Magnetism and Electricity I 1

Part I hntesbeto hroyaisi ae p n eeal in- generally one of up, concept taken new the is thermodynamics troduces of details subject the the digital of When aware or development. not historical stopwatch are their we modern of if or a construction even their clocks, of of use use the all make can of We evolution all clock. historical time. We entire beating is the ‘second’. who considering unit slave by a begins water even a one or or sundial No experiments, a balances, first uses the and one No for clocks use. clock rulers, to put i.e. immediately described, are they is and life everyday in tested rmtevr einn ftequantities the of use beginning make very we the magnetism, and from electricity introduce to way, similar a In experiments. first the in are use they and to discussed put briefly immediately are times modern in everyone to known time n Voltage and Current Electric of Measurement The etok nmcaisuulybgnwt h ocpsof concepts the with begin usually mechanics on Textbooks Remarks Preliminary 1.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © ntuet sdt esr hs uniiso neprmna ba- energy the of experimental concepts explain an the introduce on briefly then We quantities We sis. these lives. measure to everyday used our instruments from familiar are which saetepicpe fcok,blne n hroeesat thermometers and balances school clocks, from known of well present. principles as the just be are then as will contents Its dispens- become will able. chapter entire this future, the in Sometime ,and ,and mass C1.1 lcrcpower electric h esrmn paau hc a entidand tried been has which apparatus measurement The . olsItouto oPhysics to Introduction Pohl’s . temperature lcrclrssac,electrical resistance, electrical h ye fthermometer of types The . lcrccurrent electric , https://doi.org/10.1007/978-3-319-50269-4_1 and voltage length , otg measurements. voltage and current for (multimeters) instruments measuring ital dig- of and supplies) (power sources voltage and current practical modern, of use make to choose structors in- when difficulty no see also We it. with to dispense not prefer we book, the of part this in treated topics the to introduction good a gives chapter this since However, extent. great to a least at today, case the already perhaps is This C1.1 1 3

Part I , . 1.3 S 1.1 , a horse- 1.2 in wires and cables. NS electric current N SN Mag- Magnetic We want here to start by elucidatingrents. the We characteristics remind the of reader electric of cur- two old and1. well-known observations: Between the “north pole” andcan the make “south pole” the of pattern a of bariron the magnet, filings we or magnetic powder. field For lines example, we visibleonto could a by put a smooth sprinkling horseshoe surface magnet andlightly sprinkle the to iron distribute filings them. around We it, obtain tapping the pattern shown in Fig. 2. A magnet exertsiron a objects. mechanical In force both on cases, otherusing the iron magnets field filings and lines give which rise on to we soft impressive can patterns. make In visible Fig. shoe magnet is “attempting” to rotate a compass needle. In Fig. netic field lines. The horseshoe magnet rotates the compass needle in a counter- clockwise direction. field lines, represented by iron filings Figure 1.2 Figure 1.1 1.2 Electric Current In everyday life, we speak of an The Measurement of Electric Current and Voltage 1 4

Part I 1.2 Electric Current 5

Figure 1.3 Magnetic ﬁeld lines. A key is

attracted to a horse- Part I shoe magnet.

N S

a horseshoe magnet is pulling a piece of iron (a key) towards itself. We are intentionally using a somewhat primitive mode of expression here. After these initial remarks, we now want to consider the three defining characteristics of electric currents: 1. An electric current produces a magnetic field. A wire which is carrying an electric current is surrounded by concentric circular magnetic field lines. Figure 1.4 shows these field lines using iron filings on a glass plate. The wire extended down through the page, perpendicular to the plane of the paper. It has been pulled out of the hole in the center of the picture after the field-line pattern was produced. – This magnetic field due to the current can give rise to a great variety of mechanical motions. We offer six different examples (a through f):

Figure 1.4 The circular magnetic field lines around a current-carrying wire – b). SN 1.6 N S A C + is hanging loosely; it a, the bar magnet 1.6 CA into the magnetic field of NS is hung parallel above a long, CA S N AC SN a). The rod is hung from two flex- A C ab 1.7 ). When a current through the wire is b). (Fig. 1.5 CA 1.7 . SN SN (Fig. a and b). 1.8 CA A rigidly fixed bar A fixed conductor and a movable, and a suspended, points to the north. (Figs. SN N CA CA magnet (wire) When no current is flowing in the wire, the endmagnet of the flexible conducting ribbon made of woven metal movable bar magnet For that reason, it is called the north pole of theWhen magnet. the current is turned on, the north pole of therotated magnet out is of the plane ofpaper the towards the observer. Figure 1.5 straight wire switched on, aperpendicular torque to acts the wire. on the magnetb) and This it process rotates can until be it reversed. is In Fig. a) A bar magnet (compass needle) Figure 1.6 is fixed. Beside it, a woven metal ribbon is flexible and freeribbon, to move. it moves When so amagnet, i.e. current that it is winds it passed itself is into through a this oriented helix around mainly the perpendicular magnet (Fig. to the c) We bring a straight conducting rod ible conducting ribbons likeswitched a on, trapeze it swing. isthe deflected double along When arrow one (Fig. the of current the is directions indicated by d) We replace the straight conductor(coil). by a When helically-wound conductor theaxis current is switched on, the coil rotates around its a horseshoe magnet e) Thus far, wefield have from considered a arrangements current-carrying wherefield conductor of the interacts a with magnetic permanent magnet. the The magnetic latter field can be produced instead The Measurement of Electric Current and Voltage 1 6

Part I 1.2 Electric Current 7

Figure 1.7 Aﬁxed ab horseshoe magnet SN

and a straight conduct- Part I ing rod CA hung like a trapeze on ﬂexible woven-metal ribbons

S S C C A A N N

Figure 1.8 A ﬁxed horseshoe ab magnet SN and a rotatable A conducting coil CA.The A current leads to the coil are made of ﬂexible woven- N S N S metal ribbons. This is also the schematic of a “rotating-coil current meter” (ammeter) or “rotating-coil galvanometer”.

C C

Figure 1.9 The mutual attraction of ab two current-carrying metal ribbons CC

by a second current-carrying conductor. In Figs. 1.9a and b, the current arriving at point A is split into two branches. At point C,they again flow together. The conductors along the branch AC are two woven-metal ribbons under a slight tension. With an electric current (Fig. 1.9b), they are attracted and are pulled together until they nearly touchC1.2. C1.2. In liquid conductors, e.g. mercury, this magnetic Figure 1.10 shows a variant of this experiment which is often used force can lead to a pinch-off for technical applications. The two flexible ribbons are replaced by of the liquid column (this a fixed and a rotatable coil, which both carry the same electric current was demonstrated in the 21st (Fig. 1.10a). The movable coil orients itself parallel to the fixed coil edition of this book, p. 242). (Fig. 1.10b). .It )in ). 1.3 10.3 a glass tube 1.13 ). – All of the above dealt (in analogy to Fig. Fe Fe 1.12 NS shows a simple demonstration experiment AC AC 1.12 exhibits similar effects of magnetic fields and heating”; see Sect. OULE The magnetic field of a current flowing in a liquid conductor The Afixedcoil heated by ; paper pointers have been attached to the ends of the needle. . It is pulled into the magnetic field of a wire coil (solenoid). SN CA 1.11 The conductor which is carrying an electric current is heated liquid conductor which is suspended so as to which illustrates how aa wire current lengthens (“J as a result of the heating by Fig. – So much for ourmagnetic examples field of of mechanical an motions electric produced current. by the 2. can be heated until it glowscent white-hot, light as bulb. can Figure be seen in any incandes- with solid conductors; we considered metal wires forA the most part. heating. To demonstrate the magnetic field, in Fig. linear expansion of awire (solenoid) and a soft ironFe core be movable Figure 1.13 Figure 1.12 f) Finally, we use a piece of soft iron Figure 1.11 (water with a small amountneedle of sulfuric acid added) is detected by a compass an electric current At the right is a fixed coil, at the left a coil which is free to rotate. The leads to the rotatable coil areinstrument made for of measuring woven-metal current ribbons. or voltage, This including is alternating a current schematic (Sect. of a rotating-coil Figure 1.10 ab The Measurement of Electric Current and Voltage 1 8

Part I 1.2 Electric Current 9

Figure 1.14 Electrolysis of water: The O H electrolytic production of hydrogen (H2) 2 2

and oxygen (O2) when a current is passed Part I through a dilute solution of sulfuric acid (this is a snapshot taken two seconds after switching on the current)

+ – AC

Figure 1.15 The precipitation of lead crys- +– tals at the cathode when an electric current is passed through an aqueous lead acetate solution A C

filled with acidified water is shown. Above it is a small compass needle. Two wires serve as current leads (C and A). – Apart from the magnetic field and the heating effect, we can observe a third effect in liquid conductors: 3. In liquid conductors, an electric current causes chemical changes. These are termed electrolytic. – Examples: a) In a vessel containing acidified water, two platinum wires dipping into the water serve as electrodes C and A (Fig. 1.14). When current is flowing, small bubbles of oxygen rise from electrode A, and hydrogen bubbles appear at electrode C. By convention, the electrode C where hydrogen is produced is called the negative pole (“cathode”). The other pole A is the positive pole (“anode”);“C” for cathode, “A” for anode. We thus employ here an electrolytic definition of the difference between the negative and the positive poles in an electric circuit. b) In a vessel containing a solution of lead acetate (“sugar of lead”), two lead wires dipped into the liquid serve as electrodes. When an electric current flows, a delicate “lead tree” made of tiny joined crystals is formed at the negative electrode C (Fig. 1.15). In this case, the electrolytic effect leads to the precipitation of metal out of the solution. Finally, instead of a solid or liquid conductor, we consider a conducting gas. The U-shaped tube shown in Fig. 1.16 contains the noble gas neon. Again, two metal electrodes C and A serve as current leads. A small framework above the tube carries a compass needle SN.We attach the leads A and C to a current source; immediately, we can observe all three effects of the electric current: The magnetic needle rotates to a new position; the tube is heated; and a brilliant red-orange + A S numerical C N . This would ap- physical quantities amperage or definition. It will not suffice for eals profound changes in the gas, : We can characterize the effects of an current strength qualitative are produced with all conductors. used for the measurements. the concept of “electric current” to summarize of a measurement procedure, and . – Here, we must be careful to keep two things sep- invent denote metallic current is a compass needle. , but its strength is independent of its direction. unit The noble gas neon as A 1 SN and definition technical setup and a C results of this paragraph direction For quantitative specification, instead ofone simply using often the speaks word ‘current’, of the pear to be superfluous;of we pressure’, do or not acite call measured a a time reason measured for the this pressure ‘intensitya usage the of in ‘strength the time’, case etc. of an But electric we current: can A current has Exception: In the case of superconducting materials, there is no heating effect. leads, and a gaseous conductor in atube. U-shaped glass Figure 1.16 In the present casesetup of electric of currents, the we measuringple: begin We instruments. with can the construct technical an Theystrength “ammeter” of can which the be allows current us directly kept on to a quite read scale. off sim- the 2. the 2. heating arate: 1. The 1 3. “chemical effects” (in theductors. broad sense) in liquid and gaseous con- Or, expressed differently: We observetogether, these and three phenomena, often rigorous physics. All concepts which areical employed processes to describe and phys- states mustdefined be by associated a with measurementvalue procedure, i.e. products of a light emitted all along thesuch as tube we rev might observe from chemicalThe processes in a gas flame. electric current in a conductor by three phenomena: 1. A magnetic field, their cause. – This is a The Measurement of Electric Current and Voltage 1 10

Part I 1.3 The Technical Design of Current Meters or Ammeters 11 1.3 The Technical Design of Current C1.3 Meters Or Ammeters C1.3. The technical design Part I of measurement instruments used in the laboratory today, for the most part with a dig- For the construction of such measurement apparatus, one can use ei- ital display, is based on the ther the magnetic or the thermal effects of an electric current: principles of vacuum and Current meters using magnetic effects (symbol shown in Fig. 1.17) solid-state physics. Such in- are based on the arrangements illustrated in Figs. 1.5 through 1.11. struments will therefore be Magnetic force is employed to move a pointer along a scale. The rest treated initially here as “black position of the pointer is determined by a spiral spring or some sim- boxes”, as is usual for com- ilar device. Rotating-coil instruments play an important role. They plex apparatus in general. are based on the scheme shown in Fig. 1.8.

The magnetic ﬁeld usually forms a radially-symmetric pattern; cf. Fig. 1.18, which shows two designs.

Figure 1.19a shows the coil Rc of such an ammeter with a mechanical pointer. For very sensitive instruments, a light-beam pointer is employed: The movable part of the instrument (rotatable coil) carries a small mirror M to reflect a light beam (e.g. a laser beam) (Fig. 1.19b). Instruments of this type are often called mirror galvanometers.C1.4 C1.4. The galvanometer, which today may appear rather old-fashioned, is simply a particularly sensitive A rotating-coil instrument, of which many are still in use. For demonstration exper- Figure 1.17 Schematic symbol for a current meter or ammeter. The same iments in the lecture hall, principle will later be used for instruments which are calibrated for use as a galvanometer still offers voltage meters or voltmeters. some advantages, since it is suitable both for the measurement of weak electric Figure 1.18 Radially symmetric magnetic currents as well as for cur- fields in rotating-coil ammeters; above with N S rent impulse measurements the poles outwards, below with poles in- (as a ballistic galvanometer wards. The magnets are shown with shading with a long response time). and soft iron is black. Two short circular Airgap Rotating coil segments mark the intersection of the rotat- Furthermore, it can be used ing coil with the plane of the page. to demonstrate damped os- S N cillations in a clear manner, including the aperiodic limit. of Rc C M C1.5 B is es- number b A illigram. ampere : C 1180 m : electrolytic effects 1 ) (in one second about Rc D through the cross-section 3.6 G e Z a G A mass M ally deposits 1.1180 milligram of sil- C1.6 ). and are of precipitated material C 8.2 A during which current flows m t and C Mass elaboration of the unit of current called the Time 2 , one determines their total elementary charges). The measurement of this number with the . The rather strange decimal value is due to the historical Two designs for rotating 1 A s; now obsolete. is the mirror for a light- 18 D or Ammeters 10 M in rotating-coil ammeters also provide the “restoring Rc 24 B : required precision by direct counting (single-electronpossible counting) today; is therefore, still one not lets eachbe single transported elementary by electrical a charge carrier, namely aof silver atom, the and carriers instead of the – There are naturallypere’. other procedures The for modern the definitioncarrying realization is conductors based of (Sect. on the the unit force ‘am- between two current- The electrolytic of the current path within a given time (Sect. 6 pecially satisfying in a conceptual sense. Itcurrent states in is principle: That called electric ‘onefined ampere’ number which of elementary corresponds electrical to charges the passage of a de- or Sometimes called the coulometric method, after the older unit for electric charge, That current which electrolytic ver in one secondas was 1 defined ampere as thedefinition. unit of current and is denoted spiral, flexible current leads. beam pointer. torque”, i.e. they rotate the coilto back its zero position when noflowing. current is or “galvanometers”. A coils Figure 1.19 1 coulomb 2 Many current meters,a in pointer deflection particular which rotating-coil is directly instruments, proportional to show the current; one 1.4 The Calibration of Current Meters The calibration of apparatus foris the based measurement on of the electric currentsa arbitrary unit definition for of electric a current.comprehension measurement and procedure The teaching simplest and was measurement based procedure for on the electric currents. It makes use of the quotient , , n ; 117 2.1 24 118 mg : : 6 1 As mol 4 D ,Eq. base units 10 1 for silver) 23 87 g ). ). : ’s principle of ) lacks precisely 2.11 mol, so that 10 C . See also Vol. 1, = elementary charges 65 : DC 9 is the valence of 18 02 : z z 6 10 = ARADAY The Measurement of Electric Current and Voltage PTB-Mitteilungen D 18 (see Sect. 24 1 : Exercise 1.1 Exercise 1.1 n Q 10 unit: ampere second, A s)the to amount of substance deposited is given by equivalence, according to which the ratio of charge Q 6 one elementary charge. Silver has a molar mass of 107.87 g of Ag. This is anof application F where the ions ( ( transport 107 C1.6. The ampere as theof unit electric current strength belongs, along with the meter, the kilogram, the second, the kelvin, the mole andcandela, the to the of the SI (Système Interna- tional d’Unités), which are defined in terms of aurement meas- procedure. All the other units are derived from these seven base units. See the C1.5. Each charged silver atom (the ion Ag Vol. 2 (2007); English, see http://physics.nist.gov/cuu/ index.html Comments C2.14. and 2.15. ( 12

Part I 1.5 The Electric Voltage or Potential Difference 13 determines the quotient

Current Ampere Part I D D ; measured in I Deﬂection Scale division to be constant and calls it the calibration factor of the instrument.

1.5 The Electric Voltage or Potential Difference

In everyday life, we speak of a voltage between two points, for example between the poles of a flashlight battery or the two contacts of an electric socket. – We give here the two defining characteristics of the electric voltage : 1. A voltage can produce an electric current. – This needs no further explanation. 2. Two bodies between which a potential difference or voltage is present are subject to mutual forces. These are often called electrostatic forcesC1.7. C1.7. The analogy between mechanical force and electric This can be demonstrated with a force meter, e.g. a balance. In voltage is often emphasized Fig. 1.20, we see a light-weight balance beam made of aluminum. by referring to the voltage It rests on a knife-edge on the metal post S. At the left end of the as the “electromotive force” beam is a metal plate C, and at the right end, as counterweight, some (E.M.F.). See also the foot- small sliders R made of paper. Below the metal plate C is a sec- note in Sect. 9.1 ond similar fixed metal plate A; the spacing between the two plates is a few millimeters. The plate A and the post S are each connected via a wire to the poles of a current source. When contact is made, the balance beam immediately begins to move. The voltage between the plates A and C produces an attractive force (Sect. 3.4). So much for the qualitative properties of electric voltage or potential differences. For the purposes of physics, we must define a measurement procedure for voltages. Here, again, we consider the definition of the measurement procedure separately from the technical design of the instruments used for making the measurements. We will start with the latter. Both of the characteristic properties of electric voltage can be used to construct voltmeters, and we thus distinguish between instruments which carry a current, and static voltmeters (“electrometers”). We will discuss the two groups separately in Sects. 1.6 and 1.8.

Figure 1.20 C A “volt balance”. R R A B is an insulator. + B –

S of C B attached A C B C A . A loop S A B is penetrated by below. An electric A Q ) shows an image of its field ), obsolete; strictly speak- whichservesasamovable ). Here, again, a metal rod C by an insulator 1.21 1.24 1.23 ). Its operation is similar to that A causes the fibers to approach the A 1.22 (Fig. (Fig. field electrometer ) and C1.8 (Fig. 2.6 on needle bearings. It C C with needle bearings indicates the force which meter. The metal housing C volt A static voltmeter ( A static voltmeter with a gold-leaf Voltmeters (instruments with glass housings are , e.g. by connecting them to a current source. The gold-leaf gold-leaf electrometer field electrometer double-fiber voltmeter C C and . The rod carries a strip of gold leaf voltage applied between fine platinum wires ormechanical fibers tension is by hung the from small the quartz rod; stirrup it is held under to the walls). Theis spacing observed of by a the microscope. fibers Figure thus increases; this increase walls of the housing (or more accurately, the wire loops is isolated from the metal housing is proportional to the applied voltage. c) The practically useless; see Sect. can be used for voltagesto between about a 10 000 few hundred volt. up pointer is attracted bycan the be wall read and off deflected; a scale. its angleb) of The deflection of the gold-leaf electrometer,aluminum but pointer instead of the gold leaf strip, an pointer. The voltage toA be measured is applied between the points a metal rod, electricallyB isolated from the housing by an insulator ing, a gold-leaf pointer with an aluminum pointer Figure 1.21 These instruments make use of the(i.e. “static” by forces the caused electric by charges a collectedoperate voltage by the on voltage; the see below). sameby They principle the as voltage a to smallonly be balance: three measured of The the can many force be different caused designs read used offa) for a such The instruments: scale. We mention Figure 1.22 1.6 The Technical Design of Static ), are intuitively 1.8 The Measurement of Electric Current and Voltage 1 C1.8. Here again, we refer to the remarks in Comment C1.3. The ‘electrometers’ described in this section, which measure voltages with practically no current flowing during the measurement, as well as the ‘rotating-coil instruments’ described below (Sect. clear and simple in theiration, oper- but are largely obsolete today. We will represent them in circuit diagrams by a circle containing ‘V’ (for voltmeter), ‘E’ for electrometer, i.e. a voltmeter inno which current flows during the measurement; or ‘G’ (for galvanometer) and ‘A’ (for ammeter). This will spec- ify where relevant whether a static (‘E’) or a current- carrying instrument (‘V’, ‘A’, ‘G’) is meant. 14

Part I 1.7 The Calibration of Voltmeters 15

Figure 1.23 Model of a “double ﬁber electrometer”. Its B measurement range lies between 30 and 400 volt. Part I A C A

Figure 1.24 The field of view of a double- fiber voltmeter with platinum-coated quartz fibers

302010 0 0 10 20 30

of view, with a scale. The double-ﬁber voltmeter is suitable for projection. Because of its rapid reaction to applied voltages, it is very convenient to use in showing demonstration experiments.

1.7 The Calibration of Voltmeters

The calibration of these instruments is based on the conventional definition of a measurement procedure and a unit for the potential difference. The simplest measurement procedure makes use of a series circuit of N identical batteries (Fig. 1.25), and asserts that the voltage between the ends of the circuit is N times larger than that of asinglebattery(G.S.OHM, 1827). Within the large number of current sources, one particular battery (voltaic element) is chosen as the “normal element” and its voltage is assigned a fixed value.C1.9 The C1.9. The volt is a derived unit of potential difference or voltage is 1 volt (V), and all voltages unit; see Eq. (1.12). How- are quoted in multiples of this unit. ever, for practical reasons, it is realized by different methods, e.g. by using normal Figure 1.25 A series circuit of 6 batteries. elements or, for the highest (The positive electrode is always denoted by +–precision, by making use of a longer bar symbol) a quantum-mechanical effect (the JOSEPHSON effect in superconducting junctions). , I R U 3 and CA as an (1.2) (1.1) holds I % C1.10 . ,forany 4 .Acur- I D constant by ,defined (when it is = U : U ’s law 1.26 . This special resistance const HM D rheostat of an object. Under V = resistance or m , we find for the con- density : 1.26 resistor a metallic conductor , e.g. a length of wire wound on of 500 volt/ampere. Stated I = const device U rs at constant temperature D R I of the conductor has a constant . One then says that O = voltmeters are in principle simply re- D is defined as the I through the conductor V = I U ; this is also called a between the ends of the conductor (e.g. in volt/ampere). This ratio volt/ampere is . – We make use of a series of different current galvanic 1.28 U /volume I . Current m = onductors, a fixed relation holds between voltage CA U sends a current through independent of I (examples: fluorescent tubes, electric arcs, irradiated , B I I Voltage a value for the quotient = to be constant. We thus measure the Electrical Resistance U . Then we divide the corresponding values of D I I CA = of R U The word “resistance” has three different meanings within the field of electro- The quotient mass by the quotient for that conductor: In words: The resistance U value R, i.e. thebetween current its I ends are strictly through proportional to the one conductor another and the voltage U case is found for metallic conducto empirically-discovered ‘law’ or to name it after its author. ductor a spool, as seen in Fig. find as the quotient Suppose that with the setup shown in Fig. e.g. a ribbon or stripthrough of the metal. conductor, while Thebetween ammeter the its measures voltmeter ends the registers current the voltage named and abbreviated bywith the international symbol convention as the “ohm”, rent source magnetism: Firstly, it refers to the quotient of voltage and current, arbitrary conductor. Secondly, it refers to a constant ambient conditions (pressure,materials. temperature But it etc.), is it still not is usual constant to consider for the many relation 3 4 This can be demonstrated with the setup shown in Fig. 1.8 Current-Carrying Voltmeters. Current-carrying or calibrated ammeters. The recalibrationthat is in made metallic possible c and by current. the fact In general, we define every conductor as an electrical value crystals, photocells). Only in special cases does one find a The electrical resistancethe in current general depends in a complex way on variable). Thirdly, resistance means, as inoppositely everyday to life, the a velocity force of which a is moving directed electric charge, similar to a frictional force. sources (e.g. various batteries orcurrent accumulators) and thereby vary the m, . 1 .Its 8 is .As specific m =% 1 10 1 .Wefind lity factor . Then the A l 55 D 7 and inversely resistivity : l 1 10 or specific conduc- D : conductivity l 45 % A : or 6 % C: The Measurement of Electric Current and Voltage ı D D and a length is a property of the mate- 1 reciprocal an example, for copper at 20 tance called the proportional to R C1.10. This type of conductor is also called an “Ohmic conductor”. Suppose that it has a cross-sectional area A % rial and is called the resistance The proportiona value of its resistance is proportional to 16

Part I 1.9 Some Examples of Currents and Voltages of Varying Strengths 17

Figure 1.26 The measurement of a resis- V tance U=I (e.g. of the ﬁlament CA of an

incandescent lamp) or the demonstration Part I CA of the special case that OHM’s law holds (the A conductor CA could be a ﬂat metal ribbon held at constant temperature for this demon- B stration)

brieﬂy, the conductor CA then has a resistance R D 500 ohm. If two resistances (cf. footnote 3 on previous page) are connected in series, their overall resistance is equal to the sum

R D R1 C R2 : (1.3)

For parallel circuits, the overall resistance R is given by the equation

1 1 1 D C (1.4) R R1 R2

(G. S. OHM). So much for the deﬁnition of electrical resistance and OHM’s law.

OHM’s law now allows us to re-calibrate an ammeter to make it into a voltmeter. – The usual ammeters contain a wire through which the current to be measured ﬂows, for example rotating-coil instruments (Fig. 1.19). We know the value of its resistance, deﬁned by the ratio

volt R D x D x ohm I ampere x is here a numerical value. As a result, we need only to multiply the ampere calibration by the factor R D x volt/ampere in order to convert the ampere calibration to a volt calibration (and the instrument into a voltmeter). We repeat: Current-carrying voltmeters are simply re-calibrated ammeters. We therefore draw them in our circuit diagrams according to the scheme shown in Fig. 1.17, but with the letter ‘V’ to indicate ‘voltmeter’. The measurement instruments described in Sects. 1.3 to 1.8 operate on readily recognizable physical principles. This is a great advantage for the student of physics or electrical engineering.

1.9 Some Examples of Currents and Voltages of Varying Strengths a) Voltages of the order of 1 V are found between the poles of dry cells or batteries for doorbells, ﬂashlights etc. A A circuit kV are 3 B G are connected to G B potentiometer 2 V or is the full output voltage C C 1 ). CA V) are available (e.g. van de 2.9 6 10 shows a convenient setup for such D .The G 1.28 voltage divider 1MV , which is typically wound with a length of C1.11 D CA . Then by moving the slider along the resistor, we can G ). The technical design of Schematic of a voltage di- ). The two poles of the current source 10.3 1.27 a potentiometer circuit. produce any desired voltage from zerothe to ends the of source 1 voltage and between 2. – Figure Many experiments require variable voltages.by These a can trick be which permits obtained age. selecting some fraction One of a makes maximum use volt- of a used. e) For physics research, generators which produceof voltages up MV to tens (1 megavolt of the current source.this Between voltage one and end sothus and attach forth the a for middle wire 1 other isa to one-half fractions metal one slider of end of the the length. resistor and a We second can wire 2 to fairly resistive metal wire madean of insulating an appropriate drum. alloy and Between ﬁxed its on ends a “variable resistor” Graaff or “belt generators”; see Sect. (Fig. a variable resistor (‘potentiometer’ or ‘rheostat’) with a sliding contact vider circuit (potentiometer circuit) resistance wire is wound ontoing an cylinder. insulat- Figure 1.27 Now, we turn to some examples of current strengths. a) Currents of theused order for room of lighting. 1 A are passedb) 100 through A common is lamps aelectric typical streetcar current or which subway is train. employed in the motors of an Figure 1.28 b) The linehundred voltage volts. between the Incf. contacts Sect. Europe, of 220 V wall sockets is is usual (for a alternating few current, c) Above voltages of several thousand voltsand (kV), there arcing will through be sparks theproduce air. an arc A 1 voltage mm long of in around air. 3000d) V For (3 kV) long-distance can transmission lines, voltages up to 10 in . ,it B G 1.27 and C The Measurement of Electric Current and Voltage 1 the schematic of Fig. must be taken into account that the current source also has an (internal) resistance which acts in series with C1.11. In calculating the voltage between 18

Part I 1.9 Some Examples of Currents and Voltages of Varying Strengths 19

mA Part I

Figure 1.29 Including a test person within an electric circuit. An ammeter of the type shown in Fig. 1.8 is used. The handles contain hidden protective resistors which keep the test person safe in the event of a malfunction.

Figure 1.30 Measurement of the current delivered by a HOLTZ inﬂuence machine or “Wimshurst machine” with a rotating-coil ammeter

c) 103 A is called 1 milliampere (mA). Currents of a few mA (3 to 5 mA) can barely be perceived when they are passed through the human body. This can be demonstrated using the setup shown in Fig. 1.29. The test person is connected to the circuit via two metal handles. The applied voltage is slowly and smoothly increased using a voltage divider as described above. d) Currents of the order of 105 A are produced by an influence machine or “Wimshurst machine”, used in the 19th century as a high- voltage generator. We can measure this current as in Fig. 1.30 using a technical ammeter. One often encounters a strange prejudice C1.12. Here, and for all here: An influence machine is supposed to produce “static electric- the applications mentioned ity”, while an ammeter measures only “galvanic” currents. In fact, later, we could replace the C1.12 there is no difference between static and galvanic electricity! influence machine by an electronic high-voltage source e) 106 ampere is called 1 microampere (A). Currents of this order (a “power supply”). How- of magnitude can easily be produced by the human body. In Fig. 1.31, ever, just as in Fig. 1.29, a test person grasps two metallic handles with both hands. They are it is then wise to limit the connected by wires to the ammeter (mirror galvanometer). Clasping maximum current by insert- the handles loosely, we observe no current. Tensing the finger mus- ing a resistance R into the cles of one hand produces a current of some 106 A. If the other hand circuit (as for example in is tensed, a similar current is observed, but in the opposite direction. Video 3.1). f) High-quality mirror galvanometers can measure currents down to around 3 1012 ampere (3 pA).C1.13 C1.13. Using electronic instruments, today we can This lower limit is determined by the BROWNian molecular motion of the measure currents two orders movable parts (rotating coil etc.). With a still higher sensitivity (a lighter of magnitude smaller. ) 1.8 illa- from its t flows during

(A) I Current . This term is analogous to ). This integral has a brief and t d I R ). The simplest example of a current b: A constant current t d F 1.32 R ( current impulse a: The current drops within a time 5 s). (This current is due to processes in the skin and not 1.32 : impulse 0 D Light pointer Observation of the weak electric currents produced by tensing

T and Their Measurement

ian motion (Vol. 1, Sect. 9.1)

(A) I Current ROWN B coil or a finer suspension),just the as zero chaotic a point manner of (albeit the much instrument more would slowly) as move the in dust particles in the finger muscles. The rotating-coil galvanometer (schematic in Fig. in the muscles!) with a mirror and lighttion pointer period is ( distinguished by an especially short osc initial value to zero.time integral over The the shaded current ( area in the figure represents the 1.10 Current Impulses Often, in the course of physicstric experiments we currents are dealing which with elec- areammeter rests constant at a over certain time;ing value a on then its constant deflection. scale the and However,flow in pointer remains only many there, of measurements, very show- currents briefly; an forsketched example with in a Fig. time dependence like that Figure 1.31 appropriate name, the the mechanical impulse is shown in Fig. Three examples of time integrals of the current or “current impulses”, measured in ampere second Time (s)Time (s) Time (s) Time

The Measurement of Electric Current and Voltage

Current Current (A) 1 I abc (A s) Figure 1.32 20

Part I 1.10 Current Impulses and Their Measurement 21 a time t. The magnitude of the current impulse is then given simply by the product of the current and the time; it is thus I t, with the unit ampere second (A s). In a corresponding manner, by summa- Part I tion (Fig. 1.32c), current impulses of arbitrary time dependence can be evaluated. This is however too tedious, so that it is done only on paper. In reality, a current impulse is a quantity which can be measured in a very convenient way. This requires reading only a single pointer position from an ammeter. The ammeter in this case merely has to fulfill two conditions: 1. At constant current, the constant deviation of its pointer must be strictly proportional to the current. This is to good accuracy the case for rotating-coil galvanometers (Sect. 1.3). Since one can consider a rotating-coil galvanometer to be a torsion pendulum (Vol. 1, Fig. 6.5), this proportionality means that the force produced in an ammeter of this type is proportional to the current flowing through it. The same is true of the (mechanical) impulse resulting from a current impulse. 2. The oscillation period of the pointer must be long compared to the time during which current flows in a current impulse. Then the torsion pendulum leaves its rest position with practically its maximum angular velocity, which is proportional to the impulse and thus also to the current impulse. For a pendulum with a linear restoring force law, the amplitude u0 of its velocity is proportional to its maximum deflection x0:

u0 D !x0 ; (1.5) where ! is the circular frequency of the pendulum (Vol. 1, Sect.4.3). We thus expect a constant value of the quotient

Current impulse D B : Impulse deﬂection I

To demonstrate this, we use a current impulse with a rectangular shape (Fig. 1.32b). That is, we pass a known current I during a short but precisely measured time t through a slowly oscillating galvanometer (its period of oscillation in Fig. 1.33 is 44 s). Any suitable sort of electric time switch can be used for this purpose.

A known electric current I of suitable strength can be provided by the circuit shown in Fig. 1.33. Using a voltage divider (Fig. 1.27), for example a voltage of 1=100 V is selected. This voltage produces a current which passes through the galvanometer and through a resistance of 106 . This current I is then, according to OHM’s law (Eq. 1.2), equal to 102 V/106 D 108 A. With this setup, we can observe the galvanometer deﬂection ˛ for different values of the product It. We then repeat the measurements with different current strengths. The times t are also arbitrarily varied between a few tenths and around 2 s. the gal- . Instead rectangular Observed de- 1.34 /=. It calibrated 199Ω Developed in 1899, it Ω A s/scale division. Thus, 6 8 ’s tube”) belongs to a con- 10 2 V C1.15 10 asshowninFig. 2 RAUN : 1 V G Current impulse . 1Ω 1 D

100

D I

A 10 I B –8 B c. C1.14 1.32 b), and at the same time we have As. gal- 7 1.32 “Frictional elec- Calibration with a Short Response Time. Cathode-Ray Tubes ). For all the measured values and in all cases, we obtain the 10 ) in the unit ˛ . ballistic 2.11 2.36 frictional electrification machine flection ampere second siderable extent in this latter category. about 2 The ballistically-calibrated galvanometer cansure now an be unknown useda current to impulse. mea- of To sealing show wax this, andmenters cat’s and we fur, the hair we improvised of use thea other. the One deflection hand stroke of across of about the one hair 16 produces of scale the divisions, that experi- is a current impulse of we have demonstrated thedeflection proportionality to of the current theshape impulse observed (Fig. for impulse a currentvanometer ballistically. pulse This of result canarbitrary be current readily generalized: impulse Every canpulses, be as put in together Fig. by adding rectangular trification machine”. The same galvanometer is used here asFig. in of the impulse deflections of a galvanometer with a slow response time (‘ vanometer’; see also Sect. same result; in our example, Figure 1.33 obsolete. The cathode-ray tube (“B 1.11 Current and Voltage Instruments Great physical achievements ofof past our decades general are and today technicalence knowledge, fair already and projects part are and the even like; topics some for of sci- them are also in the meantime Figure 1.34 We then compute the ratios . ARL (1858– Phys. Bl. RAUN B ’s tube” see RAUN The Measurement of Electric Current and Voltage , 1040 (1998); English: see 1 ERDINAND F 54 https://en.m.wikipedia.org/ wiki/Cathode_ray_tube e.g. F. Hars, “Hundert Jahre Braunsche Röhre”, diodes. For a discussion of “B been largely replaced by flat screens making use of liquid crystals or light-emitting 1918). Today, cathode ray tubes for viewscreens and television screens have point. C1.15. Named after K seen in several of thefor videos this volume, it seemsportune op- and reasonable to treat it in some detail at this is especially well suited for demonstration experiments in the lecture room, as can be listic galvanometer contains a lot of interesting physics (e.g. it is a damped oscillator; see Vol. 1, Sect. 11.10) and and voltage impulses of far greater sensitivity and precision. However, since the bal- C1.14. Of course, today we have technical instruments for measuring current 22

Part I 1.12 Measurements of Electrical Energy 23 represents a current- and voltage-measuring instrument with an extremely short response time. In the oscilloscope, which is today indispensable in many laboratories, it is used to display and record Part I rapidly-occurring electrical processes. The “pointer” is an electrically deﬂected electron beam which strikes a ﬂuorescent screen and becomes visible there. Frequencies up to 1010 Hz can be visualized in this way.

Using deﬂections in two coordinate directions, one can measure two different quantities simultaneously, for example two currents, two voltages, a current and a voltage, or current vs. time; the time can be represented either as a linear deﬂection or as an angle, etc.

1.12 Measurements of Electrical Energy

Today, we can hardly imagine everyday life without electrical phenomena, with their innumerable applications. Everyone needs at least two electrical concepts in daily life, namely electric current I and electric voltage U. Both are measured as multiples of the units ampere and volt. – Making use of these two electrical quantities, we can also measure the electrical energy. A typical setup is shown on the right in Fig. 1.35. We observe the same temperature rise when the product UIt has the same value (t is the time during which the current I ﬂows). Thus, this product is a measure of the energy E, with the unit ‘volt ampere second’ (V A s):

E D UIt : (1.6)

To distinguish it from other types of energy, it is often termed JOULE heating. Applying the electrical resistance R D U=I, we obtain the frequently-used form

U2 E D I2 R t D t : (1.7) R Mechanically, we measure the energy E in terms of the product called ‘work’,

W D Fl: (1.8)

(Unit: newton meter; an experimental setup is shown at the left in Fig. 1.35. – F is the force component parallel to the path l.)

A mechanically- and an electrically-produced energy are thus equal when they increase the temperatures of two identical calorimeters (Fig. 1.35) by the same amount. That occurs in the experiment when

FlD UI t (1.9) a )in- m (1.12) (1.10) (1.11) (1.13) is not .Atthe 1.9 l D same nu- F and its unit : force; they acceleration U : Resistance heater ). The definition 1.9 t measure energy units : falls along a path Time Fl F tities of energy, measured me- Current source AV IU I Water 1 watt second (W s) l units. Instead, we make use of Work are not defined independently of one D m 1 volt ampere second . Current newton meter base 2 ampere second s 1 D = D F D U Falling path 1kgm D and the mass 1volt F Voltage as the product on the right with the unit V A s. There- ) is fulfilled. – Or, expressed differently, we dispense 1newtonmeter The production of identical quan , the force 1.10 m = Water F Stirrer D 1 volt ampere second (V A s) and the unit 1 newton define the force as a derived quantity with the defining equation another as base quantities. Physicists use the mass to Analogously, in the fundamentala equation of mechanics, i.e. physically necessary, but isnational rather agreement: the The unit resultthat ‘volt’ of Eq. has an ( been expedient defined in inter- such a way with defining all three of the quantities on the right in Eq. ( dependently of one another as is the current to measure voltage. We define the voltage This equality of the mechanical and electric right: Input of electrical energy by heating a resistor (immersion heater). chanically and electricallyidentical by calorimeters. means At of the equalparatus; left: temperature a Mechanical increases metal energy input in block via which two a exerts stirring the ap- force and thus The electrical energy unit is thus the ‘watt second’, so that Figure 1.35 i.e. when the product on the left with the unit N m has the merical value fore, we find ‘volt’ as a derived quantity by making use of Eq. ( The Measurement of Electric Current and Voltage 1 24

Part I Exercises 25

In practice, we usually employ 1 kilowatt hour (kWh) D 1 kilo- volt ampere hour. This is a unit of electrical energy with an industrial C1.16 price of the order of 10 Eurocents. C1.16. The industrial price in Part I Germany at present (2016) is In mechanics (Vol. 1, Sect. 5.2), the concept of ‘power’ WP was de- about 8 Euroct./kWh; for ﬁned by the equation private users, it is nearly dW 30 Euroct./kWh, with an WP D (1.14) increasing tendency. In dt the U.S., these values are (work W, time t). Its mechanical unit is 1 newton meter/second, while 5.8 $ Ct. (industrial price) its electrical unit is 1 volt ampere D 1 watt. and 11.1 $ Ct. (private users), respectively, with the industrial price falling and the private-user price rising.

Exercises

1.1 For the calibration of an ammeter using the electrolytic method described in Sect. 1.4, divalent copper ions (CuCC)were used. At the negative electrode, a mass increase of 5.9 g per hour was measured. The ammeter indicated a current of 4.5 A. What was the true value of the current I? (The molar mass of copper is 63.54 g/mol.) (Sect. 1.4)

1.2 Bauxite (Al(OH)3) is used for the electrolytic production of aluminum (the so-called ﬂux-melt electrolysis process). The molar mass of aluminum is 26.97 g/mol. How long does it take to produce 1 metric ton of aluminum with a current of 4 000 A? (Sect. 1.4)

1.3 How could we increase the maximum indicated value of a voltmeter with the internal resistance Ri D 1 from 0.15 V to 15 V? (Sect. 1.8)

1.4 The maximum indicated value of an ammeter with the internal resistance Ri D 1 is to be increased from 0.05 A to 10 A. For this purpose, a resistor Rshunt is connected in parallel to its terminals (a so-called shunt resistor). Find the correct value of Rshunt. (Sect. 1.8)

1.5 In Fig. 1.26, we could replace the electrometer (static voltmeter) by an uncalibrated rotating-coil ammeter with a ﬁnite OHMic resistance of RV. RV is known. The voltage U is measured using the voltmeter, and the current I with the ammeter. How can we determine the resistance R of the conductor CA from these measurements? (Sect. 1.8)

1.6 A voltage Ua is applied to two OHMic resistances R1 and R2 which are connected in parallel, and the total current I is measured. If the resistances are connected in series, the voltage must be increased . < D N has that I 1 I R ,if ) 2 5 V is con- ) : between the R 1 1.9 I . Calculate 1.12 U D are connected in 0 10 V,the heat pro- . Find the current 2 1 U 5m R R D . An ammeter with 10 V, is applied. Find D U . (Sect. ”, 13th ed., p. 220), so and 1 Z 5 D 1 P R Q R = U D 2 Ws/kg; see Vol. 1, Fig. 14.3). P Q 6 R 10 45 : between the cathode and the anode (the 2 U D ) V L Optik und Atomphysik and the ratio of the amounts of heat produced 1.8 ) 2 I ). With an applied voltage of 1.12 and cells connected in series, each producing a voltage of 1 . (Sect. 2 , see also Exercises 2.11, 2.15 and 2.16; , see also Exercise 2.7. I is three times the heat produced in N 2 ) D 3.7 3.8 ) R in order to obtain the same current. Determine of cooling water per hour. The electron current in the source 1 a To illuminate a lecture room, 20 light bulbs are used, each An electric motor has an efficiency of 80 %, i.e. it converts In an X-ray source (see Fig. 19.10), the anticathode (anode) A battery with an open-circuit voltage of a) Two wires with the resistances 3 R 1.9 1.12 U 5 : 25 A. Explain the result and calculate the voltage which is flowing through the circuit. b) The wires are now con- : operating at 220 V andconsists 5 A of (DC, from a storage battery). The battery 1.8 (Sect. 1.9 80 % of the electricalAt energy an that operating is voltage fed ofof to 220 mass it V 1.5 into DC, kg mechanical it at energy. is a supposed velocity of to 2 lift m/s. a How weight large is the current it draws? (Sect. 1.10 consists of a hollow water-cooled cylinder. In100 operation, cm it evaporates series ( poles of the battery when this current is flowing. (Sect. 2 V. Each cell has an internal resistance of cooling water is atheat room of temperature vaporization is (20 °C) when it enters, and its I nected in parallel and the same voltage, the currents 1.7 the value 2 1.11 1 %, see R.W. Pohl, “ The efficiency with which X-radiation is generated is very low ( that the energy transferred to(Sect. the X-radiation can be neglected here. as a function of time in the two wires, For Sect. For Sect. is 10 mA. Find the voltage to 4 nected to a conductor of resistance a negligible internal resistance is used to measure the current, 0 duced in The Measurement of Electric Current and Voltage 1 26

Part I hl eraeo hi pcn erae h otg.Tetwo The plates the of voltage. spacing the the in decreases changes the spacing follow voltage, voltmeter the their the of increases of fibers plates decrease the a between distance while the Increasing 1. voltage: measured the of value strong the on a is there that show eim setal olwn t itrcldvlpet estart We electromag- of the development. field of the historical concepts the of its with treatment basic following systematic the essentially a of concepts, netism, give these some to of introduced want use we making we Now, end, electromagnetism. this of To concepts difference). (po- voltage tential and current electric for instruments measuring important . rlmnr Remarks Preliminary 2.1 Field Electric The 1 Figure Forms Different Observations. Basic 2.2 in summarized was book this in chapter first the Sect. of purpose The .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © V. the 220 in of power used by were value replaced which been a have batteries they quote storage today, supplies. of decades; often many set for large we hall a lecture text, of Göttingen voltage the output In the is This voltage. output adjustable readily w diinleprmnswt h eu hw nFig. in shown setup the with experiments additional Two the the important! (Fig. to removed to wires been connecting have plates the source as- current after two this even contradicts of remains the experiment voltage cause The The of source. The sumption: current connection a plates. the of two poles apparently output the is between voltage V 220 this of voltage a indicates ot r aewt insulators with made are ports h tagtowr ici hw nFig. in shown circuit straightforward the (a voltmeter static a eter to wires, other two using i w ie oacretsuc tavlaeo 2 V 220 of voltage plates a these at connect source We current a “ground”. to the wires from two via and other each from oa,teeaecmecal-vial urn ore “oe upis)with supplies”) (“power sources current commercially-available are there Today, ..n urn osdrn h esrmn) etu have thus We measurement). the during flows current no i.e. , 1.1 2.1 twsitne opoieabifoeve ftemost the of overview brief a provide to intended was It . fEeti Fields Electric of hw w aall a ea plates metal flat parallel, two shows olsItouto oPhysics to Introduction Pohl’s lcrcfield electric nuneo h pc ewe h plates the between space the of influence B ota hyaeeetial isolated electrically are they that so , and 2.2 2.2 lcrccharge electric ih) hsi extremely is This right). , ,left. obefie electrom- double-fiber A C2.1 , and https://doi.org/10.1007/978-3-319-50269-4_2 h voltmeter The C hi sup- Their . . 1 2.2 n then, and (right) le)tr “condenser”. term older) (and simpler the use will we chapter, this In details. more for below See charge. electric store to ability their quantifies which value tic characteris- the is latter the “capacitance”; and ratings voltage standard with ponents com- commercially-available for particular in used, often also is “capacitor” term the technology, electronics In condenser”. “parallel-plate a case this in “condenser”, a called is ground) the from usually (and another one from insulated electrodes two of arrangement an Such C2.1. 2 27

Part I E E 100V ; at the right as B AA E 220V 220V CC BB ) ) electrometer Video 2.1 voltmeter (Sect. 1.6) B static : parallel-plate condenser; A parallel-plate condenser with insulators A disk made of some arbitrary material between CA ). The voltage drops to a fraction of its original value. When AC AC 2.3 is employed.) indicates that a a shadow image. The diameter of the plates is about 22 cm at the left while itcurrent is connected source, to at the the rightconnection after is the broken (here andfollowing in figures, the the symbol formeter the (circle with volt- ‘E’, for the condenser plates ( with a remarkableoriginal precision. value, we When againexample the find 220 V. the spacing initial is value returned of to2. the its voltage, Without touching in the our rial plates, into we the slide space a between(Fig. thick the disk condenser of plates some (metal, mate- plastic etc.) Figure 2.3 Figure 2.1 Figure 2.2 we pull the disk out again, the original value of 220 V is restored. ). 13 The Electric Field 2 Video 2.1: “Matter in electric fields” http://tiny.cc/s9fgoy This video shows how inserting various materials into the electric field of a parallel- plate condenser which has been disconnected from the current source reduces the voltage between the condenser plates; the voltage is restored when the material is withdrawn (cf. Chap. 28

Part I 2.2 Basic Observations. Different Forms of Electric Fields 29

Figure 2.4 Two gold-covered quartz ﬁbers which have spread apart (the distance between the spread ﬁbers re-

mains small compared to the spacing of the condenser Part I plates A and C) CA

– +

In this space between the condenser plates, unusual forces act, which otherwise do not occur; an example is shown in Fig. 2.4:Twofine metal fibers (gold-covered quartz threads) will spread apart when brought into this space between the plates (we will discuss this effect in detail in Chap. 3; see also Fig. 2.39). We can amplify these phenomena by increasing the voltage: We replace the power supply by an influence machine or Wimshurst machine, already mentioned in Chap. 1 (Sect. 1.9); it produces several thousand volts. Then we sprinkle some small fibrous particles, for example small tufts of cotton, between the plates. The fibers stick at one end to the plates and stretch away from them. Sometimes they fly across the gap from one plate to the other, following straight-line paths in the center and curved paths at the edges of the plates. (This can be seen with particular clarity in the shadow image!) We investigate this remarkable behavior of the fibrous particles in more detail. We try to observe it systematically in the whole of the space between the plates. To this end, we repeat the previous experiments “two dimensionally”: Fig. 2.1 shows at the right a vertical section through the two plates C and A. We replace it in Fig. 2.5 by two metal foil strips glued onto a glass plate. Between them, we apply a voltage of around 3000 V. Then we scatter some sort of in-

Figure 2.5 The electric ﬁeld lines in a parallel-plate condenser, made visible with gypsum crystals (this and all the following images of electric ﬁeld lines have not been retouched at all)

C A ). . 1.4 (“lines 2.7 and through two new con- 2.6 1.1 A . A C electric field lines . . In the beginning, one should electric field condenser on the basis of experience. Otherwise, , 1777 ILCKE electric field W ARL , just as we have done for the basic concepts of the “me- A similar image as Elec- C , from a drawing by 2.6 A An electric field introduces a preferred direction into a region : and a sphere or OHSEPH J On the basis of our observations thus far, we introduce cepts 1. Two conductors betweenference which (voltage) there are is called an a electric potential dif- 2. The space between theselines two or objects, lines in of which force, we contains can an detect field We have to derive theexperience basic concepts of the “electrical world” from chanical world”. We have fornomenon example of come “weight” to through many understandout experiences the in such phe- daily experiences life. and With- chanics. observations, we In could a not similarconcept deal way, with of we me- the will have to become familiar with the approach the subject in ais quite completely all naive right and to unbiased visualize manner. the electric It field lines as visible chains of we will never be able toena. penetrate into the world ofof electric space; phenom- such a directionlines is make not this present clear in in “empty” a space. pictorial way The field (showing the paths of motion of gold-leaf shreds). Thisthe is principle of “electrostatic spraying” used for painting. in Fig. tric field lines between a plate C awire Figure 2.6 sulating fibrous dust particles,the e.g. glass powdered plate gypsum and crystals, tap onto into it peculiar gently. lines; The small we crystals see orient a themselves pattern of the Figure 2.7 of force”). They appearlines superficially which we to made resemble visible using the iron magnetic filings field (Figs. We can vary this experimentreshape in one a of variety the ofobtain two ways. “two-dimensional” plates patterns For into as example, shown a in we circle Figs. or a line (wire). Then we The Electric Field 2 30

Part I 2.2 Basic Observations. Different Forms of Electric Fields 31

Figure 2.8 Electric ﬁeld lines between two spheres or

between two parallel wires Part I

Figure 2.9 Sketch of the electric field CA lines between wires and the wall of the room when the positive pole of the current source is grounded, i.e. connected to the earth by a conducting link dust particles. Later on, it will be no problem to distinguish between the field lines and this rough, descriptive image. We will offer four additional examples of condensers of different types and show the corresponding patterns of their electric fields: 1. Two adjacent spheres or wires (Figs. 2.8 and 2.9).C2.2 C2.2. The concept of a field line has thus far simply been The image in Fig. 2.8 shows how the field between the poles of our electric used to describe the experi- sockets looks. Often, one pole of a current source is permanently con- mentally observed patterns nected to the surface of the earth by a conducting wire. The field then (forces on small particles) looks like the sketch in Fig. 2.9. In this drawing, the positive pole A is as- which allowed us to deduce sumed to be “grounded”. When wires are open in the room, we sometimes the existence of a vector field. see the beginnings of a “field line pattern”; one of the wires will have attracted a large amount of dust and looks like a hairy caterpillar. Along the In graphic representations wall adjacent to this wire is often a dusty strip; it marks the other ends of of fields, as in Fig. 2.9, one the field lines on the wall. in addition makes use of the density of the field lines to 2. In Fig. 2.10, at the right we see a “carrier of electricity”, that indicate the magnitude of the is, one of the plates of a condenser; it could be a metal disk A or field (the “field strength”). a sphere. The other plate is formed by the surface of the earth, the walls of the room, the furniture and the experimenter. Figure 2.11 illustrates a dainty shape for such a carrier, a “spoon” on an insulating handle. Later, in Fig. 2.48, we will see the field of a spherical carrier of electricity. 3. An antenna and the hull of a ship (Fig. 2.12). We can see how the field lines run from the antenna to the masts and the hull. 4. Figure 2.13 shows the pattern of field lines in and around a static voltmeter. A voltmeter (or “field electrometer”) of this type is simply a condenser; one of the condenser plates takes the form of a movable pointer (cf. Figs. 1.21 and 1.22). A review of the electric fields that we have observed shows us two things: ”. C , one of the ILCKE A , and the observer is the other plate A Electric field lines between a carrier of electricity (in earlier Electric field lines around Electric A “spoon” on an insulating ) 12 first to investigate the parallel-platenamely acts condenser, as said one in of 1757, the plates “The conductor times called a “conductor”) and its surroundings. J. C. W a static voltmeter, or an electrometer as condenser field lines between the antenna and the hull of a ship (a“historical” truly picture, since the antennas for the short-wave radio signals used today are just short dipoles; see Chap. handle as a “carrier ofcompactly, electricity” a “charge or, more carrier” Figure 2.13 Figure 2.12 Figure 2.10 Figure 2.11 The Electric Field 2 32

Part I 2.3 Electric Fields in Vacuum 33

1. All of the field lines always run perpendicular into the surface of the condenser plates. Part I 2. Among all the possible electric fields, two have a particular geometrical simplicity. In a sufficiently flat parallel-plate condenser, the field is homogeneous (Fig. 2.5).C2.3 The field lines are straight, par- C2.3. The experimental proof allel, and have a constant density. A spherical carrier of electricity, of the homogeneity of the at some distance from the other pole of the condenser, gives rise to field can be carried out in a spherically-symmetric field (Fig. 2.48). principle by the method indicated in Figs. 2.22–2.24 In the following, we will refer for the most part to the homogeneous using influence charges (bal- field of a sufficiently flat parallel-plate condenser. The direction of listic galvanometer), whereby the field will be defined as usual by the convention that the field lines the two carriers of electricity point from the positive towards the negative plate. must be small compared to thesizeoftheplate(seealso Sect. 2.13). 2.3 Electric Fields in Vacuum

(ROBERT BOYLE, before 1694). All of the experiments described in the previous section would give the same results in high vacuum or in air. An electric field can exist in empty space. Air at atmospheric pressure has little effect on the observations of electric fields. Its influence is seen only when very precise measurements are carried out (apart from spark discharges and similar phenomena). At atmospheric pressure, a difference of 0.06 % is found for some characteristics of the field as compared to measurements in high vacuum. This result, which has been verified by numerous observations, is understandable in terms of the molecular picture of the composition of the air. Figure 2.14 reminds us of the most important facts: It shows room air with a linear magnification of around 2 106,as an instantaneous image. The molecules are drawn as black points. Their spherical shape is arbitrary and unimportant. Their diameter is around 3 1010 m. Their average spacing is about ten times larger. The volume occupied by the air molecules themselves is thus practically vanishingly small compared to the volume of empty space around them.2 We add to this static picture of the air a fast exposure with an exposure time of around 108 s(Fig.2.15). The flight paths of three molecules are drawn in, but here only at a 6 104-fold magnification. The straight lines are the “free paths” between two collisions (about 107 m). Every kink in the paths corresponds to a collision with one of the other molecules (not shown). Their average path velocity at room temperature is about 500 m/s. 1 m3 of room air contains around 3 1025 molecules.

2 In one cubic centimeter of room air, there are about 3 1019 molecules. The diameter of each individual molecule is of the order of 3 1010 m. If they were strung together, they would form a chain which could be wound roughly 200 times around the earth’s equator. ) –9 m 2.11 type of m 10×10 –7 one different types of charge 05 two 0 1 2 3×10 1733), following a suggestion by . From among the many possible AY F D . We will have to distinguish two types U m and 9 C F. D 10 3 , (Göttingen 1778), who denoted them by the 4 An the ends of the field lines, there is something D d , a potential difference of 220 V has been established HARLES The mean free path of A schematic in- 2.16 poles of the current source. Then we insert a disk-shaped ICHTENBERG -fold magnification. A cross- 6 -fold; see also Vol. 1, Sects. 16.3 4 10 and 10 They could serve equally well for distinguishing 4nmisshown. mathematical symbols and move it backAt and the forth end in ofsurface the of each the direction motion, plate of briefly. we the Each allow double touch the reduces arrows. carrier the disk voltage between to touch the demonstration experiments, we give two examples: 1. In Fig. carrier of electricity or charge carefully between the plates (Fig. between the plates of aC condenser by connecting them briefly to the G. C. L gas molecules in room air6 (magnified stantaneous image of room aira2 at section of width Figure 2.14 and 17.10) (one positive and one negative), as for an excess or a deficit of only charge. of charges (C 3 2.4 The Electric Charge We continue our experimental observationsrive of at electric the fields following and conclusion, which ar- previous was sections: already anticipated in the which can be transferred or filledwill from call one it conductor to “electric another. charges” We Figure 2.15 The Electric Field 2 34

Part I 2.4 The Electric Charge 35

Figure 2.16 A charge carrier transports charges Part I CA

– +

E 220V

Figure 2.17 Filling electric charges from 220V a current source (Sect. 2.2) onto the plates of a condenser bb

E the plates. The carrier transports negative charges from the left to the right plate, and positive charges from the right to the left plate. 2. In Fig. 2.17, top, we see the C and poles of the current source; at the bottom of the ﬁgure, there is a parallel-plate condenser connected to an electrometer, but now without a voltage between the plates. We then move two small charge carriers in the direction of the arrows along the dashed paths. We can observe that a voltage is produced between the condenser plates, and it increases with each additional motion of the carriers. Now we cross the paths, going from the pole to A and from the C pole to C: Then the voltage decreases again;

Figure 2.18 A simpliﬁed version of the ex- 220V periment in Fig. 2.17. The positive charges are + – transferred to the left-hand condenser plate via a wire, while the negative charges are brought to the right-hand plate by a “charge carrier” (or “spoon”).

E ). + 2.18 insu- 2.19 The first E LRICH U 220V C1.10 – . CA RANZ ition between the two conductivity ). For clarity, we mention this oth still longer. With hard rub- 3.6 , i.e. it will also cause the field to decay, (see Sect. (1759). (1729). The fact that the trans electrons EPINUS RAY A G An object is bridging the gap between the HEODOR TEPHAN T S time only if they were very coldno and radiation were of located in any a kind. perfectThe vacuum characterization with of materials as ‘conductors’ or ‘insulators’ is due to An electric field could exist between two objects for a practically unlimited kinds of material is continuous was first recognized by F members of the series areend called are good insulators (very conductors, poor while conductors). those atThere its is no “perfect” conductorductor and is no so “perfect” ideal insulator. thatlator No conducts it con- to destroys some the extent fieldeven if instantly. this And takes every a very long time. fact here; we will return to it later. two condenser plates 2.5 Fields in Matter We produce an electric fieldthen in a bridge condenser the in gap the between usual manner the and plates by some object (Fig. charges of the “wrong” sign are transportedshows to a the plates. simplified (Figure version of this experiment.) Every electric charge cansize be is decomposed reproducibly into always tinyfirst the elementary portions same charge whose carriers (“elementaryand to charges”). were be called discovered The had a negative sign several seconds, with cardboard orber cl or plastic, it takesof quite the a field few minutes, occurs only and after with some amber,In the hours or decay this days. way, weis can called the order series the of decreasing different electrical materials in a series, which We repeat the experiment numerous timese.g. with in different that substances, orderor metal, plastic, wood, amber. cardboard, cloth, Inthe electric glass, each field case, hard decays; the rubber theQuantitatively, voltage result however, between we the is find plates qualitatively striking drops the differences:the to zero. Metals same: field destroy very quickly, sogether that in the an fibers unmeasurably short of time. the voltmeter With wood, collapse this to- process lasts Figure 2.19 The Electric Field 2 36

Part I 2.6 Charges Can Move Within Conductors 37 2.6 Charges Can Move Within

Conductors Part I

We refer directly to the preceding experiments and ask the question: How can the objects placed in an electric field destroy it? A first answer, which suffices for many purposes, can be seen by comparing Fig. 2.19 to Fig. 2.16. In Fig. 2.16, electric charges were transported from one plate to the other using a carrier – the negative charges from the left to the right, and the positive charges from the right to the left. This permits the charges to combine in pairs and remain close together. Then their electric field no longer acts outside their immediate location, and it disappears in the space between the condenser plates. In Fig. 2.19, the field disappears when the gap between the condenser plates is bridged by an object. This allows us to arrive readily at the following conclusion: Charges can somehow move, pulled by the force of an electric field, through material objects. The positive and the negative charges approach each other and combine as pairs. Briefly stated: In conductors, electric charges are mobile. In insulators, one must then assume that there is no appreciable charge mobility within the material. This is confirmed by experiment. One can demonstrate that electric charges stick on or in insulating materials in numerous ways (see for example the 21st edition of POHL’s Elektrizitätslehre, Chap. 25). We limit ourselves here to two examples: 1. We repeat the charge transfer experiment shown in Fig. 2.18,but now instead of a metal disk as charge carrier on the “spoon”, we use some good insulator, e.g. plexiglas. Furthermore, in Fig. 2.20, we use a different voltage source, a small influence machine, and as a static voltmeter, we employ the field electrometer which was shown in Fig. 1.22. These two carriers of electricity behave in quite different ways. A conducting metal spoon need only touch the terminal of the voltage source at one point, both for taking on charge and for dispensing it. The carrier made of insulating material, however, gives rise to only very small deflections of the instrument if it is touched at just one point. In order to transfer larger quantities of charge, we have to rub the whole surface of the carrier on the pole of the current source and on the condenser plates to take up and release the charge. We have to “spread around” the charges onto the surface of the carrier in order to collect them, and to dispense them, we have to “scrape them off” again. 2. We can also make spots of electric charge on the surface of insulating materials. These spots can be made visible just like grease spots on a piece of cloth, by dusting them. For example, we can put an insulating plate of glass between a sheet of metal and a pointed wire. The sheet is connected to one pole of a current source with a high voltage, e.g. an influence machine. Then a small spark is allowed to ition, . The overall A conductor always Influence will prove . 2 in the visible region. The shows an image of such 4 > C2.5 . 2.21 of conducting material into the influence ). Figure 2.9 , 1757). In our experiments on field de- can also be short piece C2.4 ILCKE figure” (Göttingen, 1777) W figure” ) ARL An electric spot. This kind of Trans- C ); as a result, the charges flow away where the layers have been irradi- 25 Video 2.2 ICHTENBERG ICHTENBERG OHANN Electric spots can also be made by spraying charges onto a thin layer of insu- electric field, so that itthen does observe not the touch phenomenon the of condenser plates. We can later to be our most importanttion tool coils, for radio detecting electric antennas fields etc.). (induc- our At this experiments point, which we we consider describe thecontains in result positive advance: of and negative charges,state, but in there its are usual “uncharged” exactly the same amounts of the two plates using a (moreperiments, or we less now good) put conductor. a Continuing these ex- cay up to now, we have bridged the gap between the two condenser projection) “L ferring charges using charge carriers made of different materials (at the left isinfluence an machine which was especially designed for shadow readily produced on photographic emul- sions, which then, instead ofwith being a dusted powder, are developedway in ( the usual Figure 2.20 ated: Dust particles stickprinciple only of to xerography, the used spots in copying which machines. were not irradiated. This is the 4 lating material whose optical indexlayers of become refraction conducting is on irradiationChap. (as described in detail in the 21st ed the charges on theproduce surface an of electric the field which glassplate. extends If plate out we are dust into a invisible; the fineglass space powder, but for surface, above example they flowers the we of sulfur, can ontoby the see the that powder the which ends sticksa there, of white just room the wall as field (cf. with lines Fig. a“L an are electric marked wire above 2.7 Influence and Its Explanation (J Figure 2.21 jump from the other pole of the source to the pointed wire. Initially, - www. 1770). ICHTEN ’s figures” HRISTOPH (1742–1799), C EORG from the Historical . The Electric Field ICHTENBERG 2 ICHTENBERG BERG L Physics Institute of the Uni- versity of Göttingen. A number of original examples of his apparatus can still be seen today in thetorical His- Collection at the 1st professor of experimental physics at the University of Göttingen (from images made by L the latter case, an imagethat like in Fig. 2.21 isThey seen). are compared with the which are formed when negative electric charges are either sprayed onto a plate, or are removed from it (in http://tiny.cc/99fgoy Various figures are shown of this book. See e.g. powerlabs.org/wimshurst. htm machine). Its principles of operation were described in detail in previous editions high voltages, as we have already mentioned several times in relation to the influence machine (or Wimshurst C2.5. Influence plays an important role in producing C2.4. G “L Video 2.2: to a lightning stroke, leaving an optical change in the plexiglas so that their paths become visible. by grounding at the edge of the disk, they flowedalong out branched paths, similar branched figures as on the surface. Energetic electrons were first injected to aof depth ca. 1 cm into the disk; then removing electrons from the interior of a plexiglas disk leads to the same kind of Collection. At the end of the video, we demonstrate how 38

Part I 2.7 Inﬂuence and Its Explanation 39

Figure 2.22 Inﬂuence. Two ﬂat disk- Field direction shaped carriers of electricity ˛ and ˇ are

touching each other within an electric ﬁeld. Part I

- +- +

C αβ A

Figure 2.23 The two carriers ˛ and ˇ Field direction have been separated, still in the ﬁeld

-++-

C α β A

“charge” of an object refers simply to an excess of charges of one or the other sign. To demonstrate influence, we make use of the homogeneous electric field of a sufficiently large and flat parallel-plate condenser AC (Fig. 2.22), and show the individual steps in the experiment in terms of two-dimensional field-line patterns. We use a metal plate as the conducting object. It is made in the form of two disks with insulating handles; the disks touch each other at several points when they are held together. The flat surfaces of the disks are oriented perpendicular to the field lines. We can then make the following observations:

1. We separate the two disks within the field and observe that the space between them is field free; fibrous powder shows no tendency to stick to them or to orient itself (Fig. 2.23). Interpretation: Field . . β ˇ – and 2.23 E and ˛ 2.24 + α Field direction have to end on the 2.23 and . There are always “influence shape of objects placed in them 2.22 2.25 on: As a result of field decay within ; they are then connected to a static volt- ). The voltmeter indicates a voltage and +++ 2.24 1.24 and is opposite to the direction of the original field in the . The fields thus mutually cancel each other in Fig. 1.23 2.24 Example of influence with distortion of the electric field Explanation of influence. The carriers of AC ––– ), tightly joined and thus undetectable. ab are compatible with each other? Answer: The direction of the and C electricity which have been removedprove from to the be field electrically region charged Figure 2.24 Figure 2.25 decay means that charges,move within pulled the by conductors until the no forces field remains in between an electric field, Where do these charges come from?they The inevitable were conclusion is already that present in( the conducting disks, however as pairs 2. We keep theregion, two as disks seen separated in and Fig. remove them from the field meter (Figs. a conductor, the field lines in Figs. thus an electricwith field; opposite the signs. Interpretati two disks each carry an electric charge, The homogeneous electric fieldclarify was our intended experiments only onfields to influence are simplify behavior. and inhomogeneous and Inis the general, arbitrary. electric Thendistorted, the for field example as lines in are Fig. not only interrupted, but also charges” which collect at the points where the field lines are inter- condenser 2.23 field in Fig. surfaces of the disks.a negative The charge, right-hand and disk the left-hand in disk these3. a images positive charge. acquired How can we show that the field-line patterns in Figs. The Electric Field 2 40

Part I 2.8 The Free Charges on the Surface of a Conductor 41 rupted. These can also be detected individually in every case; one need only break apart the conductor at a suitable junction while it is in the ﬁeld. This is suggested in Fig. 2.25b by the dotted line. Part I

2.8 The Free Charges on the Surface of a Conductor

Now, continuing our experiments, we again bring a conducting object into an electric field. Previously, we bridged the gap between the two condenser plates with the object; the field decayed immediately, and we concluded that the charges within conductors can move freely. The second time, the object was located separately within the field region, not touching the plates, and we observed that the charges can be pulled apart by influence. Now, as a third case, we allow the object to touch only one of the electrodes which produce and delimit the field. We ask: How do the mobile charges distribute themselves within the conductor? We will find that the answer is that they move to the outer surface of the conductor and remain there. We deduce this from a two-dimensional model experiment with field lines made visible as dust patterns. In Fig. 2.26, the two black circles indicate the poles of the current source. The field between them originally looked like the field in Fig. 2.8. Now, however, we have attached a conductor in the form of a hollow sheet-metal cup to the negative pole; it has a small hole at its top. We see that all the field lines end at the outer surface of the cup. In its interior there are no field lines, and thus also no field-line ends or free charges. This model experiment of course requires additional confirmation through further experiments. We describe three of these: 1. Fig. 2.27 corresponds to our model experiment; but in addition, the positive pole of the current source is connected to the housing of our static voltmeter. The voltmeter acts as a condenser (Fig. 2.13), so that it can accept and store electric charge. The positive charges pass

Figure 2.26 Pattern of ﬁeld lines between a sphere and a“FARADAY cup” with a small opening

+ – , . – AITZ On the cage”). . 2 1 3 ). We first move 220V 2.28 ARADAY 2.11 ) + E 1.22 , RANKLIN F ENJAMIN Shielding against electric fields by a metal mesh (J. S. W On the bottom inner surface , suggests a general method for obtaining such a field- 2.26 This is illustrated by the arrangement shown in Fig. 1745. The voltmeter is similar to the one in Fig. 1755) of a conducting box whichon is all nearly sides, closed or ofarenofreecharges(B a conducting cup, there Figure 2.27 Figure 2.28 free space: Thea region conducting to shell, be closed on shieldedinfluence all charges need sides. to only Then the be the outerfaces field surrounded surface remain indeed of by moves the completely shell;therefore free field but free. its of inner The charges,closed; sur- shell and or a shield housing its need madelarge not inner will of even volume suffice. be wire completely This is mesh kind whose of openings shield are is called not a too “F through the connecting wire,ported by while a the small “charge negative transport spoon” charges (Fig. are trans- we can not collect eveninner a surface small of charge a from conducting its cup, inner there are surface. no freeA electric practical charges application: Often,from one electric wishes fields. toFig. shield a The certain phenomenon space of influence, explained in the spoon along path 1 andsame observe behavior a is deflection found of on thecharge voltmeter. path The at 2. all In along contrast, the pathnected spoon 3. via transfers This a no result conducting is wire astonishing; to the the cup current is source, con- but nevertheless, The Electric Field 2 42

Part I 2.8 The Free Charges on the Surface of a Conductor 43

Figure 2.29 Removing and return- 2 ing charges with a “charge spoon” Part I 1 α

+ –

220V

Without the “cage”, the static voltmeter would exhibit a large deflection. Within the cage, it indicates no voltage at all. The field cannot penetrate into the interior of the cage. We can increase the voltage C2.6. Earlier editions of this of the influence machine and allow loud sparks to jump between the book contained the following balls and the cage, but the interior will remain free of fields and no remark at this point, and the sparks will be seen there. For sparking to occur, an electric field must author often repeated it in be present. Protection by Faraday cages plays a significant role in the private: C2.6 laboratory and in technical applications. “Technical applications 2. In a second experiment, we put a metal cup onto our static volt- make use of Faraday cages meter (Fig. 2.29). The housing of the voltmeter remains connected as protection against light- to the positive pole of the current source, while the cup can be con- ning strikes. They are used nected briefly to the negative pole to charge it. The voltmeter then for example around depots indicates a voltage of 220 V. We touch the outer surface of the cup for explosives or flammable with our “charge spoon” and move the spoon about 1 m away to a. substances, in the form of The voltmeter indicates a reduced voltage, i.e. some of the negative a wire mesh with relatively charges stored on the cup and the voltmeter have been transferred by large openings. However, the spoon to a. Then we bring the spoon along path 2 to the interior another safety rule is that no surface of the cup and fill all the negative charges back. The volt- ungrounded water lines from meter again indicates 220 V. The spoon can hold no charges as long hydrants are to be passed as it is connected to the interior wall of the cup, so that it is uncharged into the enclosure; otherwise, when we take it out again. a lightning bolt could jump from the wire cage onto the water line and thus enter the Figure 2.30 The production of a high building, causing a calamity. voltage between the cup and the housing In practice, much sad expe- of the voltmeter (the experimenter must be rience has been accumulated aware of the meaning of Eqns. (2.3)and with such arrangements!” (2.20)!) A closed space with an insulated conductor which enters it is a condenser. This should be kept in mind, in particular when we are E dealing with the rapidly oscillating electric fields + – associated with electric vibra- 20V tions. is B and A van de (“static V. 6 10 . There is no opposing frictional electricity 1 . They are used e.g. for low- C , where it is removed by brush 2; 2.31 is visible through two windows. ( C 2 B In the interior of the cup, all of the C pole of a small battery. The other pole of C or belt generators are used; they are constructed A A belt generator for high voltages without corona discharge . We move the spoon back and forth between the negative High Voltages is connected to the 2.30 electricity”) to charge the belt, employing the friction between brush 1 One can leave off the small battery and use A . all of it flows out ontoof the outer this surface type of have the been sphere.several constructed Belt meters with generators in spherical electrodes diameter, of sothe up that field-free to the interior experimenter while making can observations. sit safely in the battery “sprays” negative chargesa via moving a carrier conducting brush ofdriven 1 by electric onto a charge; small electric this motor.the is The interior charge an of on endless the the belt belt hollow is sphere which carried into is pole and the innervoltage surface indicated of by the the cup. voltmeterup as to We much can around as thereby 400 wefiber V, increase wish, the the voltmeter. for maximum example allowed Explanation: charge voltage on for the the spoon double- electric is field deposited there to each limit time theis amount used of technically charge in delivered. This thedescribed trick in construction the of following high-voltage section. generators,tion It as of is influence also machines. important for the opera- as shown in the schematic in Fig. Figure 2.31 Graaff generators 2.9 Current Sources Delivering For voltages of up to around 5 million volts (5 MV) in air, 3. Finally, webut using carry a out currentFig. a source third with only experiment a with low the voltage, same e.g. setup, 20 V as in energy nuclear physics researchThe field (production is of generated artificial between isotopes). two large, spherical electrodes C an insulator; at the lowercan right be is produced, an corresponding electric to motor.) a voltage Sparks of up to 30 cm long losses. The interior of the sphere The Electric Field 2 44

Part I 2.10 Currents During Field Decay 45

and the belt. This is in fact simply a newer version of the old frictional electriﬁcation machine (OTTO VON GUERICKE, 1672). The rotating car-

rier of electricity is no longer a sphere or drum, but rather an endless belt Part I (WALKIERS DE ST.AMAND, 1784, R.J. VA N D E GRAAFF, 1933). Carri- ers of electric charge in the form of belts can be made larger than disks, spheres or drums, and thus permit greater charge separation distances and C2.7 higher voltages. C2.7. Modern van de Graaff generators for research or isotope production have their high-voltage parts enclosed 2.10 Currents During Field Decay in a pressure tank containing an insulating gas (often SF6) to prevent sparking; they Based on our observations, we have attributed the decay of electric can produce up to 25 MV. fields to a movement of charges within matter (conductors). We will Even higher voltages can be now try to gain some more detailed knowledge of this charge move- obtained by operating two ment, and we find that: During the decay of the field, an electric or more in tandem (i.e. in current flows through the conductor. We can observe this current series). Replacing the rub- using a technical ammeter, e.g. a mirror galvanometer with a rapid ber or textile belt by a chain response time. We use a large condenser as shown in Fig. 2.32, con- with alternating insulating 2 sisting of 100 pairs of plates (with an area of all together about 8 m and conducting links (the and a spacing of 2 mm between the plates; cf. Fig. 2.52). We con- “pelletron”) permits a higher nect a voltage of 220 V to this condenser, as usual. Then its field is operating velocity and thus caused to decay by a piece of conducting wire. The galvanometer is still higher voltages and cur- connected in this wire in series with a piece of wood. The wood acts rents. See e.g. https://www. as a poor conductor and slows the decay of the field, so that it takes aps.org/publications/apsnews/ about 10 s. During this time, the galvanometer deflection indicates 201102/physicshistory.cfm. that a current is flowing. The time evolution of the current can be recorded with the aid of a stop watch and is shown in Fig. 2.33.The

Figure 2.32 The slow decay of an electric ﬁeld through a poorly-conducting wooden link (the static calibration factor of the gal- G 7 vanometer is BI 2 10 A/scale division)

Figure 2.33 The current which ﬂows ×10–6A during the decay of the ﬁeld (recorded 5 using a rapidly-responding galvanometer 4 as in Fig. 1.31) I 3 2

Current Current 1

02 46810 s Time t 2 .Of O 2 H 2.16 . 1 1 2.35 C and E E 2.34 A nser discharge) process. The ). We now measure this cur- 1.10 (see Sect. ) 2 , an arbitrary example of the time dependence of 2.33 1mm < During field decay through During field decay through and Current from Current Impulses. The Relation Between Charge current impulse a conducting wire, an electrolyticcluded cell in in- the circuit exhibits(electrode electrolysis area a conducting wire 1, alamp small included incandescent in the circuit lights up Figure 2.34 is the rent impulse on discharging the small condenser that we have often In our experiments with(discharging electric of fields, condensers) we tous made gain to use some of several particular important field insights;location phenomena: decay of it mobile led First charges tofinally on influence, the to then outer the surfaces to currentscharge. of the This which conductors, latter flow and phenomenon now through bringsgoal, us a closer the conductor to quantitative an during important measurementunits. dis- of electric charges inWe electrical refer to Fig. the current during aarea field enclosed decay under (conde the curve is the time integral of the current, that quantitative treatment of thiscourse, curve the brief will current be flowbe given during detected through in the the decay Sect. heat of it generatesillustrate the or both field its electrolytic using could effects. the also We schematics in Figs. 2.11 Measurements of Electric Charge Figure 2.35 The Electric Field 2 46

Part I 2.11 Measurements of Electric Charge from Current Impulses 47 Part I

Figure 2.36 The technical arrangement used for the experiment in Fig. 2.32; at left, a mirror galvanometer. The window at the bottom of the column allows us to see the mirror which reflects the light pointer onto the scale. The oscillation period T of this galvanometer is long, around 44 s ( ballistic galvanometer), and the spacing of the condenser plates is 4 mm. used already, together with the ballistic galvanometer (with a long response time) which we calibrated in Sect. 1.10 (Fig. 2.36). We carry out the experiment several times with variations. In all cases, the condenser plates are set initially to the same spacing, ca. 4 mm, and a field is produced (charging the condenser) by applying a voltage of 220 V (static voltmeter!). Then the experiments are as follows: 1. The connection used to discharge the condenser and destroy the field contains only the rotating-coil galvanometer with its low- resistance coil. The field decays in an unmeasurably short time. 2. A poorly-conducting (resistive) object is placed in the circuit, in series with the galvanometer, for example a piece of wood (compare Fig. 2.32). The decay of the field now requires several seconds. 3. First, the spacing of the condenser plates is increased, increasing the voltage correspondingly. Then the field is caused to decay (condenser discharge) as before, either rapidly or slowly by using the wooden “resistor”. 4. and 5. Now, we carry out two experiments on the production of the field. We return the condenser plates to their original spacing (4 mm), but this time, we include the galvanometer in one of the two conducting wires used to charge the condenser and thus produce the field (Fig. 2.37). In the fourth experiment, we generate the field very rapidly; in the fifth experiment, we generate it by using a resistive conductor much more slowly, over several seconds. In all five cases, we observe a current impulse of the same magnitude (in this example, about 108 As;seeEq.(2.1)). We changed the time dependence of the field, the magnitude of the voltage, we observed both the decay and the production of the field. What remained unchanged? Only the electric charges that were placed on or taken off the condenser plates, negative charges on one plate and positive on as- + G t (2.1) d I /: E R . We place – 2.38 We can measure electric . A s. The spoon thus con- 10 t charge on a small “carrier” d 10 I 220V Z +– D Q through the equation Q associated with the field. Q ) Meas- The current impulse during the production of 2.36 Unit: 1 ampere second (A s), also called 1 coulomb (C) (obsolete) . tained a negative charge of this magnitude. For a quantitative investigation ofduction the with mechanisms the of aid of electricalthe similar con- 21st experiments edition in of gases, this see book, for Chap. example 15. urement of the charge on a “carrier of electricity” (the galvanometer is the same as in Fig. the field Figure 2.37 a negative charge on it by touching itcurrent briefly source. to the negative Before pole of doing the terminal that, of we the had galvanometer (which already isto connected calibrated the the in positive left ampere pole second) ofalong the some current arbitrary source. pathand Now to observe we the a move current right the impulse terminal spoon of of 6 the galvanometer (a spoon with an insulating handle), as shown in Fig. Figure 2.38 As a first example, we measure the the other. From this we may deduce that the current impulse charges Q by determining a current impulse We define the charge sociated with the decay orelectric production charges of the field is a measure of the The Electric Field 2 48

Part I 2.12 The Electric Field 49 2.12 The Electric Field Part I We follow the measurement of electric charges by measurements of the associated electric field. The principal characteristic of an electric field is its preferred direction, as indicated by the field lines. To describe an electric field quantitatively, we must therefore use a vector. We call it the electric field E.Thedirection of this vector is the same as that of the field lines, conventionally from C to .Themag- nitude of the vector can be determined by suitable experiments. One example can be carried out with the aid of two devices (Fig. 2.39):

Figure 2.39 The deﬁnition of the U l electric ﬁeld strength E

440V

1. Flat parallel-plate condensers of differing plate areas A and plate separations l,and 2. Some sort of indicator for the electric field (a “field electroscope”). The indicator need only be able to identify two spatially or temporally separated electric fields as equal. It need not measure them, but rather only verify the equality of the two fields. As an indicator, we choose two small5, fine gold-covered quartz fibers, such as we have already seen in Fig. 2.4. We place them with their plane parallel to the field lines and observe the spread of their tips on a scale using an optical projection method6. During the experiments, we can vary the voltage between the condenser plates at will. To accomplish this, we use the familiar voltage divider circuit (Fig. 1.27). We will investigate a series of condensers with varying plate areas A and plate separations l. By regulating the voltage, we adjust the spread angle of the electroscope fibers to be the same each time; this means that the same force is acting on them and thus the same electric field is present each time. We find in this way a simple experimental result: The electric fields are the same as long as the ratio U=l, i.e. (voltage/plate spacing), is the same. The surface area of the plates plays no role. The homogeneous electric field of a flat parallel-plate condenser is determined uniquely by the quotient U=l. For this reason, one makes use of the ratio U=l to arrive at a first definition of the electric field strength (the magnitude of the

5 Otherwise, they would distort the field too strongly; compare Fig. 2.25b. 6 For thought experiments, a different indicator would be preferable, namely a tiny charged carrier of electricity attached to the arm of a force meter. . 4 s of m ∆ We U 3 (2.2) (2.3) s C ∆ : C2.9 2 s the field ∆ 1 3 ::: s E ). ∆ 2 C C2.8 E 3.8 2 1 . Then the sum E U m U E computes . For the generally C E 1 U 2.15 , ..., 2 E ; (compare Sect. , leads to a relation that is often 1 U E E to D n determine for this substitute or as of the condenser plates is equal to s C along this line, the field must be l d s s m between the condenser plates s )). E U m 2.2 Z E field of the same strength and direction ). Spacing at arbitrary points within an arbitrary electric C 2.2 E Voltage , we have connected the arbitrarily-shaped con- ::: D path integral of the electric field along an arbitrary C E 2.40 2 s In the vast majority of cases, one The path integral of the electric 2 . Examples can be found in Sect. E ) in a parallel-plate condenser: C E 1 s E The path integral changes its signalong if the it curve. is It computed is inin positive the when the opposite the direction direction curve of is the followed field, for i.e. the from most part 1 Field strength We already know this latterbetween the sum, condenser however: plates. Therefore, It we is must simply have the voltage E We use as unit 1 V/m. The next step leadswith to the homogeneous electric an field important of acan generalization. parallel-plate measure condenser, one the By field comparing field Figure 2.40 strength E most important electric field, the homogeneous fieldplate of condenser, a this flat parallel- computationthe is defining dispensed equation ( with by referring to In metrology, the measurement of electric fieldminor strengths plays role. a very vector or, in words: The curve is equal to the voltage U between the ends of that curve. will make frequent use of this relation in the following pages. practically homogeneous. Denote the componentsdirection of of the the field path in the elements field: Its individual smallThey regions are are replaced practically by still the homogeneous. ducting electrodes of athe general condenser individual by path a elements broken line. Within useful: In Fig. a parallel-plate condenser. We the “surrogate condenser” its fieldvoltage/plate direction spacing (Eq. and ( the value of the ratio The vector nature of the electric field ) of 2.3 . be- 5 curve. ), Eq. ( : s 3.8 d closed magnitude voltage U for simplicity potential differ- E (Sect. Z OHL ' D The Electric Field 2 . With the definition of the ignores the question of the sign here. Since he isonly dealing with the However, it holds for all the electric fields which are treated in the present chapter. They are characterized by the fact that their path integrals are independent of the curve chosen, and are therefore zero along any Mathematically, such electric fields can be represented as the gradient of a scalar(potential field field). They are termed “conservative fields” or “potential fields”. C2.9. The sentence in italics here is by no meansis trivial. also It not true inwill general, be as shown in Chap. C2.8. P tween two points, it suffices here to use the voltage as a ence U becomes, taking the sign into account, ' 50

Part I 2.13 The Proportionality of Charge Density and Electric Field Strength 51 2.13 The Proportionality of Charge

Density and Electric Field Strength Part I

In all of the electric fields that we have considered thus far, the field lines had ends, and electric charges were found to be sitting at those ends. Therefore, a quantitative relation between charge Q and electric field strength E is to be expected. We search for it experimentally by looking at the geometrically simplest field, our old friend the homogeneous field of a parallel-plate condenser. In Fig. 2.41,wesee a condenser; let the area of each of its plates be A, the voltage between them be U, and their spacing l. Then the magnitude of the electric field between the plates is E D U=l, as seen above. In the figure, a ballistic galvanometer can be connected to the condenser. RIt has been ballistically calibrated and measures the current impulse I dt when the condenser is discharged (when contacts 1 and 2 are closed). We thus measure the magnitude of the two equal positive and negative charges Q which were on the condenser plates (e.g. in ampere second). We repeat these measurements several times with various values of the plate area A and the field strength E D U=l. The result of all these measurements is Q D " E ; (2.4) A 0 or, in words: The surface density Q=A of the charge on the condenser plates is proportional to the electric field strength E ("0 is a constant of proportionality). We find the same simple relation for the surface density Q0=A0 of the influence charges.InFig.2.42, we see again the influence experi-

Figure 2.41 The proportionality of the ﬁeld strength and the surface charge density G 12

Figure 2.42 The measurement of the displace- CCAA ment density D as the surface density Q0=A0 of 0 α β the inﬂuence charge Q (U 8000 V) α β

––++ ; its vec- C2.11 (2.5) HARLES ,theyhave .Thelargest is also a β lies parallel to E (quoted e.g. in D 2.43 . As its unit, we D 0 A = are being measured. . 0 0 Q : Q ) takes on the form: α D permittivity of vacuum 2.4 As Vm D 12 : are still in contact; at the right, E or the ,Eq.( ˇ 0 10 " D , with , one makes use of a condenser with and D electric constant 2.41 ˛ 854 D : 8 , which cause hardly any distortion of the 0 . The displacement density It relates a charge density A D 2 C2.12 0 m " = . This leads us to conclude that E , in a vacuum (and for all practical purposes also in on is 0 in 1785. to an electric field E, measured in terms of a voltage ) in a homogeneous field, now using rather thin metal 0 electric field constant " ˇ 0 , with areas Q " 2.7 ˇ The in- )). displacement density D and measured via a current impulse, through a constant of pro- ˛ C2.10 2.5 . and ), 2 ˛ OULOMB m 1.33 = For precise measurements of the electricple field constant, condenser instead of sketched the in sim- Fig. The word “displacement” isthe unfortunate. displacement of It the charges isence when intended . the to field is remind interrupted us due of to influ- . This can be seen by the fact that the influence charges depend on (Eq. ( E tor the inclination of the disks relative to the direction of induced (influence) charge is found whento the the disks are field perpendicular vector The surface density of these influencecalled charges the has its own name; it is It is called the official (SI) name is simply the being measured as the “impulse deflection” of a ballistic galvanometer. Its calibration was carried out as in Fig. air), one finds the value: use for example 1 A s field. At the left in the figure, they have been separated, still in the field. In Fig. fluence charge Figure 2.43 This is the essential content of the law discovered by C A. C Inserting the displacement density ment (Sect. disks (quoted e.g. in V/m). For the factor the disks As been taken out of the field and their charges portionality ’s ” )) ’s D 3.8 )or )which )). 2.5 ’s law 2.9 equation is a natural OULOMB : 3.5 OULOMB 0 suggests in the ’s law). The A " d equations in AUSS has been fixed ’s formulation (Eq. ( .) . 0 ) with that equation E OHL " E AXWELL I 2.14 2.17 AUSS Q 2.15 OULOMB ) is a special case of the The Electric Field D AXWELL is also called the “electric D 2 For the history of the 0 2.4 Q " M and use the expression for the force on a chargeelectric in field, an F discovery of C law (as given in Eq. ( (The differential form of the first M can be seen if wethe compare equation for the field strength of a charged sphere (Eq. can be found at theSect. end of by law today (see Comment C23.1). integral form, and is also known as G constant which must be determined experimentally. But we should mention that the value of connection to C (or G of C law, see e.g. J. L. Heilbron, p. 470, cited in theSect. footnote in It is the first of the C2.12. When the quantities current and voltage are considered to be independent of each other, asdo we here, then C2.11. Equation ( C2.10. P 21st edition that one could simply use the expression “electric field quantity instead of “displacement”. D flux density”. ( general relation ( is discussed in the following section: 52

Part I 2.14 The Electric Field of the Earth. Space Charge and Field Gradients. MAXWELL’s First Law 53

Figure 2.44 The same experiment as in Fig. 2.41, but using a condenser with a corona s G

ring Part I

220V

a “corona ring” S (Fig. 2.44). The surface charge density is measured only on the inner part of the condenser, thus avoiding disturbances due to the inhomogeneous stray ﬁeld around the edges of the plates.

The experimental fact summarized in Eq. (2.5) can be interpreted in three ways: 1. One considers the readily-measurable quantity D as a useful aid to the measurement of electric fields E, employing E D D="0; 2. One can treat the displacement density D simply as an abbreviation for the product "0E, which occurs frequently in electromagnetism; 3. or, one can treat D as an independent, second quantity which is equivalent to E for the quantitative description of electric fields.C2.13 C2.13. The introduction of two vector fields, E and D, In this book, we will place all three of these possibilities on an equal which differ only through footing. a constant of proportionality "0, might seem unnecessary here. For the complete description of an electric field 2.14 The Electric Field of the Earth. in vacuum, E is indeed sufficient. D is therefore not even Space Charge and Field Gradients. mentioned in some textbooks. MAXWELL’s First Law However, in the presence of dielectric materials, the simple relation D D "0E no The earth is always surrounded by an electric field E (G. LE MON- longer holds. Then it is often NIER, physician, 1752). This field points downwards and is per- found to be helpful to employ pendicular to the surface of the earth in flat regions. The field can both of the field quantities be readily detected and measured by making use of Eq. (2.5). We (see Chap. 13). employ a flat parallel-plate condenser which can be rotated around its horizontal axis (Fig. 2.45). It is set up in the open. Each plate (made for example of aluminum sheet metal) has an area A of about 1m2. The plates are analogous to the small disks in the influence experiment. They are each connected to one terminal of a ballistic galvanometer which is calibrated in ampere second. We alternately orient the plane of the plates in the vertical and the horizontal, that is alternately parallel and perpendicular to the field. With each change of orientation, the galvanometer indicates a current impulse Q D ). G (2.6) 2.46 (Fig. . Its total negative 2 ) is responsible for the m 3 14 space charge is the displacement den- : 2 : 10 m A As x E V m = @ @ 9 1 0 Q : " 130 10 D of 5 A s. Where are the corresponding D 5 x e D 15 0 : @ D " @ A 10 1 D D D 6 llistic gal- % E D A s. The ratio of these charges (A s/m 9 % D e A A cloud of positive space charge A measurement of the dis- (Video 2.3) ). The electric field strength would be practically the same of the electric field of the earth. Averaged over time, one finds of around 10 2.48 t D d I The earth has a surface area charge is thus strong “slope” (the gradient) of the field. We find rier of the negative charges.localized on The innumerable, corresponding invisible positive carriers charges incarriers the are all atmosphere. together These form a cloud ofThe positive volume density positive charges? One could thinkwould of be the fixed dealing stars; with ina the that charged usual case, we radially-symmetric sphere electric at(Fig. field a of great distanceat several from kilometers other altitude above objectsface the itself or earth’s (the surface charges earth’s as radiuscase. on is Already the 6370 at sur- km!). an altitudearound But of 40 1 this V/m. km, is At the not an field strength altitude at has of all dropped 10These km, the to observations it lead is us only to ato a few a new V/m. fundamental kind relation between of charge and electric electricwhich field field. we and have The thereby treated fields thus farbody were delimited which on carried both electric sides by charges.we a have solid In a the solid body case on of only the one earth’s side, field, namely the earth itself as car- above the negatively-charged surface of(approximated the here earth as a planar surface) placement density of the electricthe earth field using of a rotatable parallel-plate condenser connected to a ba vanometer Figure 2.45 sity Figure 2.46 R or ,so ı The Electric Field 2 denser is rotated by 180 field of the earth” http://tiny.cc/69fgoy In the experiment, the con- Video 2.3: “Measuring the electric that the current impulse is doubled. 54

Part I 2.15 The Capacitance of Condensers and Its Calculation 55

Figure 2.47 The relation between a ﬁeld D gradient and a space charge ∆x D+∆D Part I

Derivation: In Fig. 2.47, two homogeneous ﬁeld regions are sketched, with the cross-sectional area A and displacement densities D and (D C D), one above the other. D is thus assumed to increase by the amount D on passing down the vertical path element x. Then from Eq. (2.4), it follows that Q " E D D D (2.7) 0 A or E D Q " D D D %; (2.6) 0 x x Ax since Q is the charge contained within the volume A x.Itismarkedby the C signs in Fig. 2.47.

Equation (2.6) is a special case, limited to a gradient in one direction (along the x axis); usually, this relation is written in the general mathematical form (obtained in the limit x ! 0 and in three dimensions): % div E D (2.8) "0 and is called the “fundamental equation of electrostatics”. It is one of the four MAXWELL equations (see Sect. 6.5), and it describes the general relation between the vector ﬁeld E and the charge density % (Exercise 2.12). In integral form, it is given by I 1 E dA D Q ; (2.9) "0 where the (2 dimensional or surface) integral is to be carried out over the closedR surface area A which encloses the charge Q. The surface integral E dA is also called the electric ﬂux.

2.15 The Capacitance of Condensers and Its Calculation

Combining the two equations

D D "0E (2.5) and Z E ds D U ; (2.3) ) ) of 2.5 2.2 A = (2.12) (2.14) (2.13) (2.11) (2.10) /: Q 70 pF C2.14 D : (and a radially- is located on the farad Q 11 C2.15 V As 11 10 : 10 : ; 2 or 7 2 . 7 1 : 2 C . Inserting both into Eq. ( C ). A charge l from the center of the spherical R . The capacitance is defined for D l A = Q . In its homogeneous field, the C R 2 0 C U his will be treated quantitatively in 4 " m 1 1 2.48 on the electrodes 1 2 C C D D D Q between the electrodes 10 D R C D ). E m U D 3 C 1 C 14 : capacitance 3 is equal to the surface charge density 10 Charge for several resistors, it is worth the trouble to : 2 circular plates of 20 cm diameter and areas of D 4 As ARADAY Voltage 12 2.8 D Vm , at a spacing of 4 mm: 10 2 C m 2 86 : 8 10 D flat parallel-plate condenser spherical condenser electrode with a radius r numerical example C 14 : A 3 is the electric charge on one of the electrodes (which define the In a series circuit, we find derive the capacitance ofparallel. several We condensers readily obtain connected for in two series condensers in or parallel 2. A symmetric electric field; see Fig. Its unit is 1 ampere second/volt (A(F) s/V), (after and Michael is given F the name ‘farad’ Q boundaries of the electric field),charge and (of opposite has sign) the on samethe the magnitude other other negative. as electrode. Often, the One weof is speak the positive conveniently, “charge but and on less the rigorously, condenser”,and and “discharging” correspondingly the of condenser “charging” (t the following section). We give examples ofwith the geometrically capacitance simple of electric fields: several condenser1. designs A electrode, it gives rise to a displacement density Here, as in Sect. we can compute the distributiontrary fields. of We the thus electric arrive at field theimportant physically strength concept as well of of as the arbi- technologically every condenser as the quotient surface of the sphere. At a distance gives displacement density the charge on the two condenser plates (electrodes). Equation ( then gives the field strength ) by imagining that 2.14 The Electric Field 2 pairs of concentric spherical surfaces are brought into the field, and then computing the density of surface influence charges (displacement densities) which would be present on them. For those who prefer an experimental derivation, we refer to Com- ment C2.16. C2.15. One can try toEq. derive ( C2.14. When two condensers are connected in parallel, their two negative electrodes are equivalent to a larger electrode whose total area is the sum of the twoelectrode individual areas, and similarly for the positive electrodes. They thus act like a larger condenser, and the overall capacitance of the circuit is just the sum of theual individ- capacitances of the two condensers. In a series circuit, the voltages on the individual condensers simply add, and since the capacitance is proportional to the reciprocal ofvoltage, the the reciprocal of the overall capacitance is just the sum of the reciprocalsthe of individual capacitances; this is analogous to resistors connected in parallel. 56

Part I 2.15 The Capacitance of Condensers and Its Calculation 57

Figure 2.48 The radially- symmetric electric ﬁeld lines

between a negatively-charged Part I sphere and very distant positive charges

and, from Eq. (2.5), the field strengthC2.16 C2.16. The experimental derivation of the important Q E D : (2.15) equation (2.15) starts with R " 2 4 0R the experiment illustrated The voltage U between the charged sphere and the very distant in Fig. 2.49 (Video 2.4), boundary of the field (e.g. the walls of the room) is obtained from which can be carried out with Eq. (2.3) by integration. Then spheres of different radii. Measurement of their capac- RZD1 RZD1 Q dR Q itances C(r) as a function U D E dR D D : (2.16) R 2 of the radius r of the sphere 4"0R 4"0r RDr RDr yields Eq. (2.17), and Q Equations (2.10)and(2.16) together yield the capacitance of a spher- U D " (2.16). ical electrode (spherical condenser): 4 0r Substituting Eq. (2.15), it D " C 4 0r (2.17) follows that 10 RZD1 4"0 D 1:11 10 As=Vm : U D ER dR The capacitance of a sphere is proportional to its radius. RDr In Fig. 2.49, we measure the capacitance C of a globe which has been (concerning the sign, see hung on an insulating cord to verify Eq. (2.17). A voltage of 220 V Comment C2.8). The im- suffices for this measurement. portance of this equation is that it is experimentally veri- The earth’s radius is r D 6:37 106 m. It thus forms with the system of fied in principle for all radii, fixed stars (as counter electrode) a condenser whose capacitance according including extremely small to Eq. (2.17) is 708 microfarad (F). values of r (“point charges”). Thus, by superposition of In an analogous manner, we can compute the electric fields of more many such point charges, the complex electrode shapes, as long as they are sufficiently symmet- electric fields and potentials ric, as well as the spatial distribution of the field strength and their (Chap. 3)ofarbitraryknown 7 capacitances charge distributions can be calculated. 7 Examples:

r1r2 2 concentric spheres: C D 4"0 ; (2.18) r2 r1 a 2 coaxial cylinders of length aW C D 2"0 : (2.19) ln.r2=r1/ ), E : in am- 2.20 (2.20) G which is 220 V). The D U (Video 2.4) propeller aircraft .FromEq.( r C2.17 B matter transport llistic galvanometer is produced: It blows away : insulator) B !): : ; r r U D 1.33 R D ) yields the field strength directly on r E 2.16 electric wind pole of the current source ( C on the surface of a sphere and the radius of to rotate. The voltage between the propeller and the )and( G E 220V 2.50 2.15 − + E of the sphere are inversely proportional to each other. Measurement of the capacitance of a condenser formed by r The air which streams away fromin the from point the is sides; replaced it by is airA in which counter-force turn comes acts ionized on and accelerated the awaysketched point. from in the It Fig. point. can for example cause the “propeller” Apart from details, the same process occurs as with a The propeller or fanblows accelerates it the air backwards whichrected as enters to a from the jet. the jetit of sides air and The to accelerates maintain counter-force the a which aircraftresistance constant on to is velocity its takeoff oppositely and motion in (Vol. later 1, the di- allows Sects. face 5.11 of and 10.11). the inevitable frictional walls of the room need be only a few thousand volt . associated with an electric current. from the point and is a first example of the Therefore, in the neighborhoodelectrodes, of even corners quite and low pointsThe voltages of give air condenser rise becomes totion” conducting high of at air field molecules) high strengths. and is fieldglow no longer (“corona strengths an discharge”) insulator. (“field indicates A fundamental violet-colored ioniza- molecules. changes In in addition, the an air Any sharp corner or point canical to surface first order with by considered a as a small spher- radius of curvature curvature the field strength a sphere and the flooris of connected the briefly lecture to room. the To charge it, the cardboard sphere pere second was carried out as in Fig. the surface of the sphere (where Figure 2.49 For an overview of complexnation fields, of we Eqns. give ( a useful tip: The combi- negative pole of the(it current was source “grounded”). was (The previously calibration connected of to the the ba earth D farad. r 11 farad. The V. The cur- 10 3 11 , corresponding A s. Doubling G 10 8 )is3 4 10 D 2.17 The Electric Field C 27 m) is charged up to 2 : Video 2.4: “The capacitance of a sphere” http://tiny.cc/jaggoy The globe (of radius a voltage of 10 0 capacitance calculated from Eq. ( rent impulse on discharging it produces a galvanometer deflection of 3.7 scale divisions in the ballistic galvanometer C2.17. A planned application of this matter transport is a rocket drive for spaceusing flight ion beams. the charging voltage doubles the measured charge. The capacitance is found to be to 4 58

Part I 2.16 Charging and Discharging a Condenser 59

Video 2.5: “The Electric Wind”

http://tiny.cc/maggoy Part I Right: Ionic wind (particularly effective as a shadow image). – An instructive variation: Hang a light-weight condenser made of a pointed and a ring-shaped electrode mounted rigidly, with two thin wires which serve as Figure 2.50 At left: Pinwheel (ANDREAS GORDON, 1712–1751) (Vi- electrical leads and sus- deo 2.5). pension. This “pendulum” swings whenever the jet of the electric wind is blowing 2.16 Charging and Discharging through the ring. a Condenser

The time dependence of discharging a condenser was already shown in Fig. 2.33. In order to investigate it quantitatively, we use the circuit shown at the upper right in Fig. 2.51. As can be seen there, the total voltage is the sum of the condenser voltage and the resistor voltage:

Q U D U C U D C RI ; (2.21) C R C

R c

t A U d d U I C = c I

E Uc Uc 1 U U c 1 U'c 1 1–) e)≈0.63 A B 1 1 e ≈0.37 e ≈0.37

τr τr τr τr 0 5×10–3 10–2 s 0 5×10–3 s Time Time

Figure 2.51 Production and decay of the electric ﬁeld in a condenser as a function of time, i.e. charging and discharging the condenser. An oscilloscope is used here to measure the voltage (shown as a static voltmeter in the circuit at upper right). From its time dependence (curve A), we obtain by differentiation the time dependence of the current shown above (Eqns. (2.23)and(2.24)). B: Discharging begins already 0 < D 6 D 3 D D D 3 at Uc U.(C 10 F, R 10 , r relaxation time RC 10 s) – (2.24) (2.23) (2.22) ), or of A . B 2.1 7 ). By rotating relaxation time t t RC RC 2.53 C e e 0 0 B Q Q RC RC : is called the D t Q + d I d D RC I R D or C ). A variation on this multi-plate r or C Q Ã 2.52 and D t t RC RC U ), as one can verify by substituting into C2.18 0 e e U 0 0), Q are insulators) 1 D D B D Â of the circuit. It can be used to measure resistances U 0 U Q Q Shadow image of a rotary vari- The design of multi-plate D Dielectrics and Their Polarization ) (simple differential equations can best be solved by fol- Q 2.22 time constant for charging ( on discharging ( or whose value is greater than the order of magnitude of 10 several pairs of plates (Fig. condenser is the rotary variable condenser (Fig. This differential equation has the solutions able condenser condensers. Usually, they have three instead of the one pairshafts of shown mounting here. ( Eq. ( lowing this recipe: Guess thethe original solution equation). and The test time it by substituting into Figure 2.53 2.17 Various Types of Condensers. Thus far, weThey have used consisted condensers either of of practically a only pair two of types. parallel plates (Fig. Figure 2.52 and thus, 0, OULE ), and ). D R 2.22 11.2 The Electric Field 2 since then the condenser would be discharged within an inﬁnitely short time, without the energy it contains being converted into J C2.18. This result clearly does not hold for which would lead to oscillations (see Sect. heat. The current in thiswould case be determined by other properties of the circuit which are not taken into account in Eq. ( 60

Part I 2.17 Various Types of Condensers. Dielectrics and Their Polarization 61

Figure 2.54 Charging a Leyden jar Part I

one set of plates so that different fractions of the plate areas overlap, one can vary the capacitance of the condenser. Technical condensers often have liquid or solid materials (“dielectrics”) between their plates rather than air. We mention here three types which are frequently used: 1. The well-known and venerable Leyden jar8. Figure 2.54 shows at the right a primitive version: A glass cylinder has a sheet of tinfoil glued inside and another outside. Its capacitance is usually in the range of 109 to 108 F(1to10nF).

A small inﬂuence machine produces currents of around 105 A (Sect. 1.9). With this current, it can charge a Leyden jar with a capacitance of 108 F in 30 s to a voltage of about 3 104 V(Fig.2.54). A spark gap with two balls at a distance of 1 cm, wired in parallel to the Leyden jar, can serve as a rough voltmeter. At around 30 000 V, a spark jumps the gap, accompanied by a loud ‘snap’. The time during which a spark of this kind jumps is ca. 106 s. This can be registered with a rapidly-rotating photographic plate (or with a photo detector and an oscilloscope). The current in the spark must therefore be 30=106 or 3 107 times larger than the current from the inﬂuence machine; it must be around 300 A. This large current causes a strong heating of the air in the spark gap, with the resulting noise (thunderclap principle).

2. The paper condenser. We lay out two strips of metal foil C and A, separated and covered by two strips of insulating paper P, P , then roll up the whole packet and press it together (Fig. 2.55). Newer variants use plastic foils with vapor-deposited metal electrodes on each side. They can be fabricated for voltages of up to several thousand volt. 3. Electrolytic condensers. In these, the insulating separation layer (dielectric layer) is produced electrolytically and has a thickness of the order of 0:1 m. Condensers of capacitances up to 103 F, or even 1 F are commercially available today, and can be used at up to several hundred volt. The treatment in this and the following chapter is limited to electric ﬁelds in vacuum, which is practically the same as in air. Matter in an electric ﬁeld will be discussed in Chap. 13. Nevertheless, in the three

8 See: J.L. Heilbron, “Electricity in the 17th and 18th Centuries: A study of early modern physics”, University of California Press, Berkeley, CA (1979), p. 309. ). 2.3 (2.25) by a rough . , positive on P 2.56 2.56 . It in turn produces P C ). dielectric constant of a condenser m C A of the dielectric. Numerical values of the empty condenser 0 gap by a conducting object (Fig. Exercise 2.13 , and its C . Their spacing, equal to the thickness of the ce in conductors; in Fig. 2 . 13.1 : A finished paper condenser with a capacitance of Capacitance completely filled with a dielectric polarization Capacitance : The condenser is partially unrolled. The two metal foils D 10 μF dielectric constant , its , is about 0.02 mm ( " At the left electric polarization of the molecules P at the right F; dielectric will be given in Table With a given charge onin the its capacitance electrodes is of accompanied thethe by condenser, gap a an between decrease increase its in electrodes itslar voltage. with effect a Filling to dielectric partially thus filling produces the a simi- a “polarization of thesurfaces, whole just as dielectric”. with influen Then charges appear on its the left and negative on thecannot right. be But used the to polarization separate of charges an like insulator the influence in a conductor. is called the two-dimensional model. Theby molecules small are conducting spheres. represented arbitrarily is Influence called within an individual molecules The conductor cancels thethe field field in lines itscharges by interior. are the induced It amount onfluence. thereby of its shortens its surfaces: thickness. this is AtIn the the an phenomenon same insulator of time, face or in- as dielectric, they the dofield in charges causes a cannot a metal. shortening move ofcan to Nevertheless, the simply an the field insulator assume lines; sur- in inindividual that an the molecules. influence simplest electric case, is one This operating is on the illustrated scale in of Fig. the each have an area of about 4 m 10 paper strips Figure 2.55 types of condenser described above, we have intentionallythat anticipated topic. We thereforethe introduce three new concepts at this point, A good insulator causesIt an can electric be field “penetrated” to byThus decay an its electric only name, field “dielectric”. very over slowly. a longThe period ratio of time: The Electric Field 2 62

Part I 2.17 Various Types of Condensers. Dielectrics and Their Polarization 63

Figure 2.56 A model experiment to ex- a plain the polarization of a dielectric in

terms of polarization of its individual Part I molecules (Exercise 2.14)

– + CA

Imagine that the polarized insulator in Fig. 2.56 is split into two parts along the cross-section ab perpendicular to the field lines and the two halves are removed from the field: Then each half contains the same number of C and charges, and is thus wholly uncharged. The polarization charges are also called “bound” charges, to distinguish them from the “free” charges on the condenser plates or the surfaces of conductors in the field. The model experiment in Fig. 2.56 contains extensive but not essential simplifications. In reality, the molecules are not spherical, and the charges do not move to the ends of the molecules. More details are to be found in Sect. 13.9. In any case, a rather unspectacular experiment, sliding an insulator into the gap between the plates of a condenser (Fig. 2.3), has led us to an important result: In the interior of matter, charges are present. They can be displaced by an external electric field. This results in an “electric deformation” or polarization of the molecules themselves. What happens when an object is electrically charged according to this picture? From Sect. 2.7, this can only mean that an excess of charges of one sign is present on the object. But how many charges of both signs are present, whose difference is observed in charged bodies? We offer an example: An amount of water of mass M D 1 kg has a volume of V D 103 m3 and, if it were spherical, a radius r D 6:2 cm. From the molar mass of the water molecules, M=n D 18 g/mol, we find the amount of substance in the sphere to be n D 55:56 mol, and thus the number N 25 of water molecules to be N D nNA D 3:3410 molecules, each with 10 electrons. Since the charge of a single electron is 1:602 1019 As (Sect. 3.6), the sphere contains charges Q of both signs, each totalling 5:4 107 A s. Between this sphere and the walls of the laboratory, we could – with some difficulty – produce voltages of U > 106 V. Then, from Eq. (2.17), a charge q D 6:9 106 A s would sit on , ; h Q % = q 2.15 V(see 5 are con- above it; .Letthe ). h ). x 2 10 of the volt- (Sect. of electrons C > molecules at 3 V e 2.14 2 D x 2.15 C N m for the case that and above the earth’s U 25 1 x C 0for C be of the order of the 10 and thereby produced 0 (Sect. q of the H F (Sect. D V Q , the electrolysis of water D % 45 : 2 N ), we assume that the cloud x 2 D 2.35 2.8 has a constant volume charge D 2 and V C h N and then be that of the earth’s surface, and . When the condenser is connected, 2.46 D h U x (that is, we have disturbed the electric D of hydrogen gas on discharging through 3 x Q between the earth and a point at x U between a negatively-charged sphere of radius as a function of the height between C h Q E U U . This means that, although we would say in the D ). ). 13 q . 1.4 10 2.15 Q ). up to a height of 3 of the condenser be, if it were charged up to a voltage of : % 1 nF is connected to a static voltmeter which was previously Two condensers with the capacitances In the experiment shown in Fig. The voltage An uncharged parallel-plate condenser with a capacitance of For a simple application of Eq. ( : C 2.15 1 nF (a small Leyden jar) and 1 cm and the distant walls of the room is 0 D D D 1 2.3 300 K and atmospheric pressure is C nected in series. Find their overall capacitance is carried out by discharging aitance condenser. How large must the capac- 2.2 see also Sect. Video 2.3). Determinethe the sphere, surface expressed charge as density the at surface the number surface density of (Sect. meter (Sect. 2.5 r the electrolysis cell? The number density the voltage decreases by 10 %. Find the capacitance C 2.4 charged and indicated a voltage and d) the voltage total charge equilibrium in the object,when but the only object by hasmolecules a or an very atoms, exceedingly can small the small amount). charge difference mass, Only i.e. for individual laboratory that a largereality electric we charge would was have removed present orfraction on added of only the an its object, unimaginably positive in small or negative charges a difference of 220 V and produced 10 mm Exercises 2.1 the surface of the sphere, either positive or negative. The ratio of positive space charges in Fig. density negative surface charge density the height, both known. Then, find a) the volume charge density b) the electric field surface; c) the voltage is then 1 The Electric Field 2 64

Part I Exercises 65

2.6 A coaxial cable of length l has an inner conductor of diameter 2r D 2 mm in a 5 mm thick plastic insulation with the dielectric constant " D 3, which carries the outer conductor. Find the capacitance Part I per length unit, C=l (Sects. 2.15, 2.17).

2.7 Two point charges Q1 D Q and Q2 D3Q are at a distance of 1 m. At what point along their connecting line is a) the electric ﬁeld strength E D 0, and b) the potential ' D 0 (besides at inﬁnity)? (Sects. 2.15, 3.8).

2.8 A rotating-coil galvanometer (Fig. 1.19) has a resistance of 100 . When a charged condenser (capacitance 0.1 F, voltage 10 V) is discharged through this galvanometer, it shows a ballisticR deﬂec- tion of 10 scale divisions. Find the voltage impulse Udt which would produce a deﬂection of one scale division (Sect. 2.16).

2.9 A condenser of capacitance C is discharged through a resistor of resistance R. a) Determine the time t1=2 in which the voltage decreases to one-half of its initial value; b) how large is the capacitance C of the condenser if R D 1012 and the voltage decreases by 20 % after 10 seconds? (Sect. 2.16).

2.10 A neon lamp (Fig. 1.16) lights up at a voltage between 160 and 220 V; the current increases linearly with voltage over this range, following the equation I D .U 157 V/=7:63 k (the “differential resistance” dU=dI is thus 7.63 k). The lamp is connected to a condenser charged to a voltage of 220 V; its capacitance is C D 50 F. Find the time that it takes for the lamp to go out (Sect. 2.16).

2.11 A condenser (C), an OHMic resistor (R), a battery (U0)and a switch are connected in series in a circular circuit. The switch is closed, so that the condenser is charged through the resistor up to the voltage U0. Find the energy Wbatt supplied by the battery and compare it with the energy WC stored in the condenser. What role is played by the resistor R? (Sects. 2.16, 3.7).

2.12 A rotary condenser (Fig. 2.53) has semicircular plates with a radius of 5 cm and spacings of 1.1 mm. How many plates will be needed to obtain a maximum capacitance of 500 pF ? (Sect. 2.17).

2.13 Determine the dielectric constant " of the paper strip P in Fig. 2.55, if for the dimensions given there, the capacitance obtained is C D 10 F (Sect. 2.17).

2.14 Find the effective dielectric constant " in the model experiment illustrated in Fig. 2.56, in which metal balls of diameter d are arranged on a cubic lattice with a lattice constant 2d within the otherwise empty space between the condenser plates (Sect. 2.17). . , " % https:// is ﬁlled by the b l h ). spacing of the plates) is as- ). 3.7 D 3.7 , l , ) contains supplementary material, which The online version of this chapter ( 2.17 2.17 is applied to the plates, the liquid rises ,sothatavolume U h (Sects. ) is dipped vertically into a liquid (density l h ) so that the lower edges of its plates just touch " surface area and D A is applied? (Sects. , l , and spacing U b A parallel-plate condenser (with rectangular plates of height The volume between the plates of a parallel-plate condenser A D , width V a Electronic supplementary material 2.15 doi.org/10.1007/978-3-319-50269-4_2 is available to authorized users. the liquid. When a voltage dielectric constant sumed to be completely ﬁlled with a liquid of dielectric constant ( 2.16 liquid. Find the height between them to a height How large is the forcea with voltage which the plates attract each other when The Electric Field 2 66

Part I nEeti Fields Electric in Energy and Forces esat sawy,fo xeietlrsls Figure results. experimental from always, as start, We Experiment Fundamental The 3.2 However, and capacitance quantities. voltage, electric current, electric other and for mass, various as length, find well time, can as measuring one for temperature teaching, instruments or of research assortment for laboratory an physics every In 1. Remarks Preliminary Three 3.1 oprsta oc otefreta ecall we that force generally the one to force, a force th of that and measurement compares rarely the only requires investigation all, an at if found, .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © .Terlto ewe force between relation The 2. computed are rather quantities. but directly, other measured from not are forces general, In et ..Tecrepnigeprmnsyedteresult the yield experiments corresponding is The method 3.2. One Sect. in invidious teaching. most physics the of of all one is problem This experimentally. derived .Eatytesm iuto sfudi lcrmgeimfrthe for electromagnetism in found Sects. is in derived situation equation same fundamental the Exactly 3. fun- applications. this numerous t for is justification real equation The damental precision. acceptable barely a with aeasalba aac.Tecriri lcda h etrof center the at placed is carrier The balance. beam small a case iksae hrecarrier charge disk-shaped a lcrclqatte (charge quantities electrical iigterlto ewe h ehnclqatt ‘force’ quantity mechanical the between relation the giving in h nljsicto sfudol ae ntrso l fits of all of terms in later only found applications. is general justification final the tion, E efudol ae ntescesso its of successes the in later only found be o a D olsItouto oPhysics to Introduction Pohl’s D Q F Q m F ˛ lcrcfield electric , F ,mass ntelvro oc ee,i this in meter, force a of lever the on or or nmsl ntahn as If labs. teaching in mostly en F F m D 3.2 D ecie ndti nVl 1, Vol. in detail in described n acceleration and , Q m and E E a oc meters force ; .Aan o hsequa- this for Again, ). weight 3.3 , : https://doi.org/10.1007/978-3-319-50269-4_3 ui newton). (unit a 3.1 r obe to are a obe to has F shows and 3 67

Part I b). be- and .Its A A U 2.25 0). After and condenser C (Fig. D 2 spacing of the 1 both U ( D . Todothis,we I ˛ l ( l = S to apply a voltage U I D + E A G , so that the field strength of the ho- α U C is produced with the aid of a current source abc A . This produces a new field-line pattern, as in A –– The carrier, when itself uncharged, should have and III and C , below). The carrier is now pulled upwards by the V C 3.9 3000 V 220 V , we use the current source it should not distort the field by influence + C3.1 The fundamental experiment on the force between a charge and a); The field- . 3.2 3.2 3.1 This field would pull the chargedIn carrier order towards to the prevent plate that, whichbetween is the the nearest. carrier plates (in must an be unstable placed equilibrium). precisely at the center In addition (Fig. are mounted on insulating posts. is a small round weight which keeps the center of gravity of the balance . It operates at the voltage part c of the figure.and It a is (cf. the Fig. result of a superposition of the patterns b field from the condenser plates. mogeneous field in the condenser is tween the plates C line patterns in the fundamental experiment of Fig. charging the carrier, we willof observe Fig. the field pattern shown in part b connect it briefly to the negative pole (contact 1) and I plates to the positive pole of the current source electrodes). With this setup, we proceed as1. follows: We transfer a negative charge to the carrier 2. a parallel-plate condenser, between the two electrodes Figure 3.1 Figure 3.2 shape and its orientationchosen perpendicular for to a the reason: fieldonly a lines minimal have effect been onA the form of the electric field between C and an electric field. The (insulating)of quartz Al balance sheet beam metal carries on twoS its riders right made side and canbeam swing between below two the limiting knife-edge stops. bearing. The condenser plates (electrodes) The field between Forces and Energy in Electric Fields 3 C3.1. In textbooks one often finds the term “a sufficiently small test charge”. In carrying out the experiment, one has to make compromises in order to have sufficient sensitivity and precision for the measurement. 68

Part I 3.2 The Fundamental Experiment 69

3. We measure the force F using the balance. Furthermore, we measure the charge Q on the carrier. This is done with the calibrated ballistic galvanometer (carrier ˛ is contacted by the wire 2!). Finally, Part I we measure the voltage U and the spacing l between the condenser plates. 4. From each set of four corresponding quantities (the charge Q,the force F, the voltage U and the spacing l), we compute the product Fl and the product UQ and ﬁnd experimentally, within the error limits, that they are proportional to each other: Fl QU: If we again set the constant of proportionality equal to 1, we obtain FlD QU: (3.1) (The direction of the force will be discussed in the following section.) In this equation, the term on the left is a quantity of work; therefore, the term on the right must also be ‘work’. That is, one can measure work not only mechanically as the product of force times distance, but also electrically, as charge times voltage; or, written as an equation, W D QU: (3.2)

The charge is the time integral of a current; when the current I is constant over time, we can write Q D It (Eq. 2.1). Inserting this quantity into Eq. (3.1) yields FlD UIt ; (3.3) an equation which we have already encountered in Sect. 1.12. In the experimental setup which we have used (Fig. 3.1), the electric field was homogeneous. Its field strength is given by E D U=l. Inserting E into Eq. (3.1)gives C3.2. The complete equation in vector form will be given F D QE (3.4) in the following section. Of course, Eq. (3.4) also holds . F Q E /:C3.2 e.g. in newton, in ampere second, in volt/meter when other units are used; it is an equation relating physical quantities, as always. In words: The force observed is proportional to the carrier charge Q and also to the field strength E of the condenser field (which is as- C3.3. This is not due only to sumed not to be modified by the small charge on the carrier (Fig. 3.2, the compromise mentioned part a). E is not for example the field strength of the actual field in Comment C3.1; rather, present during the measurements (part c)!C3.3 the “test charge” also pro- Equations (3.1)to(3.4) are often used. We set the proportionality duces an electric field, whose factor in Eq. (3.1) equal to 1. This means that the three quantities magnitude is in fact rather ‘work’, ‘charge’, and ‘voltage’ cannot be measured independently of large near the carrier (1=r2, one another; instead, work and charge are used to measure the voltage Eq. 2.15). However, this field as a derived quantity. This makes it possible to use the same energy cannot exert a force on the units for mechanical and electrical measurements (see Sect. 1.12). carrier (test charge) itself. ) is Q be 3.5 E (3.5) (3.7) (3.6) E forces ,which test object as a vector quantity s d E : ; s : F Q . These forces lead to the F d E . As a result, in mechanics, D Z in an arbitrary field, e.g. an E E 1As 1 Q 1Nm Q at rest Z ), , then they perform the work as a defining equation and place s D D Q a or are parallel and point in the same 3.4 s m D d F s E D d E Q F and Z F D . The vector nature of the electric field 1volt(V) E D Z F move a charge U refers to a small charge on a E vector Q Q D ), and then we can define the field F C3.4 of the Electric Field 2.11 direction. it at the beginning ofcan the treatment, follow or the also, path inwants sketched electrodynamics, they to out derive in the thisexperiments fundamental section. will empirical have to However, facts resign whoever quantitatively themselvesmore to from following tedious a path, longer and makingresources. use of the currently-available technical In this relation, does not affect the form of the field already present. Equation ( contains the definitionFor of a the positive direction charge, of the electric field vector: Theoretical treatments need only describebut the not results to of demonstrate experiments, them quantitatively they can use the equation preferred directions which are soof graphically apparent electric in field the patterns lines, and they require that an electric field represented as a already apparent in its definingto equation. To agree show upon this, we a first measurement have procedure for defining the charge (Sect. as a derived quantity with the unit If forces 3.3 The General Definition The essential qualitativewhich property they exert of on electric electric charges fields are the using the fundamental equation ( inhomogeneous field, along a path and from this, the voltage follows: charges. ), since a force 3.5 : The “test charge” negative 3.1 Forces and Energy in Electric Fields 3 C3.4.This is also found experimentally from the fundamental experiment in Fig. was negative, the field directed from above to below, and the resulting force was directed upwards. This agrees with Eq. ( opposite to the field direction acts on 70

Part I 3.4 First Applications of the Equation F = QE 71 3.4 First Applications of the Equation

F =QE Part I

The application of Eq. (3.5) is in general not at all simple. Usually, the carrier distorts the initially-present electric field due to influence, even when it is not itself charged. The field takes on a complicated form. In such cases, one has to compute the field strength at each surface element of the uncharged carrier; then, after charging the carrier, one must multiply the charge on the surface element by that field and sum over all such products. In reality, the situation is still more complicated, since the charge on the carrier will shift the charge distribution on the electrodes that produce the field by influence (imagine a point charge in front of an uncharged metal plate: It induces a mirror charge of opposite sign due to influence). Integra- tion of the MAXWELL equation (2.8) is in general possible only with the aid of complex computer programs. This tedious procedure can be avoided only in a few limiting cases; we offer two examples: 1. The forces between two small spheres at a large distance R from each other. A sphere carrying the charge Q has, by itself, a radially- symmetric field (compare Fig. 2.48). At a distance R, it produces an electric field of strength

Q E D : (2.15) R 2 4"0R

After bringing up the second sphere with its charge Q0, the ﬁeld becomes quite different. For the special case that the two charges are equal, Q D Q0, it can be seen in Fig. 3.3 (where the charges have opposite signs), and in Fig. 3.4 (for charges of the same sign).

Figure 3.3 Field lines between charges with opposite signs. As in Fig. 3.4, also, they were generated by vector addition of the ﬁelds of the individual charges (the direction of the ﬁeld lines is by convention from C to , or beginning on C, ending on charges).

Figure 3.4 Field lines between charges of the same sign. The corresponding negative charges can be thought of as residing on the distant walls of the room. , ) ) It Q 3.5 3.5 2.4 ( (3.8) ( (3.9) (3.10) (3.11) C2.10 . ). In this way, OULOMB 3.5 . We again use it to by C : E )and( 2 2 . The field is still homo- A ). It acts on the charge A R : Q 2 ; = 0 s 2.15 0 0 A E ) by itself produces the field 3.4 Q 2.10 " 1 2 0 Q 2 2 ; Q s 2 " R 0 Q 1 QQ C3.5 E " 0 D D 0 ; Q C3.6 D 1 A " Q 2 D 0 D˙ 4 EA QE . The field lines can be thought of as D " 0 0 1 2 F 1 " " F D 3.5 D Q D D F E F D Q F . A plate (with charge )), i.e. to combine Eqns. ( All the field lines on the upper side of the plate now 2.15 C3.7 The attraction of two con- attraction of the two plates (electrodes) in a flat parallel- The charges on the second plateright modify the side). field drastically (Fig. Nevertheless, most treatments of electromagnetismbeginning. put it at the very 2. The plate condenser sketched at the leftending in on Fig. distant charges ofor opposite other sign distant on surfaces. the Compare walls Fig. of the room was the culmination in 1785 of a century of experimental research. geneous at not-too-large distancesThere, in its front magnitude of is and behind the plate. one has to assumealone the (Eq. ( original, undistorted fieldwe obtain of the the magnitude of first the sphere force and for its direction, the resultthe that two it points charges along (spheres). thepulsive line connecting For force, charges and of forwas the opposite first same signs, sign, stated an it in attractive is the force. a form re- This law denser plates. The increasedfield density lines of in the the right-hand imagethe reflects increased field strength. Figure 3.5 In order to apply the equation vanish. What remains is theparallel-plate well-known condenser. homogeneous field of a flat Now we change thedenote meaning the field of of the the symbol complete condenser. Then we find on the second plate with the force This field is to be used in applying Eq. ( is ; R con- = R R R .Inthis 0 0 ’s law OHL Q 2 R ,and 0 QQ Q 0 1 " is the force which )) in vector form OULOMB towards 4 F 3.8 Q D Forces and Energy in Electric Fields exerts on F 3 the unit vector which points from where Q siders only the magnitudes here. ily understand by considering insulating plates with fixed charges instead. faces, each facing the other plate. This however has no influence on the new field distribution, as one can read- no longer remain on both faces, but instead are now concentrated on the inner sur- C3.7. Since the charges are free to move over thefaces sur- of the plates, they For simplicity, P C3.6. The two charges have opposite signs, which gives rise to the attractive force. C3.5. C specified. formulation, in addition to the magnitude of the force, its direction and sign are also (Eq. ( is 72

Part I 3.4 First Applications of the Equation F = QE 73

M Part I BC 4×10–8 A Farad + G

Figure 3.6 The mutual attraction of two condenser plates C and A. B is an insulating post, M is a screw micrometer with a mm scale and a vernier, G is a weight. Numerical example: A D 20 20 cm2 D 4 102 m2; plate spacing l D 10:2mmD 10:2 10 3 m; voltage U D 7500 VC3.8 C3.8. Instead of the inﬂu- ence machine and the large 8:86 1012 As 5:63 107 V2 4 102 m2 F D condenser which is needed : 4 2 2 Vm 1 04 10 m to smooth its voltage out- Ws D 9:6 102 D 9:6 102 N : put, one could of course use m a high-voltage power supply. When the electric force becomes greater than the weight of G, the plate A Note the robust kitchen scale begins to move upwards. which permits a sensitive measurement of the forces. or " U2A F D 0 ; (3.12) 2 l2 i.e. the force is proportional to the square of the voltage U and inversely proportional to the square of the spacing l of the plates. Figure 3.6 shows a setup for testing this equation. It is intended in particular to give a feeling for the orders of magnitude involved. For precision measurements, one must use a ﬂat parallel-plate condenser with a “corona ring” here also (Fig. 2.44).

AccordingtoEq.(3.12), the force increases in inverse proportion to the square of the spacing of the condenser plates. Therefore, to produce large forces, condensers with very small spacings have been constructed. A highly-conducting plate is set onto a poor conductor, both with smooth surfaces. Figure 3.7 shows a metal plate M in contact with a lithography stone St. Both have a surface area of around 20 cm2. The stone’s weight is about 2 N (mass m D 200 g). When a voltage of 220 V is applied, the stone “sticks” to the metal plate. It can be lifted together with the metal plate using the handle. Of course, this condenser is not insulated. In our

Video 3.1: “Forces in an Electric M Field” 220V St http://tiny.cc/xaggoy Note the protective resistors! Figure 3.7 The mutual attraction of two condenser plates which are made of Note also that the experiment a good conductor M and a poor conductor St. As a result of the unavoidable in Fig. 3.7 is reversed: the roughnesses on their surfaces, the spacing is very small at some points, and stone is lifted and pulls the the electric ﬁelds are therefore very large there (Video 3.1) (Exercise 3.1). metal plate below it upwards. ) 2.20 (3.13) ( (3.14) (3.15) ,and : 3.7 .Thespray pressure on the 3.8 reduction of the sur- ). We could readily use it : A 2 2 : 1.29 : r U 2 2 2 : 0 r 2 E U " r 0 U 0 , it acts like a 2 " 2 1 ) and obtain for the " (see Vol. 1, Sect. 9.5). In the pres- D A ﬂows across it. The human body only

D r r 6 D 2 E e : 3.13 e Surface area = p p 2 D p . Then on its surface, the ﬁeld strength is D U 0 which acts perpendicular to a surface p F of the water decreases. The fact that the water coalesces into ) . The surface tension by itself produces a pressure that h Force

D 3.11 is generally defined by the ratio of Charged Bodies. Reduction of Their Surface Tension p is the field strength directly on the inner surfaces of the plates. . The voltage between the sphere and the distant carriers of r E example, a current of several 10 thus make the stone “stick” onto the metal plate. as a “connecting wire” instead of the metal wires shown in Fig. feels currents of more than 3 to 5 mA (Fig. The reduction of thestrated surface in many tension ways, by e.g. an using electric the field setup can shown be in demon- Fig. nozzle of a glass containeras emits the a depth jet, which becomes a stream of droplets droplets is a result ofbetween its surface the tension. water Now, and weImmediately, the produce the water walls an again electric of flows field the out of room the using nozzle as an a influence smooth machine. jet. This is a convenient but lax way of putting it. The pressure is not ‘directed’, but the opposite charge is For the homogeneous field of afrom parallel-plate condenser, Eq. we ( thus find We want to applyradius this equation to the case of a charged sphere of Here, We insert this value into Eq. ( surface of the charged sphere This pressure is directed outwards face tension is directed inwards, rather the associated force. 1 Pressure 3.5 Pressure on the Surfaces ence of an electric field,reduced the to remaining inwardly-directed pressure is Forces and Energy in Electric Fields 3 74

Part I 3.6 GUERICKE’s Levitation Experiment (1672). The Elementary Charge, e = 1:602 10 19 As. 75

Figure 3.8 The inﬂuence of an electric ﬁeld on the surface tension of water (GEORGE MATHIAS BOSE, 1745) Part I

3.6 GUERICKE’s Levitation Experiment (1672). The Elementary Charge, e =1.602 · 10−19 As.

A particularly important application of the equation F D QE for physics is the “levitation experiment”. It is the original form of the arrangement shown in Fig. 3.1. A light-weight charge carrier is brought into a vertically-directed electric field. Suppose that the carrier is negatively charged and the condenser electrode above it is positive. Then its weight FG pulls the carrier downwards and the force U F D QE or F D Q (3.1) l pulls it upwards (compare the field lines in Fig. 3.9). In the limiting case U F D Q ; (3.16) G l there will be an “equilibrium”, and the carrier is “levitated”. Then we can compute the charge Q on the carrier from its weight FG and the field strength U=l. For demonstration experiments, light-weight objects which would fall only slowly in air, such as feathers or cotton, gold leaf flakes,

Figure 3.9 The elec- A tric field lines (lines of force) in the levitation experiment. One can practically “see” how the charge carrier is pulled upwards, although the field which – enters in Eq. (3.16)is the homogeneous field of the empty condenser. C , G 2 F .The 3 is not ho- UERICKE G B , one can use 3.10 C 3.10 A is often left off; then . TTO VON 2.17 3.10 ). 3.11 ray bottle; friction with its walls a - EN (1746); TTO VON Old rep- Acharged (Video 3.2) (1672) ). The electric field strength is varied by changing the ILSON m. They are charged by contact to a solid body (by “static W 3.10 : gold-leaf flakes; UERICKE Usually, however, their diameters are found from their sinking speed in air Guericke was mayor of Magdeburg, 1646–1681. See also Vol. 1, Sect. 9.9, and : downy feathers). B JAMIN G (1602–1686), in the year 1672 (Fig. volume can be computed from the diameter and yields the weight at the left by O resentations of the levitation experiment: at the right, by B soap bubble levitated in an electric field Figure 3.10 small liquid droplets, usuallythan oil 1 or mercury, withelectricity”). diameters This of can less bean accomplished air by jet letting through them the beseparates the nozzle carried necessary of in charge. a The sp of condenser ca. plates 1 have cm. a spacing Theare motions observed of through the a charged microscope. dropletsbe in obtained The the by weight electric a field of microscopic measurement the of droplets their can diameters The levitation experiment canduced scale; be readily instead repeated of on the soap a bubble greatly in re- Fig. ( a “Plumula potest per totum conclave portari” 3 (Vol. 1, next-to-last paragraph of Sect. 10.3). 2 J. L. Heilbron, p. 216, cited in the footnote in Sect. soap bubbles etc. arethen most caught in suitable. the electric These field between carriers(Fig. two charged are condenser charged plates and spacing of the plates.mogeneous; otherwise its (The field strength fieldspacing would be of as independent the shown of plates.) the into In Fig. sink, this way, or we tosimplify can remain the cause setup, levitated the the (i.e. carriers upperthe to to plate ceiling rise, “float” in of Fig. at the room ation takes experiment fixed on was height). its first function. demonstrated To by In this O form, the levita- Figure 3.11 Forces and Energy in Electric Fields 3 Video 3.2: “Soap Bubbles in an Elec- tric Field” http://tiny.cc/9aggoy 76

Part I 3.6 GUERICKE’s Levitation Experiment (1672). The Elementary Charge, e = 1:602 10 19 As. 77 after multiplication by the density of the liquid and the acceleration of gravity. A very fundamental result emerges from such experiments with small, but still readily visible charge carriers (R. A. MILLIKAN, Part I 1910; he continued the classic experiments of J. S. E. TOWNSEND, 1897, and J. J. THOMSON, 1898): An object can accept or release electric charges only as integral multiples of the ‘elementary charge’ e D 1:602 1019 A s. In spite of numerous efforts, no one has ever been able to observe a smaller charge than 1:602 1019 A s on a positively- or negatively-charged object. For this reason, we call the charge e D 1:602 1019 Asthe elementary electric charge. It is the smallest individually observed negative or positive electric charge. For example, an electron possesses exactly one negative elementary charge.

The ‘Millikan experiment’ is not difficult to carry out. It should be found in every beginning physics teaching laboratory. It is most impressive when observed individually through the microscope. With a projection microscope, even gentle air currents in the condenser disturb the observations. They arise from the intense light source which is required for projection.C3.9 C3.9. Today, this disturbing effect can be easily avoided The cathode-ray tube (or oscilloscope) already mentioned in Sect. by combining a television 1.11 can be conveniently used to demonstrate the force on individ- camera with the microscope. ual electrons in an electric field (Eq. 3.5). The schematic is shown in Fig. 3.12. The electrons (of charge e and mass m) which emerge from the hot cathode C pass through a tube F and through a hole in the anode A. This tube and the anode together serve as an electric lens and form an image of the small opening in the anode on the fluorescent screen S. Along their paths from the anode A to the screen S, the electrons pass through two parallel-plate condensers whose field directions are rotated by 90° relative to each other. We show in the figure only one of these, namely DD0. Using its electric field, the electrons can be deflected within the plane of the paper (the other condenser produces deflections perpendicular to the plane of the paper). In this manner, one can combine the two perpendicular deflections (“x and y axes”). The deflections are proportional to the field strengths E produced by the voltages between the condenser plates. For the deflection x of the path (Fig. 3.13), we find

1 e y2 x D E : (3.17) 2 m u2

Derivation: Let the electron pass along the length y of the condenser at a velocity u in the time t D y=u. During this time, the electron falls through D 1 2 a distance x 2 at. Here, a is the acceleration of the electron in the direction opposite to the electric ﬁeld E which produces it. From the fundamental equation of mechanics, the force F D maand thus with Eq. (3.4), eED ma. Inserting the values of a and t yields Eq. (3.17). (Compare this also with the parabolic ﬂight path treated in Vol. 1, Sect. 4.9.)

Finally, we make one more not-unimportant remark: A dropper bottle can dispense its medicine only in “elementary quanta”, namely ) 3.11 ( S remain un- , so that the l 0 . It does this =" D 3.14 (‘deflection plates’) 0 D D E x and D ; and their spacing A 2 A E 0 2 " y , suppose there is an electric field of V D just behind a negatively-charged collimator cylinder”). In modern versions, the cathode C F D D´ . The one plate attracts the other and pulls , performing work in the process, for example A Al l EHNELT and thus the field strength D (“W Q CF V F 200V 800V . How much energy is contained in this field? The deflection of electrically-charged beams by the homoge- C3.10 A cathode-ray tube with a heated cathode. The latter consists E V 30 changed. For the work performed, equal to the energy previously since the charge serve to deflect the electron beam. itself is not imaged onfied the image fluorescent of screen, the but cathode. instead The a condenser strongly plates demagni- of a glowing tungsten filament (focussing) tube neous electric field of a parallel-plate condenser with the constant force magnitude We imagine that the field is produceddenser. by a Let charged the parallel-plate con- area of its plates be In an empty space of volume as individual droplets. This ofthat droplets course exist does independently not inside prompt thelikan us levitation bottle. experiment to doubtless Similarly, the shows assume that Mil- thereto is the a lower divisibility limit of electricsame charges. subdivision of But charge itof within by objects! the no interiors means The (or proveswithin existence on the of a the individual, charge surfaces) countable carrierassumption. charge remains “quanta” for the moment only a very useful 3.7 The Energy of the Electric Field Figure 3.12 Figure 3.13 work of lifting according to the schematic of Fig. field volume is it along a distance Physics onduct- blockade” effect, , Jan. 1993, p. 24). Forces and Energy in Electric Fields OULOMB 3 “single-electron transistors”; see M. A. Kastner, Today C3.10. This assumption appears to have been confirmed only recently by the following observation: The electrical properties like currents or capacitances of metallic or semic ing particles of the order of 100 nm in diameter can be varied over orders of magnitude by changing their electric charges (the “C 78

Part I 3.8 The Electric Potential. Equipotential Surfaces 79

Figure 3.14 The derivation ∆l of the energy of an electric

ﬁeld. The plates are not con- Part I nected to a current source.

stored in the electric field, we then find " W D F l D 0 E2A l ; e 2 C3.11. This means that it or, integrating ( l ! dl) over the full spacing of the plates from 0 to holds also for inhomoge- l, neous electric fields. In " a volume element dV, con- W D 0 E2V (3.18) e 2 taining an electric field E,the 12 energy "0 D 8:86 10 As=Vm : 1 2 dW D "0E dV Equation (3.18) holds quite generally, in spite of our having derived 2 it for a specific case.C3.11 is stored. Often, one quotes instead the energy density: Only small amounts of energy can be stored in practice in the form of D 3 3 dW 1 2 electric fields. For example, in one liter ( 10 m ) at a technically D "0E : practicable field strength of E D 107 V/m, the stored energy is only dV 2 0.44 W s. From this, it follows that in general Z Equation (3.18) for the energy of an electric field is often written " differently, i.e. by making use of Eqns. (2.4)and(2.2): W D 0 E2dV : e 2 V 1 We D QU ; (3.19) 2 C3.12. The fact that the two equations (3.19)and(3.20) and continuing with the aid of Eq. (2.10), hold not only for empty parallel-plate condensers, 1 2 We D CU : (3.20) but in general, i.,e. for ex- 2 ample when the condenser Here, Q denotes the charge on a condenser of arbitrary form, U its contains a dielectric, can be voltage and C its capacitance.C3.12 derived from the following considerations: The energy We stored in a charged condenser is completely con- 3.8 The Electric Potential. Equipotential verted into JOULE heat by Surfaces discharging the condenser through a resistor R.Ifwe insert Eq. (2.23)forI into To represent electric fields, in addition to the field-line patterns, effec- the expression for the power, P 2 tive use is often made of the electric “equipotential surfaces” (contour WeDI R (Eq. (1.7)) and in- maps). In Fig. 3.15, we show an electric field between a plate and tegrate over time, we obtain a wire which is parallel to it. Directly above the plate, there is a small these two equations. . V W two 10 : (3.21) 0 U equipo- conven- Q ,then 30 V a 0 D point in and is de- Q is the volt- s b is positive. U c W 50 V and d E point within the field; of the field n )), where in order to bring it from QE 3.2 single on it is positive. (At the same time, has a positive charge potential difference 0 and therefore the carrier is moved Q , this is the plate. This electric (Eq. ( = (test charge). This carrier can be ). If one refers to a voltage as a po- 3.15 the force ) is negative, since the direction of the W 0 U to , which always applies between 0 3.15 Q 2.3 D C Q on the carrier Q . The set of all such points which can be ' . The reference point is often connected against n difference (in Eq. D s must be performed ' ), the numerator and the denominator have the same ;inFig. voltage U d ; that would require performing the work “potential” is thus a name for the voltage be- U ;::: ' a c is also called the Charge E ; 3.21 0 has been performed. The carrier is then at one of b R Q ; , and in other fields produced by two charged bodies, can be specified at only a U a performed . D 0 ' Q 3.15 A schematic drawing of electrical W voltage QU the field into regions of increasing potential; i.e. D W potential Work .InFig. Therefore, in Eq. ( sign and the voltage opposite directions.) a negative reference point to the equipotential surface. Then field points by convention from the path integral against the work Justification: If the carrier in Fig. is the voltage between the equipotential surface and the to the earth’s surfacelar (“grounded”); point then in the the potentialearth’s field at surface. means a The the particu- tween voltage an between arbitrary that point within pointpoint, a and and field the the and equipotential a surfacestial are conventional reference surfaces of constant poten- the positive sign of theerence potential point. implies a negative charge at the ref- age between the starting andcarrier end is points moved. of the Weferent path then starting along points repeat which above the the the same surfacepoints of experiment in the but other plate with and regions dif- work different of end the field. Each time, we stop when the tional reference point noted by the symbol tension or U tential, this implies that a reference point was previously agreed upon moved to the point equipotential surfaces Figure 3.15 the end points The charge carrier with the charge this is not true of the points (it is the potential In electric units, this work is reached by performing the same amount of work is called an tential surface To characterize an equipotential surface, we make use of the ratio Forces and Energy in Electric Fields 3 80

Part I 3.9 The Electric Dipole. Electric Dipole Moments 81

(often taken without specific mention to be the earth’s surface). Un- fortunately, the words ‘potential’ and ‘voltage’ are often used loosely as synonyms and not correctly distinguished. The potential differ- Part I ence between two points within a field is the voltage between those two points.C3.13 C3.13. The relationship between the voltage U or Example: Consider the potential distribution (potential field) of a charged potential difference ' (be- metal sphere (carrying a charge Q at radius r). Assume that the refer- 0 tween two points 1 and 2), ence point (where the potential is zero)isatR D1. The electric field R strength within the sphere is zero, and outside the sphere, it is given by and the path integral E ds, 2 E D Q0=4"0R for R r (Eq. (2.15)). Then we find in the region out- is thus found (in agreement side the sphere the potential: with the above justification in petit type) to be ZR ZR Q 1 1 Q Z2 ' D E dR D 0 dR D 0 ; (3.22) 2 4"0 R 4"0 R D ' D : 1 1 U E ds 1 and within the sphere, In Eq. (2.3), for simplicity, 1 Q the sign was not taken into ' D 0 D const : (3.23) 4"0 r account.

Thus, for Q0 > 0, the potential decreases as 1=R outside the sphere and is constant in its interior. The equipotential surfaces are concentric spherical shells (for R r).

3.9 The Electric Dipole. Electric Dipole Moments

The fundamental equation F D QE for the appearance of forces in an electric field requires not only the field itself, but also a body carrying an electric charge. Considered superficially, this would appear to contradict our long-established experience: We observe forces on light-weight uncharged bodies in electric fields. Imagine some scraps of paper in the neighborhood of a piece of amber which has been rubbed with fur; or the dancing dolls below a glass plate which has been charged with static electricity. To understand these phenomena, we need two new concepts:the electric dipole and the electric dipole moment. We suppose that in Fig. 3.16, two “pointlike” electric charge carriers with the charges CQ and Q are attached to the ends of an extremely thin and ideally insulating rod of length l. This dumbbell-shaped object is called an electric dipole. Its electric field is similar to those shown in Figs. 2.8 and 3.3. We further imagine this dipole to be oriented as in Fig. 3.16 with its long axis (dipole axis) perpendicular to the field lines of a homogeneous electric field. Then a torque acts on it:

l M D 2QE D QlE: (3.24) mech 2 (as –F F .The E –Q (3.26) (3.27) (3.25) S F E l . The force R of the dipole . All of these –F along the con- F E +Q is the vector that /; dipoles have been S . This then leads in R E N C r )),andthatitvanishes 0 to A p is perpendicular to where . : : 3.24 p F E E X is defined to be to the point of action of D S p p F D D 0 mech M electric dipole moment p direction mech mech in an applied electric field M M X the means that the mechanical torque which acts ; its D . is then completely unimportant. 0 is mounted at the end of a spoke mech Ql E and assumed in Eq. ( R S C3.14 M mech produces a torque vector R is important, not the M F S 3.16 ,arod For a torque, only the length An electric dipole which is oriented and is parallel to 3.17 F p or The idealized dipole asever, defined positive above and is negative charges not ofized found equal at in magnitude a can reality. particular be separation How- a within local- matter measurement in procedure a can number ofof be ways, a used and structure to of defineexperiment this the in type. mechanics: dipole moment Such a procedureIn makes reference Fig. to an This vector product individual torques can be added vectorially,distances in from spite the of mutual axis their of different rotationthe of overall the (observable) body. torque: We thus obtain length of the spoke of the lever arm perpendicular to electric field lines Figure 3.16 drawn in Fig. when couple on the electric dipole is maximal when points from the point of action of Now we suppose that in some arbitrary body, formed by some sortperiences a of torque localization of charges. Each of them ex- length of the spoke Figure 3.17 We call the product (unit: ampere second meter, A s m).be The described electric as dipole a moment must necting line between the two charges from general to Forces and Energy in Electric Fields 3 C3.14. For the vector product, see Vol. 1, Comment C6.1. 82

Part I 3.9 The Electric Dipole. Electric Dipole Moments 83

Figure 3.18 An electric dipole in an inhomogeneous electric ﬁeld (the ﬁeld direction is upwards (Cx direc- +Q F0=Q·E0

tion), and E decreases with increasing x) Part I

–Q

Fu=Q·Eu

Here, p denotes the total, macroscopically observable electric dipole moment of the body which is the vector sum of the individual, unknown microscopic dipole moments within it. This overall dipole moment can always be represented in terms of an idealized dumbbell-shaped dipole: Two pointlike charges CQ and Q are ﬁxed at a separation l. The rod in this dumbbell points along the direction of its electric dipole moment.

This defining equation suggests a measurement procedure for electric dipole moments (which is however unimportant in practice): We mount an object on a rotatable axis perpendicular to an electric field and find its rest position. Then we rotate its axis by 90ı out of the rest position and measure the required torque as the product of force and force arm (lever). This torque is then divided by the strength of the homogeneous electric field E to give the overall dipole moment p.

So much for an electric dipole or a body with an electric dipole moment in a homogeneous field. The field acts on the dipole, producing a torque, and orients the dipole with its axis parallel to the field direction, as long as its rotation is not hindered. The same holds true in an inhomogeneous electric field. The dipole in Fig. 3.18 is supposed to have already oriented itself along the field direction (positive x direction). In addition, however, in an inhomogeneous field, something new occurs: In an inhomogeneous field, along the direction of increasing field @E=@x,thereisa force

@E F D p : (3.28) @x

Derivation: The upper C charge experiences the force QEo upwards, while the lower charge experiences the force QEu, downwards. Combining these, the overall force

F D Q.Eo Eu/ (3.29)

acts on the dipole. Furthermore, we have

@E E D E C l : (3.30) o u @x

Equations (3.29)and(3.30) together yield Eq. (3.28). ). 2.56 ). This is , giving rise 3.19 . Every material ; in an insulator, depolarization factor electric fields parallel molecules surfaces all ) of the dielectric (Fig. y material body which is brought , the individual are made visible with fibrous powder. In ILBERT within G . The parallel orientation is energetically fa- 4 “electrification” C3.15 boundaries of the field (e.g. on the plates of a condenser), (or ILLIAM fields, all bodies are in addition pulled towards re- : The field displaces the positive and negative charges ) has its smallest value along the long axis). Forces on an uncharged Electric Dipole Moments. Pyro- and Piezoelectric Crystals field-line patterns 13.6 influence polarization Electric dipole moments produced by influence At the delimiting well-conducting bodies immediately become charged,carriers”, and then, they as fly “charge over toitself the opposite with electrode, opposite where sign. thecharging In process requires repeats the some time case (several of seconds).sticks insulators to During the or this electrode. time, poor This can the conductors, be objectusing seen the small very cotton clearly balls. in a shadow projection The polarization of the oblong body attains its maximum value in this orientation, 1600, physician in London andof originator the word “electric”). how e.g. inhomogeneous gions of higher electric field strength, regardless of their shapes. vored relative to all other possible orientations (Fig. oblong object made of metal,lator or which an is insu- mounted onperpendicular a to rotation the axis plane of(a the “versorium”: page W object in an electric field; top, a small, and with it, the induced(see dipole Sect. moment (this is because the 4 Figure 3.19 its dipole moment withoutthe question: referring to How can experiments.practice? we We produce must Now electric distinguish between dipoles we two within ask cases: matter1. in body acquires an electric dipole momentdue to in an external electric field, relative to each otherinto within it. ever In a conductor, theyto move a to its 3.10 Induced and Permanent We have thus far introduced the concepts of the electric dipole and there are displacements As a resultence, of oblong this bodies “induced” orientto dipole themselves the in moment, field produced direction by influ- , ”, 31 Vier . Forces and Energy in Electric Fields 3 https://en.m.wikipedia. org/wiki/William_Gilbert_ (astronomer) Jahrhunderte Spannung Physik in unserer Zeit 159 (2000); English: See C3.15. See A. Loos, “ 84

Part I 3.10 Induced and Permanent Electric Dipole Moments. Pyro- and Piezoelectric Crystals 85

2. Permanent electric dipole moments. a). In every charged condenser, charges of both signs are displaced relative to one another.

As a result, most charged condensers have their own electric dipole Part I moment. It is lacking only when one of the condenser electrodes surrounds the other in the form of a closed hollow box.

Unfortunately, an electric field produces a dipole moment even in every uncharged condenser due to influence. Therefore, in Sect. 3.9, we did not start from experiments; every one of our dipoles would have still moved in the field even if they had originally not possessed permanent dipole moments. b). We put a mixture of liquid wax and resin into an electric field and allow it to solidify there. Its induced electric dipole moment (due to influence) is frozen in and thus has become permanent. As a result, the solidified object (preferably cut into the form of long sticks after solidification) acts as an electret. It appears to be a good electric insulator with positive electric charges at one end and negative charges at its other end. These charges can be measured in an influence experiment using a ballistic galvanometer (Sect. 1.10). The leads of the galvanometer are connected to metal cartridges and these are wiped simultaneously over the two ends of the electret; the induced charges separated in the cartridges by influence are then measured by the galvanometer. Electrets of this type maintain their polarization for years, as long as they are kept in a well-fitted metal protective housing; otherwise, over the course of time, they attract charge carriers (ions) from the air and thereby cover their ends with a layer of charges of opposite sign, so that their electric dipole moments are no longer detectable to the outside world. c). Pyroelectric crystals, e.g. tourmaline, have permanent electric dipole moments owing to the arrangement of their charged microscopic components (F. U. T. AEPINUS, 1756). Their directions fall along a polar crystalline axis; in the case of an oblong tourmaline crystal, for example, along its long axis. Normally, this permanent dipole moment is not apparent, due to the covering layer of ions as mentioned above. It becomes active only when one changes the elementary electric dipoles by thermal expansion or contraction.For example, if we dip an oblong tourmaline crystal of ca. 5 cm length into liquid nitrogen, then only part of the covering layer compensates the dipolar charges. The crystal now appears to be a good electret, attracting scraps of paper, etc. (Fig. 3.20).

Liquid nitrogen is often clouded by ﬁne ice crystals. They can be removed by dipping a tourmaline electret into the liquid.

Figure 3.20 A coating of powder attracted to the ends of an electret (tourmaline) https:// G C3.16 ). ,i.e.they ). mechanical between the is connected 3.21 l and a density 3.6 C 3 5cm : 0 )? (Sect. piezoelectric 8 2.12 . Find the change in the en- as compared to the video!) U ). . We want to check this assump- l 3.7 3.7 ) contains supplementary material, which The online version of this chapter ( (Sect. ). (Note that the roles of the metal plate and n = 3.4 of the droplet be if it can just be levitated by the R (Sect. 3 (compression or tension). This can be demonstrated by see also Exercise 2.7. , see also Exercises 2.11, 2.15, and 2.16; A tourmaline crystal acquires an 3.7 3.8 stored in the condenser when the spacing of its electrodes is 8g/cm A parallel-plate condenser with a capacitance A water droplet is charged with a single electron. How large Refer to Video 3.1: In the video, the brass plate is lifted by e W D % ergy of 3.3 to a current source with the voltage 3.2 must the radius electric field of the earth (130 V/m, Sect. the stone slab are reversed in Fig. electric dipole moment when presseda ballistic (detected galvanometer by or a static voltmeter) Figure 3.21 applying a voltage of 460to V, which the stone is when just lifted. sufficient Compute to the make average it spacing stick Electronic supplementary material Exercises 3.1 doi.org/10.1007/978-3-319-50269-4_3 is available to authorized users. Pyroelectric crystals are at the same time change their electric dipole moments also as aagain result using of a rod-shapedtwo insulated tourmaline electrodes, crystal. connects thempresses to One the a crystal fixes galvanometer, along and it its com- long between axis using a vice (Fig. deformation changed by the factor 1 tion. The brass plate has a volume of 5 stone and the brassvoltage plate. drop occurs For across the simplicity, gap we assume that the entire For Sect. for Sect. Forces and Energy in Electric Fields 3 C3.16. Conversely, piezoelectric crystals are deformed by an electric field (theycome be- longer or shorter). We can for example produce sound waves (ultrasound generator), or control very small length changes by an applied voltage, as in a scanningnel tun- microscope. 86

Part I utr,temtl.Haigo h odco (J conductor importance. the of equal con- Heating important of most metals. technically means the the ductors, no in lacking by are changes are Chemical characteristics in changes three chemical These 3. and effect; heating a conductor: medium. conducting 2. a the field; in magnetic currents A electric 1. of characteristics three mentioned ftecnutri omdit ig eoti edlnsa shown as lines field obtain we ring, a Fig. into in formed is conductor the If h antcField Magnetic The n ieaecneti ig (Fig. rings conduct- concentric current-carrying are straight wire long, ing a of lines by field magnetic taken The forms geometric We typical fields. the magnetic fields: of of magnetic some nature considering the into by deeper start delve to want now We the However, O Ch. (H. Fields Magnetic of Production The 4.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © antcfil utlk neeti ed–cneiti empty in exist can – field electric an like just (the – space field magnetic A current. electric every of companion inseparable the is odrit hi-iepten,poiigiae ftemagnetic the of or images filings providing iron lines. of field patterns, orientation which chain-like phenomena the into these was powder encountered of important already the is most have This The we space. (field-free) here. ordinary fact in decisive those in from can phenomena field different fields, magnetic observe a electric We like experimentally. only fields, studied Magnetic be unimportant. practically is nahlclpten f Figs. cf. pattern; helical a in r once nsre ota h aecretflw hog l of all through flows current same the a each forming that them, so that series they imagine conveniently, in more could connected source; One current are own pattern. its has field winding overall circular an give to posed a om ota eea iclrwnig r daett ahother each circu- to a adjacent around are wire windings the circular (Fig. several wind that We so form, deformed. lar somewhat be to and lob akn ne certain under lacking be also 4.2 4.1 .Nw h edln atrsfo ahwnigaesuper- are winding each from patterns field-line the Now, ). yEeti Currents Electric by ERSTED h crls pert epse cetial outwards eccentrically pushed be to appear “circles” The . C4.2 vacuum antcfield magnetic coil 80.Teitoutr umr nChap. in summary introductory The 1820). , .Tepeec farmlcls(f Fig. (cf. molecules air of presence The ). b idn h iecniuul noaspool a onto continuously wire the winding (by olsItouto oPhysics to Introduction Pohl’s 4.3 speeti l cases. all in present is odtos(superconductivity conditions and 1.4 4.4 ). ). OULE , https://doi.org/10.1007/978-3-319-50269-4_4 antcfield magnetic A etn)may heating) C4.1 2.14 ). 1 ) lines of concept the For C4.2. e omn C10.3. Comment see superconductivity, on remarks For C4.1. e omn 2.1. Comment see , 4 field 87

Part I C4.3 pole of the current source is connected to the end of the coil at The magnetic field lines from a short current-carrying coil. The The magnetic The magnetic field C field lines from three parallel circular rings, each carrying the same current lines from a current-carrying circular ring, made visible with iron filings Figure 4.1 is called its northmagnetic pole field and of a is coil, often the compass marked needle by points an along the arrowhead. direction In the Figure 4.3 One end of a compass needle normally points to the north; this end Figure 4.2 arrows denote compassHere, needles, the and their arrowheads are the north poles. the upper left. The Magnetic Field 4 like the left-hand end of the compass needle within the coil). a magnetic field which points to the right in the planepage. of The the left-hand end of the coil is its south pole (just seen to be circling inwise a sense clock- around its axis. Within the coil, it produces C4.3. If we look atfrom the the coil left, the current is 88

Part I 4.1 The Production of Magnetic Fields by Electric Currents 89 Part I

Figure 4.4 The magnetic field lines from a long current-carrying coil. In the interior of the coil, there is a (nearly) homogeneous magnetic field.C4.4 C4.4. The field distributions shown here, obtained using iron filings to trace the pattern of the field lines, are quite similar to the real field distributions, as can be seen here in the figure:

50 cm

Figure 4.5 The magnetic field of a bundle of long, thin coils. The individual coils were completely separate in this model experiment. The apparent zig- –50 zag shaped connections are only simulated by the accumulation of iron filings –50 0 50 cm between neighboring wires. This figure shows the field distribution of a coil as com- of the field lines (Fig. 4.3). The direction in which the arrowhead puted from the MAXWELL points is defined by convention to be the positive direction of the field. equations (Sect. 6.5); the ra- The same field as with a single large coil can be obtained with a bun- tio of length to diameter of dle of identical long, thin coils, if the bundle has the same cross- the coil was taken to be 5 : 1, sectional area and all the coils are carrying the same current. Fig- corresponding to the coil ure 4.5 shows a field-line pattern obtained in this way. Compare shown in Fig. 4.4 (computed it to Fig. 4.4. This experimental finding is readily understood: In by J. A. Crittenden, Cornell Fig. 4.6, we have drawn a bundle of coils in cross section; for sim- University). The inhomo- plicity, the cross sections of all the individual coils have been chosen geneous regions of the field to be square. In the inner parts of this drawing, one can see that (near the poles) are discussed neighboring currents are opposite and thus cancel. Only the thick in more detail in Sect. 8.6. black windings around the perimeter of the bundle contribute to the field. Only this current path need be considered. At the ends of the coils, field lines project out into the open space outside the coils. They emerge not only through the two open ends of 4.3 ). The shows an 4.7 .Thecoil 4.8 , square coils. a–b corresponds to long s not necessary; by the right choice coils which have no poles . 4.5 ab These regions where the field lines emerge from a coil : Accumulation of iron filings around a current-carrying coil; poles, in analogy to a permanent bar magnet. A current- Schematic of a bundle of Top . : The poles of a permanent bar magnet made visible with iron filings 4.4 It is also possiblemust to be fabricate wound in the form of a closed ring. Figure stant all around the ring,of but the this spacing i of neighboring windings, itwithout is poles possible but to with produce a coils variable cross-sectional area. example. In the coil shown, the cross section of the windings is con- middle section outside a coil will remainlines free emerge of iron only filings. at The field longer the and regions longer, called the “poles”. polesto become the As less field a and in coil less theand becomes important interior relative of the coil. Compare for example Figs. Bottom (the magnet is madesintered of in a a ceramic: magnetic field). ferritein This powder the magnet with has top the a image. samepressed binder, geometry together. shaped as The the and coil bar is composed of 100 flat slabs which have been are called its carrying coil behaves very muchor like a balanced bar magnet: on Ifa an it is compass axis suspended needle horizontally,scattered in the over the coil it, north-south will a direction. coil orient will itself attract If like them iron to filings its poles are (Fig. Figure 4.7 the coils, but also neartheir the windings. ends, out of the sides of the coils through Figure 4.6 the plane of the picture in Fig. The Magnetic Field 4 90

Part I 4.2 The Magnetic Field H 91

Figure 4.8 Magnetic ﬁeld lines in the ﬁeld

of a current-carrying Part I ring coil (torus)C4.5 C4.5. Such “toroidal” magnetic ﬁelds are used for example to contain the hot plasma for research into nuclear fusion (Tokamak principle). For these experiments, large superconducting coils with diameters of several meters are employed.

We summarize: The geometric distribution of the magnetic fields of current-carrying conductors is determined solely by the geometry of the conductors themselves. In the case of long, thin coils, the magnetic field lines within the coil are practically straight lines, apart from their polar regions. We are then dealing with a homogeneous field .C4.6 The homogeneous C4.6. The field becomes magnetic field of a long coil plays a similar role in the treatment more and more homogeneous of magnetic fields as does the homogeneous electric field of a flat the longer the coil (see the parallel-plate condenser in the discussion of electric fields. figure in Comment C4.4.) The magnetic fields of bar magnets or, in general, permanent magnets are in no way different from those of current-carrying coils; that is, we can replace the magnetic field of any bar magnet in the space surrounding it by the field of a coil of similar size and geometry. We need only adjust the distribution of the windings correctly. We will discover the reason for this similarity in Sect. 4.4. In Chap. 2,wemadeuseofthehomogeneous electric field of a parallel-plate condenser to define two quantities which allowed us to quantitatively characterize any electric field; those were the field quantities E and D. Analogously, we will now use the homogeneous magnetic field of a long coil to define two new quantities with which we will be able to quantitatively characterize any magnetic field. These are the fields H and B. Initially, we will consider only the field H.

4.2 The Magnetic Field H

Like the electric field, the magnetic field must be described quantitatively by a vector. This follows from the readily-seen preferred directions of the magnetic field lines. The vector is called the magnetic field H. S ). can and 4.9 l of the R magnitude of tension (compass nee- 1 , can be measured A , different lengths I A S ), we adjust the torque required R the spring by rotating the pointer ; it is not required to measure them, . Some of these coils have a single layer equal wind up N is the R through the field I The quantitative deter- C4.7 magnitude of an electric field, i.e. its electric field strength The needle must be small compared to the linear dimensions of the field, in order dle) connected to the axle of a spiral-spring torsion balance (Fig. The resting position of thetion needle of is the determined spiral by spring.the the earth’s (For relaxed magnetic simplicity, posi- field.) we neglect the influenceWe of insert this magnetic needleand into rotate the the homogeneous pointer field until of it a is coil in its resting position on the scale (spring relaxed). Then we until the needle isperpendicular oriented to perpendicular its own to resting position). the The field required lines (and thus spring can then be readquired off to the rotate scale; the it needle to isfield a a . position measure perpendicular of to the the torque magnetic re- be measured in volt/meter (V/m).a Correspondingly, the magnetic field, called thein magnetic field ampere/meter strength (A/m).experiments. This They require can two be components, namely confirmed by1. a long coils new of series various designs; of and 2. an arbitrary indicator for the magnetic field. The indicator need only be able toseparated identify two magnetic spatially fields or as temporally but only to determine that the two fieldsAs have the our same indicator, we strength. choose a small magnetic needle mination of the magnetic fielda of long coil (schematic!). ( coil) variable resistor or rheostat usedadjust the to current Figure 4.9 to have sufficient spatial resolution. 1 We now replace the coilrepeating by the a experiment series throughmay of the different have whole ones, different series and cross-sectional of continue coils. areas These The to rotate the compass needle to a perpendicular orientation so that it different numbers of turns of windings, while somethrough have the many coils layers. (using the rheostat By varying the current )). 3.25 and Eq. ( 3.16 The Magnetic Field 4 C4.7. This is completely analogous to the torque that an electric dipole experiences in an electric field (cf. Fig. 92

Part I 4.2 The Magnetic Field H 93

Figure 4.10 The determination of the direc- Direction of tion of the magnetic ﬁeld in the neighborhood motion of the electrons of an electron current (see also Fig. 1.5) Part I

Compass S H N needle

Current direction has the same value as with the first coil in every case. This equality of torques means that the magnetic fields also have equal strengths. In this way, we find experimentally a very simple result: The magnetic fields are equal, as long as the quantity Current I Number of turns N Coil length l has the same value. The cross-sectional area and the number of winding layers have no influence on the field strength. The homogeneous magnetic field of a long coil is determined uniquely by the ratio NI=l; or, in words, by the “current times number of turns divided by the length of the coil”.C4.8 For this reason, we use the ratio NI=l in order C4.8. One must take care to to arrive at a first definition of the magnetic field strength in a long avoid the end regions of the coil; it is: coils in these measurements; there, the field strength de- NI H D : (4.1) creases markedly (see the l figure in Comment C4.4). (The product NI of the current and the number of turns in the coil winding is sometimes abbreviated as ‘ampere-turns’). The unit of H is thus 1 A/m. The direction of the magnetic field vector is parallel to the long axis of the coil. Its sign is indicated in Fig. 4.3 (see also Fig. 4.10). The next step takes us to an important generalization. By making a comparison to the homogeneous field of a long coil, we can measure the field H, its magnitude and direction, at any arbitrary location: We replace its individual small regions (which are practically homogeneous) by the equally strong and similarly oriented field of a small, long coil and determine the vector H for this coil using Eq. (4.1)and the direction of the coil and its current. As a measurement prescription, one can proceed in several different ways: We could for example calibrate the arrangement in Fig. 4.9 and thus convert it into a magnetometer. The calibration is accomplished by varying the current I in a long coil and thus changing its field strength H as defined by Eq. (4.1). We find that the torque required to orient the needle is proportional to H. For example, the demonstration model used in the lecture hall at Göttingen has a calibration factor of 50 A/m per angular degree. a 4.11 500 A/m. about 10 cm P D N S H that the spring has S . In the great majority of 8.6 circular carrier of electric charge ’s experiment: ). Now we consider a surprising fact: C4.9 of the electric charges which produces the , the strength of the field is ’s Experiment (1878) 2.10 P the field strength. Examples can be found in OWLAND motion shows a plan view of a The magnetic field lines OWLAND calculates Produces a Magnetic Field. R . 4.12 6.3 Sect. Analogously, we can measure the earth’s magnetic field. Figure shows a schematic drawing of its fieldparallel to lines. the The surface component of which the is in earth Göttingen, is it called the has horizontal a component; value of aboutMagnetometric 16 A/m. measurementssomewhat are tedious. They time-consuming are howeverto and indispensable measure when very one therefore small needs fields; theytechnique, are as then will carried be out with discussed a in different Sect. cases, one of the earth’s field. The arrowscate indi- the direction of the field . Figure at the outer edgeure). of This an conducting insulating ring disk is (which interrupted by is a shaded narrow in slit between the fig- Figure 4.11 been tensioned by rotatingTherefore, through at an the angle point of 10 angular degrees. magnetic field. Noconductor, other e.g. details a of copper thecharges; wire, process or, acts are roughly only important. speaking,This as is as a The shown a guide by R for pipe the through moving which they flow. in front of its northalong pole. the We direction bring of the theThen needle field with we lines, its turn i.e. spring in relaxed perpendicular the to the needle the direction field of by and the read rotating field. off the the scale frame until the needle is 4.3 The Motion of Electric Charges An electric current withinalong a its conductor length consists (Sect. of moving charges With a magnetometer which has beento measure calibrated the in magnetic field this of way, a we bar want magnet at a point It is simply this , we can see that the 4.3 The Magnetic Field 4 C4.9. By comparison with Fig. geographic north pole of the earth is in fact a magnetic south pole! 94

Part I 4.3 The Motion of Electric Charges Produces a Magnetic Field. ROWLAND’s Experiment (1878) 95

Figure 4.12 ROWLAND’s experiment (the diameter of the ring-shaped charge carrier is 20 cm) Part I

a c b

M NS

Figure 4.13 The magnetic field lines H which surround a positive charge that is ++ moving at a velocity u towards the observer + u + H and b. The lower image shows the same charge carrier from the side; it is mounted on a vertical shaft and enclosed in a grounded metal housing. Between the ring-shaped carrier and the housing, a voltage of about 103 V can be applied, so that the ring then carries a positive charge Q of around 107 A s. At the point M,there is a sensitive magnetometer, indicated schematically in the figure; it could be a compass needle with a mirror and light pointer. (In its resting position, the long axis of the needle is perpendicular to the plane of the page). The charged carrier rotates at N revolutions within the time t; its rotational frequency is N=t 50 Hz. During the rotation of the charged ring, the needle indicates the presence of a magnetic field.C4.10 Thus, a charge set mechanically in motion C4.10. The magnetic field produces a magnetic field, just like a charge moving through a con- produced in this apparatus ductor. Its magnetic field lines are drawn schematically in Fig. 4.13. was 50 000 times weaker They surround the cross-sectional area of a section of the rotating than the horizontal compo- ring (enlarged in the drawing), to the right of the axle c; the section nent of the earth’s magnetic is moving towards the observer with the velocity u. field (see Fig. 4.11)atthe location of the experiment In the second part of the experiment, the charge carrier is discharged; ( 16 A/m in Berlin; see then leads are attached at the points a and b in Fig. 4.12 and a cur- H. A. Rowland, Am.J.of rent I is passed through the ring ( 105 A). It produces an identical Science and Arts, 3rd Series, magnetic field to that of the rotating charged carrier. We see that Vol. 15, p. 30 (1878)). More N sensitive measuring instru- I D Q : (4.2) t ments such as are available today, e.g. a HALL probe or We then introduce the velocity u of the carrier and its path l D 2r a SQUID (superconducting into this equation, obtaining quantum interference device), were not available back then! Nl N u u D or D : (4.3) t t l C4.11 ,the (4.4) P NI ; ’s experiment, we need only as- l u . We will return to this significant through the coil, or its number of ) yields the important relation: l Q = I D 4.2 the iron effectively increases the num- Qu I OWLAND D . 7 . Of course, it neither increases the number of A charge Q which is moved along the path l at the ve- Inserting an iron ? ; a compass needle set up there shows a small angular . How could we amplify the magnetic field and increase ˛ ). Its magnetic field is supposed to be just detectable at P ˛ Magnets are also Produced byMotion the of Electric Charges or both. In any case, we would increase the product We increase the current N 4.14 We introduce a previously non-magnetic piece of soft iron into number of ‘ampere-turns’ of the coil. Or: the coil, an ‘iron core’. This leads us to concludeber that of ampere-turns turns in the coil winding,monitor nor continuously does with it an increasethere ammeter. the must current, Therefore, be which within we microscopic thethe currents same iron, flowing sense along as invisible the paths macroscopicturns’ current add in in to the those coil. ofdifficulties: the Their windings According ‘ampere- of to the R coil. This notion causes no turns Either: sume that some sort of orbital motions of electric charges are present core acts like an increasethe in number of turns orcurrent the (i.e. the ‘ampere- turns’) in a coil (solenoid) or in words: locity u acts as a current I the point deviation the signal Figure 4.14 4.4 The Magnetic Fields of Permanent In our first experiments,of we electric produced currents magnetic inmagnetic fields metal fields with also conductors result the (wires). from the aid charges. mechanically-produced Then motion Now, of we we consider saw as that of the obtaining magnetic third fields: possibility their production the byHow oldest permanent do magnets. method the magnetic fields of permanent magnetsWe come about? again refer(Fig. to experiment and take a current-carrying coil Inserting this quotient into Eq. ( experiment in Chap. : u l Q D lt Ql D I The Magnetic Field D 4 t Q C4.11. This is the definition of an electric current: 96

Part I 4.4 The Magnetic Fields of Permanent Magnets are also Produced by the Motion of Electric Charges 97

Figure 4.15 A rough schematic of ordered molecular currents Part I

within the iron. We know that electrons are present in all material objects. We can imagine their motion within the iron to be microscopic circular (orbital) motions .C4.12 This preliminary but already quite C4.12. This idea occurred serviceable concept is called the model of molecular currents. It can to Ampère as early as 1826. be represented roughly by a drawing as in Fig. 4.15. Compare this He called them ‘circuital cur- figure with the cross-section through a bundle of thin coils in Fig. 4.6. rents’ and presumed that such currents were responsible for “material magnetism”, i.e. the These molecular currents must have been present in every piece of magnetism of permanent iron even before it is put into a magnetic field; but they are on the magnets. average not ordered. Only in the magnetic field of the coil do they become ordered: Their rotational axes all line up parallel to the long axis of the coil. The individual molecular-current orbits act like the small rotatable coil in Fig. 1.10. We can reduce the magnetic field of the coil either by pulling the iron core back out or by interrupting the current through the coil. Then the field produced by the iron for the most part disappears, but not completely. The majority of the molecular currents rotate back to their original disordered arrangement; only a certain portion maintain the orientation that they received in the magnetic field. The iron is then said to exhibit “remanent” magnetism (Chap. 14). It has become a permanent magnet (like a compass needle). Only one single point is essential in this model: The existence of some sort of orbital motions of electric charges (e.g. the electrons) in the interior of the iron. This decisive point can be tested experimentally: The mechanical angular momentum of the orbiting charges can be demonstrated and measured (Einstein-de Haas experiment, 1915). We remind the reader of the following experiment in mechanics: A person is sitting on a swivel chair and is holding a rotating object, e.g. a wheel. Its plane of rotation is in some arbitrary orientation relative to the figure axis, and the chair is at rest. Then the person moves the rotational axis of the wheel so that it is perpendicular to the figure axis (Fig. 4.16). This tipping of the rotating wheel gives the person an angular momentum, causing a rotation (of person and chair) around the figure axis. This rotation gradually comes to a stop due to friction in the bearings of the chair and with the air. Now we imagine that the person on a swivel chair is replaced by an iron rod, and the wheel by the disordered orbiting elementary charges. The iron rod is hanging as shown in Fig. 4.17 along the M 1.11 applied C4.13 s (discharging a condenser 3 ), the resonance enhancement 11.7 experiment illustrated in Fig. he unavoidable inhomogeneities of acted as a driving force for the weakly- / t . ). At its resonance frequency (see Vol. 1, H is a mirror M 4.17 Schematic of the Conservation of angular momentum . 14.9 through the coil). This methodof makes make the use molecular of currents the thatto small zero, remain fraction and parallel thus after give thethe rise current current to goes were the remanent keptthe magnetism constant, of magnetic t the field iron. iniron If the rod coil would would benetic disturb gradually field pulled the is into measurement. strongest, the(solenoid as The action). region where in the the mag- For their fundamental experiment, Einstein and de Haas Sect. 11.10 and this volume, Sect. (Vol. 1, Eq. (11.7))pendulum makes the large oscillation and amplitude readily of detectable the in torsional the laboratory or lecture a sinusoidally-varying current to thenusoidal magnetic coil, field and the correspondingdamped si- torsional pendulum formedsion by the fiber iron (cf. rod Fig. and its suspen- experiment to demonstrate the molecular currents in iron.mogeneity The of ho- the magnetic field in the coil is not sufficientthe to rotation observe using continuously flowing currents; therefore, the field-reversal or the pulsed-field method must be used. ( for the optical pointer) Figure 4.16 Figure 4.17 long axis of arotational coil. planes of When all the the orbiting currentto charges in are the oriented the axis perpendicular coil ofing is the angular switched rod momentum on, and and rotates the the (justchair). coil. like the Details The person of in iron the theSect. rod swivel experimental takes results on and the analysis result- are given in a current pulse lasting only around 10 For the practical implementation of the experiment, we could use DE ). In - t/taj/ http://www. INSTEIN effect (the “magneto- The Magnetic Field AAS 4 H be readily demonstrated in the lecture room. mechanical parallelism”), this method is also employed. In this way, the experiment can 411c/EinsteindeHaas.pdf modern demonstration experiments of the E 128 (1979), p. 545 (inglish; En- online at physics.umd.edu/gr current method was used. For the history and description of these experiments, see V.Ya. Frenkel, Usp. Fiz.Nauk C4.13. In the original Einstein-de Haas experiment, the resonant alternating- 98

Part I 4.4 The Magnetic Fields of Permanent Magnets are also Produced by the Motion of Electric Charges 99 room. From this amplitude (corrected for the resonance enhancement, see Eq. (3) in the paper by Frenkel, Comment C4.13.), the angular momentum can be obtained, and comparison to the maxi- Part I mum magnetization M0 of the iron rod (measured separately) yields the “gyromagnetic ratio”, i.e. the ratio of the magnetic moment to the angular momentum of the “molecular currents”. See also Sect. 14.9. After this experimental detection of the angular momentum of the molecular currents, we can say today with certainty that the magnetic ﬁelds of permanent magnets are also produced by the motions of electric charges.

In earlier times, magnetic substances were held to be the origin of the magnetic fields of permanent magnets. Similarly to the electric field lines, it was presumed that the magnetic field lines would begin on one body and end on another. At their ends, there would be magnetic “charges” or poles of opposite signs (called “magnetic monopoles”). All efforts to separate the magnetic charges have been futile to this day. Even the relatively primitive model of molecular currents makes this failure understandable. In this model, a permanent magnet is in the end the same as a bundle of current- carrying thin coils, and there, we know that there are only closed field lines in the form of loops without beginning or end. A refinement of this model will be discussed in Chap. 14. nFig. in ecle the called be nuto Phenomena Induction iue5.1 Figure ( coil current-carrying origin, short arbitrary the of from field field magnetic the inhomogeneous e.g. an with start We Phenomena Induction 5.2 the from follows fields electric and magnetic between elec- an relation p only motion This produce in rest charges at but charges field, electric tric rest, at observer an For Remark Preliminary 5.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © ments id ffilsi eeldby revealed is fields of kinds h tegho h edb ayn h urn hog h edcoil field the through current the varying change by and (rheostat field field magnetic the the of in strength rest at the coil induction the groups: leave We three 1. into out organized carry be we can setup, which this experiments With of series time. response a short a with voltmeter a eaiet n nte ysiigo oaigoeo them. of one rotating i or the sliding by of another position one to the relative change We 2. ie o l rcia upss–i i) Chapters vacuum air). a in – in purposes induction practical all to for relating – facts (i.e. experimental the presents elwt hi vlainadexplanation. and evaluation their with deal R OWLAND 5.1 M F (M. R eodcoil second A . nuto experi- Induction n switch). and xeiet tl lsrcneto ewe h two the between connection closer still A experiment. nuto coil induction ARADAY olsItouto oPhysics to Introduction Pohl’s J rmnwo.Ised r once to connected are ends Its on. now from 1832) , spae nti antcfil;i will it field; magnetic this in placed is nuto phenomena induction oueamgei edi addition. in field magnetic a roduce dcinci n h edcoil field the and coil nduction FC R , https://doi.org/10.1007/978-3-319-50269-4_5 h edci FC coil field the 6 hschapter This . and voltmeter 7 To the To ilthen will J ) 5 101

Part I J In the following b: lower voltages between the ends 5.2 the action. Its mag- ; it is measured e.g. in ). During short, precisely . 2 V 5.3 J during Time (s) Time V as drawn, we use a ring-shaped galvanometer in a different way –4 G induced voltage

a b J

(cf. Fig. 10 Voltage (V) Voltage 0.1 Ω 2000 Ω voltage impulse n of induction phenomena, we mea- 1.10 . 2.11 , where we measured current impulses in order 2 J voltage connected G , measured in t A rotating-coil Two ) is calibrated d of the same mag- , and is also called the U a: high voltages during a short time period. When the motion 5.5 / t R d 5.2 Its calibration in Vwhich s was is described in carried Sect. out analogously to the calibration in A s measured time intervals, we apply known voltages to the galvanometer. U R volt second (V s).the This current, quantity measured is for example analogous intreated to ampere in the second detail (A time s), in integral which Sect. of we sure the voltage impulse from the inductionwith coil using a a slow galvanometer response time (ballistic galvanometer). For a quantitative investigatio sections, we thus usefrom that the in ballistic Chap. to determine the quantity of electric charge coil made of ﬂexibleinduction coil insulated in wire. thearea magnetic by We ﬁeld, moving deform some i.e. parts this we of ring-shaped its vary windings itsIn relative to cross-sectional all other three parts. cases, we observe an of the windings of the induction coil nitude depends onone the coil speed is of rotatedFig. the rapidly, we process. observe a For voltage example, curve as when shown in is slower, we see a curve like that shown in Fig. over a longer time. The area of the. shaded regions is the time integral of the voltage volt second (V s) impulses nitude Figure 5.2 to an induction coil (cf. Fig. in volt second for theurement meas- of voltage impulses galvanometer 3. Instead of the induction coil Figure 5.3 Induction Phenomena 5 102 Part I 5.2 Induction Phenomena (M. FARADAY, 1832) 103

SS' u Part I

FC J

To the galvanometer

Figure 5.4 In the frame of reference S, the field coil FC is at rest and the induction coil J is moving at the velocity u. In S’, the situation is reversed. The galvanometer is in each case in the frame of reference in which the observer is at rest. C5.1 This principle, formulated only for uniform, linear motions, along with the fact A known voltage of suitable magnitude is obtained from a voltage divider that light always propagates circuit as shown in Fig. 1.27. with the same velocity in We observe the galvanometer deflection ˛ for various values of the prod- vacuum, independently of D ˛ uct Ut, then take ratios BU (Voltage impulse Ut/Impulse deflection ), the state of motion of the and obtain the same value in all cases, e.g. light source which emits Vs it, together form the basis B D 2:4 105 : U scale division for EINSTEIN’s theory of special relativity ( Annalen This is the ballistic calibration factor of the galvanometer. der Physik, Vol. 17, (1905), p. 891). A good account of We make use of the calibrated galvanometer and repeat the three ex- the historical development periments described above. This leads us to an important discovery: of this theory is given by In the experiments of the second group, only the relative motion be- F. Hund, “Geschichte der tween the induction coil and the field coil plays a role.Inorder physikalischen Begriffe”, to emphasize the significance of the statement printed in italics, we BI-Hochschul-taschen- 1 C5.1 mention that it led EINSTEIN to the “principle of relativity” . bücher, Vols. 543 and 544, Mannheim, 2nd edi- 1 The relativity principle mentioned here can be expressed in terms of the follow- tion, 1978. English: See ing two equivalent statements: 1. “It is impossible to determine by experiment e.g. M. BORN,“Einstein’s whether one is at rest or in a state of uniform motion”. 2. “When two experiments Theory of Relativity” (Dover are carried out under the same conditions in two frames of reference which are Publications, 1965) (avail- moving relative to each other at a constant velocity, both experiments will lead able online for download at to the same conclusions”. In order to make the relativity principle clear in terms of the setup for an experiment of group 2 as shown in Fig. 5.1, the one frame of https://archive.org/details/ reference S is chosen so that the field coil FC is at rest in it. The induction coil J einsteinstheoryo00born ); is then moving in that frame at the velocity u, e.g. to the right. In order to measure or https://en.wikipedia.org/ the induced voltage impulse, the galvanometer is located in frame S (Fig. 5.4). In wiki/History_of_special_ the other frame of reference S’, the induction coil J is at rest and the field coil is relativity . moving with an equal but opposite velocity, e.g. to the left; then the measurement is carried out with a galvanometer which is at rest in S’. In both frames of reference, the observers measure a voltage impulse when the coils are well separated from each other. Their descriptions are different: The observer in S says that the induction coil is moving through the magnetic field; the other observer, in S’, says that the magnetic field within the fixed induction coil is changing. The relativity principle now postulates, in agreement with experiment, that both observers measure an identical voltage impulse in spite of their different descriptions of the process. , ) 2 = . J 5.5 7 4.1 ( A 2when p = J A when the two , measured in J .Furthermore, A A 5.6 D surrounds the field A J is supposed to be en- , and is equal to ı J is equal to A A ). In all cases, the induction : 5.6 l parately induction that is due to NI , right). Then D , left), while the second is completely 5.5 H ,cf.Fig. ; see the footnote). There the induction 5.5 0 , the cross-sectional area of the field coil, a S FC A , and ‘Induction in conductors in motion’ in . The first induction coil J N 5.3 is equal to A , right). A third induction coil has the form of a flat rect- . This separation is quite important for our understanding of on its outside (Fig. 5.5 5.5 FC Furthermore, we employ induction coilsnumbers of of differing geometries turns and Sect. the phenomena. Aexperiments summary from will the follow second in groupframes Sect. of will reference, be in treated terms in of detail the in theory of both relativity, in Chap. coils are oriented with their axes parallel. tirely inside the field coil (Fig. the angle between the axes of the coils is 45 coil is at rest; onlyin the the magnetic experiments field of which group 1. penetrates it changes,Quantitatively, as we will describea se change ina the change in current the through distancetions between the of the the two field field coils coil or coil, andat the the relative induction or rest’ orienta- coil: alternatively ‘Induction in in to conductors Sect. when half the areathe of field the coil induction through coil a slit projects in into its theIf side. interior the of induction coil surrounds the field coil on its outside (Fig. left), then inside the homogeneous magnetic field in(Fig. the interior of the field coil angle and consists ofthird several coil hundred turns can of eithertween insulated be two wire. pushed of into This its thethe windings, field field so coil coil, that from or partangles the else of relative side to its it be- the cross-section canbe field is be rotated lines, inside around put and the completely fixed axis in inside, a with tilted varying position (it can coil the plane perpendicular to the field lines.The Examples: induction coil of cross-sectional area independently of its orientation, etc. In order to treathomogeneous magnetic induction field quantitatively, within we afield strength first long is field make given by coil use (solenoid). of Its the 5.3 Induction in Conductors at Rest As a result, wethose can of always group 1. discuss We these needto experiments only consider together to the change with experiments our of frameinduction group of coil 2 reference and from (that the is, point in of view of the coil encloses a magnetic field of cross-sectional area Induction Phenomena 5 104 Part I 5.3 Induction in Conductors at Rest 105

J J Part I FC FC

GGUIA UA I

Figure 5.5 Setups for the experimental derivation of the law of induction (Video 5.1) Video 5.1: “Induction in Conductors at Rest” Figure 5.6 A section through http://tiny.cc/ebggoy a rectangular induction coil which For the experimental setup, is located within a homogeneous a see Fig. 5.8. magnetic ﬁeld and rotated relative to H the ﬁeld by a certain angle

Figure 5.7 An unwanted reduction of the induced voltage impulse due to G “backwards-running” (returning) ﬁeld lines from a short ﬁeld coil

In the latter case, one has to take care to avoid the error source illustrated in Fig. 5.7: A reduction of the induction signal due to “backwards-running” ﬁeld lines (compare Fig. 4.3). The diameter of the induction coil must not exceed the diameter of the ﬁeld coil by too great a margin.

Now to the experiments: The current I in the field coil is alternately switched on and off, so that its magnetic fieldR builds up or dies out. In each case, we observe a voltage impulse Udt between the ends of the induction-coil windings. We find that it is proportional 5.5 C5.2 per- (5.1) (5.2) (5.3) to the N magnetic magnetic or the . .Todothis, U ). ;and J ). When the current and the 5.6 B 5.1 : to each other; therefore, ˚ : Vs : Am law of induction l P 6 H NI constant of proportionality; opposite 0 A AH J , in a homogeneous field, at least (see also Sect. 10 t 0 N d 0 D magnetic field constant 6.1 U of the bundle of magnetic field lines t D in air is practically the same as in . Its ‘official’ name is the R 257 d A A : D J of the induction coil 0 ˚ 1 U . Using an induction coil as in Fig. t J N U d R N D are the = 0 0 H d , that is the ratio of the number of turns D l ; = I P H N ) is one formulation of the be a constant factor, i.e. a of the voltage impulses has been left out of our considerations so 0 5.1 of the Magnetic Flux Magnetic Flux Density C5.3 . of the field coil; sign l C5.4 there is a minus sign on the right-hand side of Eq. ( far, since it is notcoil of is switched decisive on, importance. theand When electric in current fields those of in through the the the field windings field coil of are the directed induction coil in the field coil iswill switched return off, to the this two question fields in are Sect. in the same direction. We The it is only necessaryrate to of ensure change that the magnetic field has a constant The value of the factor with several hundred turns in itssmall winding, steps we and employ move a its rheostatwill sliding with obtain contact a at constant induced a voltage: constant speed; then we for a limited time, one can maintain a constant voltage vacuum: constant Equation ( meability constant of vacuum Common names for 4. to the cross-sectional area 2. to the ratio flux 5.4 The Definition and Measurement For applications of the law of induction, we define the Instead of the voltage impulse then we have 1. to the current which pass through the induction coil. All of these experimental results cantion. be Let combined into a single equa- 3. to the number of turns length (see 5.5 Vs/Am 7 )), we ). ,where magni- 2.3 ::: 10 ampere . 5.1 A d and 4 (Eq. ( 56637 : 2 change of the sign D B 12 0 Z in vector notation is . This applies also to Induction Phenomena has today been fixed by D ˚ 5 0 between switching on and switching off the field or reversing the direction of motion can be seen. only the ˚ obtained in terms of aintegral: surface ignore the sign here forplicity. sim- This topic will be discussed in the following chapter (see also the paragraph in fine print below). The equations up to andcluding in- those of Sect. should thus be understood as involving only tudes Videos 3.2 (4 C5.4. The general definition of C5.3. In connection with the definition of the unit of electric current, the C5.2. As we already did in Chap. Comment C1.6), the value of law: 106 Part I 5.4 The Definition and Measurement of the Magnetic Flux ˚ and the Magnetic Flux Density B 107 and the magnetic flux densityC5.5 C5.5. The introduction of the vector field B, which differs D : B 0H (5.4) from the magnetic field H Part I only by a factor of propor- The unit of ˚ is seen to be 1 volt second (V s), and the unit of B is tionality , might seem thus 1 V s/m2 D 1 tesla (T). (Sometimes, the older cgs unit Gauss 0 superfluous here. Indeed, the (G) (1 Gauss DO 104 T) is also still used.) description of magnetic fields Using these two new quantities, Eqns. (5.1)and(5.2)takeonthe in vacuum requires only one forms of the two field quantities; Z in general, B is preferred. In Udt D NJ ˚ ; (5.5) some textbooks, the field H Z is in fact not even mentioned, and B is defined without fur- Udt D NJA B ; (5.6) ther ado as the magnetic field. P U D NJ ˚: (5.7) However, when matter, in particular magnetic materi-

For NJ D 1, i.e. an induction loop instead of an induction coil,Eq.(5.5) als are present, the simple becomes relation B D H no longer Z 0 suffices. Then, both of the D ˚ ; Udt field quantities are generally required (see Chap. 14). Voltage impulse D change in the magnetic flux; this is formally analogous to the important mechanical equation (Vol. 1, Sect. 5.5): Z Fdt D p ;

(Mechanical) impulse D change in the momentum:

Making use of Eq. (5.5), we can measure the magnetic flux of a field coil in a very simple way. As in Fig. 5.5, we surround the field coil by an induction coil, switch the current inR the field coil on or off, and measureR the resulting voltage impulse Udt.Thenwehave˚ D Udt=NJ, the magnetic flux of the field coil, as sought.

Figure 5.8 shows an example. There, NJ D 1, i.e. instead of an induction coil with NJ turns in its winding, a simple induction loop is used. From the magnetic flux ˚ of the field coil, we obtain its magnetic flux density B (also called the magnetic induction field) by dividing ˚ by the cross-sectional area of the field coil. The law of induction makes it possible to readily measure the magnetic field H or the magnetic flux density B D 0H in inhomogeneous fields as well: We need only make use of an induction coil or loop of sufficiently small area.

As an example, we measure the magnetic flux density between the flat poles of the electromagnet in Fig. 8.2. We set up a small induction coil J (Fig. 5.9), often called the probe coil, perpendicular to the field lines in the region of the field that we want to measure, and connect the ends of the coil to a calibrated ballistic galvanometer. We then observe the voltage impulse when the field is switched on or off, and divide it by the number of ‘area turns’ NJA of the probe coil. We thus find for the electromagnet of 2 6 Fig. 8.2: B D 1:5Vs/m or 1.5 tesla (that is, H D B=0 D 1:2 10 A/m). ) B (5.8) slider (or H : occurred as equally- ), the cross-sectional H about as large as a hu- xL , we are looking paral- 5.1 . Again, we make use of J 5000 A/m). In the circular ‘area G B J 5.10 2 m D 4 H A 10 H is determined experimentally by the 0 3 : We keep the field strength ), we measure the horizontal component D . At the same time, we observe a volt- 5.2 C5.6 1) D A t 5.8 xL J D d N J U N cross-sectional area A using the handle at right. This changes the area ( , we made use of fixed coils and varied the mag- D x J Z A 5.4 of a long coil us- and ˚ which can slide along them; it can be moved along an Probe coil for measuring the magnetic flux Measurement of the L ) 5.3 of the field region and the field strength ; this corresponds to A 2 Video 5.1 age impulse. Its magnitude side and are connectedsecond. to At a the ballistic right, galvanometer, the calibratedof two in horizontal length wires volt are bridgedarbitrary by distance a of the loop by field of view, we see at theangles. left two Their metal wires ends which emerge are from bent at the right field coil through slits in its left area important factors. Now, we will describegroup an 3 experiment belonging as to defined in Sect. constant, but vary the a homogeneous magnetic field. In Fig. lel to the field lines into a long coil ( one generally employs a flatman induction hand, coil with several hundred turns in its windings. In this case, As an application of Eq.of ( the earth’s magnetic field in Göttingen. For such measurements, ing an induction loop netic field strength only. In the resulting Eq. ( galvanometer to be: turns’) density of an electromagnet (one3cm turn with an area of Figure 5.8 magnetic flux 5.5 Induction in Moving Conductors In Sects. Figure 5.9 ( D and . 8m, sign : U 0 8.6 B 5.10 D l 8cm, : . 5 2400, V s/scale division. D 5 turns is used. The D J 8 A; induction coil: 40; and the ballis- : N N 10 0 Induction Phenomena D 2 5 J D : 3 with http://tiny.cc/ebggoy In this video, instead of an induction loop, a coil Video 5.1: “Induction in Conductors at Rest” large as if the inductionwere coil inside the homogeneous field. See also Sect. Exercise 5.1 induction coil at the endthe of field coil, the resulting voltage impulse is half as Note that when the measurement is carried out with the data of the setup are:coil: Field unambiguously from the experiment in Fig. C5.6. Here, again, the is not taken into account.could It however be determined the galvanometer is tic calibration factor of I N diameter 108 Part I 5.5 Induction in Moving Conductors 109

G Video 5.2: ʃ x C U dl “Induction in Moving Con- Part I L ductors” http://tiny.cc/hbggoy A Field coil: N=l D 1000=m, I D 6A,L D 6:5cm, x D 8 cm. As an intro- Current to field coil I duction, the experiment is first shown when turning Figure 5.10 Induction in moving conductors; the induction loop contains on the current in the field a slider C A of length L on one side. The magnetic field is perpendicular to coil. We then observe the the plane of the page, pointing towards the observer. It is produced by a field same voltage impulse as in coil with 10 turns per centimeter of length, i.e. N=l D 1000=m. (The measur- the experiment in Video 5.1 ing instrument (galvanometer) is outside the magnetic field.) (Video 5.2) when the current in the field coil was switched off while the coils remained fixed. it is not called a probe coil, but rather an earth inductor.Topro- (Exercise 5.2) duce the voltage impulse, the plane of the coil is oriented vertically (perpendicular to the component to be measured) and the coil is then C5.7 4 2 rotated by an angle of 180°. We obtain Bhor D 0:2 10 Vs/m C5.7. Here, the area of the (corresponding to a field strength of Hhor D 16 A/m). coil is not changed, but due With induction in fixed coils (Sect. 5.3), we were able to maintain to the change in its angle, the a constant voltage U instead of a voltage impulse, at least for a short projection of its area onto the time. We had only to give the magnetic field a constant rate of field direction is varied; or, P D = expressed differently: The change H dH dt. We can carry out a similar experiment using induction with moving conductors as described above; we can move scalar product B A changes the slider along the x direction at a constant velocity u D dx=dt.Then (in this experiment by 2BA), from Eq. (5.8), we obtain a constant voltage between the ends of the and the voltage impulse is induction loop: proportional to its change (Exercise 5.3). U D BuL : (5.9)

This is a formulation of the law of induction for a moving conductor which stays within a region of constant ﬂux density during its motion. Note that in this and the next sections, we use a fast-responding galvanometer calibrated in volts, not a ballistic galvanometer, to determine the induced voltage U.

One must avoid bringing the voltmeter or galvanometer, including its lead wires, into the magnetic field and moving it with the slider; it then might not indicate the correct voltage U. More details on this topic will be given in Sect. 7.3. C5.8. If we use instead In Eq. (5.9), the essential quantity is the velocity u with which the a metal disk which can rotate slider is moving perpendicular to the direction of the field lines and around an axle perpendicular relative to the source of the magnetic field (the field coil). We can to the plane of the page, we demonstrate this in a very dramatic way: We replace the slider wire would have constructed a so- (Fig. 5.10) by a broad metal ribbon (Fig. 5.11). It consists of an called “BARLOW’s wheel”. endless band which is moving through the field coil. It permits us to PETER BARLOW (1776– maintain a constant velocity of motion u for an arbitrarily long time 1862), “unipolar inductor”, and to observe the resulting constant induced voltage.C5.8 1823. u mag- unipolar C A , and the surface R L is perpendicular (like V U u which can be rotated using C5.9 R C

) to the direction of the field L 5.11 is induced between the two ends of R U NN homogeneous and time-independent C A ) ). The usual name for this arrangement is , we use an endless metal L ). The voltage ab VV b. Keep in mind that the field lines are not fixed on the surface Induction of a constant voltage in moving conductors within slider Induction in moving conduc- 4.7 Exercise 5.4 5.12 is sufficient. We surround the pole region of a bar magnet . of the Law of Induction .) ( 5.12 5.10 We could also connectaround the its tube long rigidly axis. to Thensufficient the the to tube bar let would the magnet become sliding superfluous; and contactssee it Fig. slip let would over it be the rotate surface of the bar itself – of the bar magnet like the bristles of a brush! of the magnet itself inthe field part lines. b). They follow circular paths, perpendicular to a radially-symmetric magnetic field (produced byThe the conductors bar (“rotors”) magnets which (shaded)). arenetic moving relative field to consist the in source part of a) the of mag- a short piece of metal tubing Figure 5.11 tors; here, as ribbon which is joined toside form the a field loop coil out- likesaw the (the blade magnetic field of is a theFig. band same as in 5.6 The Most General Form A general form ofquish our the requirement of law a of induction can be found if we relin- For qualitative experiments,Fig. the simple arrangement sketched in Figure 5.12 the tube (length loosely by a short piece of metal tubing that of the endless band in Fig. lines (Fig. a crank. The magneticthe field walls lines pass of nearly this perpendicular rotor, through and its path velocity netic field and also take the sign found in the experiments into acinduction , 411 (1962). American Journal 30 Induction Phenomena 5 of Physics C5.9. For a detailed description of these experiments, see J. W. Then, 110 Part I Exercises 111 count2. Then one can define experimentally the voltage induced in a loop:C5.10 C5.10. Equation (5.10) holds

Z Part I d for all induction experiments U D B dA : (5.10) in the three groups described dt in Sect. 5.2. The integral on the right-hand side can be decomposed into two parts: A time-dependent change of the magnetic flux density B, and a spatial change of the curve around the perimeter s of the integrated area A (the induction loop) at whose ends the voltage is measured. We find:C5.11 C5.11. For the derivation of I Z I dB Eq. (5.11), see for example U D E ds D dA C .u B/ ds : (5.11) P. Lorrain and D. R. Cor- dt s.A/ A s.A/ son, “Electromagnetic Fields and Waves”, 2nd edition, (Here, u is the velocity with which a line element ds of the perimeter W. H. Freeman & Co., San curve is moving relative to the frame of reference in which U and B Francisco 1970, Chap. 8. are measured). The direction of the surface element in the first term and the direction of the integration along the path integrals when combined have to yield a right-hand screw. With this formulation, the law of induction has the correct sign (see Sect. 6.1).C5.12 C5.12. The velocity u is by The attribution of the induced voltage to the two parts of the integral no means necessarily con- can change when the frame of reference is changed. Some examples stant along the perimeter of this: In experiments from group 2 in Sect. 5.2, only the first term curve (examples: experi- on the right is present when the position of the induction coil is used ments of group 3, or also as the frame of reference. However, if the position of the field coil Fig. 5.10). defines the frame of reference, both terms have to be taken into account (of course, then dB=dt has a different value). Or if in Fig. 5.6, we let the induction coil rotate and use the field coil as the frame of reference, then only the second term on the right of Eq. (5.11) contributes. But within the frame of reference of the induction coil, only the first term contributes (Exercise 5.6). In contrast, in Fig. 5.11, only the second term is present, since there, dB=dt D 0(Eq.5.9).C5.13 C5.13. The description of this Equation (5.10), while often used, is sometimes not very suitable for experiment in the frame of practical calculations, e.g. when the area over which the integration is reference of the metal band carried out is not uniquely determined, as in Fig. 5.11. In such cases, is most readily formulated one should return to Eq. (5.1)ortoEq.(5.8). using the theory of relativity (see Sect. 6.1).

Exercises

5.1 Using the informationR given in Video 5.1, calculate the expected voltage impulse U dt and compare it with what was observed in the video. (Sect. 5.4)

2 Compare the small-print paragraph near the end of Sect. 5.3 for remarks about the sign. 5.5 https:// which will at which the U

), if it is rotated 5.6 between the direction ' (Fig. ) is 10 cm wide and takes must be moved in order 5.5 5.3 turns in its windings, and its 5.11 5.10 . Compare the latter with what N ) 5.2 V s. a) Determine from this result . (Sect. 5.5 3 ) contains supplementary material, which The online version of this chapter ( C ), compute the voltage ,ithas of the earth’s magnetic ﬁeld. b) In a sec- A and 5.10 1 mV from the induction loop (measured h B A which would be observed with a slowly- D . (Sect. 5.6) t

U d U R V s is observed. Find the angle 2 and 200 turns in its windings is oriented with the aid 2 ) with which the slider in Fig. 10 Making use of Eq. ( The endless metal band in Fig. With the information given in Video 5.2, compute a) the ve- The measurement of the earth’s magnetic field in the Göt- An aircraft with a wingspan of 50 m is flying at a constant ) cm u 5.5 3 T. What voltage will be indicated by the voltmeter? (Sects. 25 7.3 : 5 be induced in the rotating coil from Sect. around an axis perpendicularsectional to area the of plane the coil of is the page. The cross- 5.6 ond experiment, the axis ofWhen it the is earth rotated inductor around is aof diameter 2 oriented by vertically. 180°, now a voltage impulse of the earth’s field and the horizontal(Sect. (called the angle of inclination). 5.4 rotational frequency is and of a compass needlesouth direction. so that Then the itsIf axis axis the is points earth moved inductor in toobserve is the a a now voltage horizontal magnetic impulse rotated orientation. north- of around 10 a diameter by 180°, we the horizontal component Electronic supplementary material doi.org/10.1007/978-3-319-50269-4_5 is available to authorized users. locity 5.2 to obtain a voltage of responding galvanometer, cf. Sect. was observed in the video. (Sect. 5.3 tingen lecture hall:of An 10 earth inductor with a cross-sectional area with a voltmeter having avoltage rapid impulse response time); and b) the expected band would have tobetween the rotate slip in contacts order to generate a voltage of 1 mV 5.5 altitude with a velocity ofis 960 km/h. attached to The pilot the has wingtipsfuselage a of with voltmeter the insulated which aircraft wire are leads.together. electrically conducting The The and wings vertical are and component connected of10 the earth’s magnetic field is 6 the form of a circlehomogeneous with field a at diameter the of centerturns, 1 of m. carrying a The a 50 cm magnetic current long field of field is 1.2 A. coil the Find with 1000 the frequency Induction Phenomena 5 112 Part I field Fields Magnetic and Electric Between Relation The iue6.1 Figure Sect. of 1) group (in experiment first the to return We Induction. of Treatment Detailed 6.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © nyoetr ( turn one only ao odtc h lcrcfield electric the detect to investigation: cator our continue us Let their at to ends; charges. with electric Up lines were field ends unexpected. electric only completely encountered and have new we now, quite something are circles ie nuto ntesmls osbecs:A nuto olwith coil induction An case: possible only simplest the in induction sider s eosreteidcdvlae(ihu aigissg noaccount): into sign its taking (without voltage induced the observe we dependent feeti edlnsaogcoe ah rudtetime-dependent the around paths closed field along magnetic lines appearance the field on of electric consists depend It of not loop. wire does the phenomenon of presence actual accidental the The insignificant. and tant sense profound more a follows: in as interpreted be can result experimental This ihacosscinlarea cross-sectional a with B one h ntuetrpeet h nue voltage. induced the represents instrument The . M un ..a nuto op sspoe oecoea enclose to supposed is loop, induction an i.e. turn, antcfil fflxdensity flux of field magnetic ceai fa nuto xeietwt nidcinci of coil induction an with experiment induction an of Schematic h odco,tesnl opo ie sqieunimpor- quite is wire, of loop single the conductor, The AXWELL N J (Fig. D J ) h etrfield vector The 1). 6.2 ). sScn Law Second ’s olsItouto oPhysics to Introduction Pohl’s A C6.1 A Ḃ lcrcfil ie ntefr fclosed of form the in lines field Electric U (Fig. D ln t egh tmaue the measures it length, its Along . BA P 6.1 h ielo smrl nindi- an merely is loop wire The B P stetm eiaieo h vector the of derivative time the is .Te tteed fteloop, the of ends the at Then ). : B ln oeabtaycurve arbitrary some along , https://doi.org/10.1007/978-3-319-50269-4_6 V 5.2 n recon- and , time- ( 5.2 ) givenbythe is paths closed on lines field such of demonstration impressive very A C6.1. bu h L the about know to needs one properly, it understand to However, textbooks. physics ductory intro- many in described is it and accelerated, be can electrons which in device a edu/ feynmanlectures.caltech. at (online 17 Chap. II, Vol. 1964, Physics on e.g. (See chapter. following the in introduce we which h ena Lectures Feynman The ;c.as Fig. also cf. ); http://www. Addison-Wesley , ORENTZ betatron 6 11.11 force, ;thisis ). 113

Part I E U )). (6.1) (6.2) α 5.2 .Ithasthe E in the schematic Ḃ ˛ UU U : A ) takes the following P HA , i.e. the induced voltage 5.2 0 E : s , so that the magnetic field is d J D N E followed, just equal to the voltage P BA field that is produced by the change s I . D D s U d electric )): ’s laws along a closed loop was always zero. It was, E 2.3 E I field. It summarizes the essential content of the second AXWELL . It has no other function than the wire s The function of the wire loop in The deeper signif- : The wire is a conductor and causes the electric field to times, the voltage increases correspondingly (Eq. ( d J 6.3 E N C6.2 R magnetic D form: Then, taking the sign into account, Eq. ( value (compare Eq. ( U in Fig. decay in its interior.age The acting charges all move along to its its length ends is and compressed thus into the the volt- remaining gap. In the electric fieldsintegral of which the we field have thus far investigated, the path between the beginning and the endtherefore of zero the whenever path the (see beginningother Eq. closely; and (2.3)). at the It the was end limit approached of a each vanishingThe gap, it situation vanished exactly. is differentinduction with experiment, which the form fieldsvoltage endless that along loops; we a here, encounter closedthe the in path number electric of this has turns a is finite increased value. to Furthermore, when independently of the exact path In this way ofinduced during an looking induction process at is the theobserved primary voltage phenomenon. experiment, is The the the path integral electric of that field electric that field is circled of the four M of a This equation yields the Figure 6.2 the induction experiment icance of the phenomenon of induction Figure 6.3 path integral of the electric field strength . in A d sign A B is positive A Z s ). The ;itisgiven t d A d 8.3 D ’s law. s , the path of integra- ) becomes d encloses the area 5.6 6.2 ENZ s E The Relation Between Electric and Magnetic Fields / 6 A I . such a way that d As already agreed upon in Sect. tion C6.2. Written completely and consistently in vector form, Eq. ( (it curves in a “right-hand” sense when one looks along the direction of the surface- element vector will be treated in detail once more in Sect. by L s 114 Part I 6.2 Measuring Magnetic Potential Differences 115

y @E E + x dy x @y @E Part I dy E E + y dx y y @x

dxx

Figure 6.4 Taking the path integral of the electric ﬁeld E along the perimeter of a surface element dx dy;thez axis is perpendicular to the plane of the page and points towards the reader (right-handed coordinate system). The path of integration (curved arrow) is in the clockwise sense (to the right) as seen along the z direction (similar to Fig. 6.15).

The equation can be written in a differential form and can then be applied to arbitrary inhomogeneous magnetic ﬁelds. Its derivation from Eq. (6.2) can be accomplished by taking the path integral along the perimeter of a surface element dx dy. This computation is explained in Fig. 6.4.One thus obtains (taking the sign of BP in Fig. 6.2 into account!):

@E @E y x DBP @x @y z

or, after taking account of the other components, in vector notation, P P curl E DB D0 H : (6.3)

In words: At each point within a magnetic field, a time-dependent change in the field strength or direction produces an electric field. The latter is a vortex field, and the curl of the field E is equal to the negative rate of change of the field B. (For the definition of the “curl”, see Vol. 1, Sect. 10.7).

The third of MAXWELL’s laws contains another, analogous relation between the two ﬁelds, but with the roles of E and H reversed. Its experimental derivation is our next goal.

6.2 Measuring Magnetic Potential Differences

We know from ROWLAND’s experiment (Sect. 4.3) that every motion of electric charges represents an electric current, and every current produces a magnetic field. We can indeed measure the magnetic field using the current. But we still lack a general experimental framework for the relation between the current and the resulting magnetic field, as described in MAXWELL’s third law. We can arrive at it by measuring the magnetic potential difference. ) s ). -th 2.3 n Phil. 6.6 (6.4) ( turns , : l = n .Its s l Arch. f. Elek- N HATTOCK , , wound for ex- D n N . Let us denote the n 3 s A magnetic potentiome- field, we can define the ∆s TEINHAUS 3 . is the (rectangular) cross- : H into a broken curve with s : A d s d 6.5 ;:::; H 2 and W. S E s magnetic magnetic potential difference Z Z . Then there are ; 2 2 D n 1 is increasing or decreasing, then a cer- H s s D ∆s H as the mag U OGOWSKI U H turns in its winding and the length 1 N H magnetic potentiometer . The magnetic potentiometer extends along the 1 n ∆s H )). This is composed of the sum of the contributions field, we found that the electric potential difference or .Ithas 5.1 s Schematic of a magnetic potentiometer (A. P. C ; :::; 2 , 141 (1912)) , 94 (1887); W. R 1 H (Eq. ( . 24 electric ; t 1 d . H U . This path is divided up in Fig. as components of the field in the direction of the path elements Mag path integral of the field whole path path element has the length sectional area of the windings of the potentiometer (the sign holds in this element. If the field tain voltageR impulse will be induced in the magnetic potentiometer, from each of the path elements; thus, if trotech Its unit can be seen to be 1The ampere. magnetic potential difference canple instrument, be the measured with a very sim- Its unit, as usual, is 1 volt. In a corresponding fashion, in a In an Figure 6.5 voltage is given by the path integral of the electric field: segments (path elements) ample on a long strap.brought It out is at wound the with two center layers of and the the leads outer are layer of windings (Fig. ter is in principle simply a very long induction coil (A coil with only aintended, single also layer represent a would, large, in flatloop addition induction coil of to formed helically-coiled the by wire). long a coil single as We want to elucidateThe the operation magnetic of potential this differences magnetic is potentiometer: to be measured along a path The Relation Between Electric and Magnetic Fields 6 116 Part I 6.3 The Magnetic Potential Difference of a Current. Applications 117 for a decreasing field H), the voltage impulse will be: Z Part I U dt D 0 AH1Ns1=l C 0 AH2Ns2=l C0 AHnNsn=l ; (6.5) Z

U dt D 0 AN.H1s1 C H2s2 C H3s3 CHnsn/=l ; (6.6) Z ÂZ Ã

U dt D 0 AN H ds =l D 0 AN Umag=l ; (6.7)

Z l Umag D U dt : (6.8) 0 NA This induced voltage impulse, measured for example in volt second, after multiplication by the the apparatus constant l=.0NA/, yields directly the magnetic potential difference as sought, in ampere. The apparatus constant need be determined only once; A, l and N by direct 6 measurement, while 0 D 1:257 10 Vs/Am.

We employ a magnetic potentiometer which is 1.2 m long. Its apparatus constant has the value 5 105 A/V s. The coil has 9600 turns, each with a cross-sectional area of 2 cm2. The induced voltage impulse is measured using the ballistic galvanometer described in Sect. 5.2; it can be calibrated as shown in Fig. 5.3 (J is the magnetic potentiometer). (cf. Video 6.1) Video 6.1: “The Magnetic potentiometer” http://tiny.cc/obggoy 6.3 The Magnetic Potential Difference The potentiometer shown of a Current. Applications in the video has practically the same dimensions. Its apparatus constant is The operation of the magnetic potentiometer is explained in Fig. 6.6. 4:58 105 A/V s. The magnetic potential difference Umag of a field coil is to be mea- ! The ballistic calibration sured between the points 1 and 2 along the path 1 2. The po- factor of the galvanome- tentiometer is shaped so that it corresponds to the path from 1 to 2; ter has the value BU D then the magnetic field is varied by switching the current in the field 1:1 104 V s/scale division coil between zero and its maximum value, and the resulting induced (Exercise 6.1). voltage impulse is observed with the galvanometer.C6.3 In this way, we reach the following conclusions: C6.3. We see here that the operation of the magnetic po- 1. Along an open path (Fig. 6.6), the magnetic potential difference tentiometer involves simply depends only on the position of the end points 1 and 2, and not on the an application of the law of shape of the path itself. The path can even include loops, as long as induction (Eq. (5.5)). The they do not enclose any currents. magnetic potentiometer is 2. In Fig. 6.7, the path which the potentiometer follows is closed, and just an induction coil with it encloses no currents. The resulting magnetic potential difference is a particular shapeR with which zero. voltage impulses U dt can be measured. 3. In Fig. 6.8, the path defined by the potentiometer encloses a current I once within a closed loop. The magnetic potential difference I I a II 12 II the magnetic potential difference 6.8 To the To galvanometer To the galvanometer G Galvanometer in the conductor which is enclosed in the path I 50 to D I Enclosing a cur- Aclosedpathfor Operation of (Video 6.1) is again independent of the exact shape of the path (circular, mag Figure 6.6 rent one time with a magnetic potentiometer ( the magnetic potentiometer which encloses no currents the magnetic potentiometer 100 A, a 2-volt storagetery bat- is sufficient to provide the current) U Figure 6.8 Figure 6.7 rectangular etc.). 4. Quantitatively, we find in Fig. equal to the current The Relation Between Electric and Magnetic Fields 6 Video 6.1: “The Magnetic Potentio- meter” http://tiny.cc/obggoy In the video, the potentiometer which encloses a current is moved relative to the current-carrying conductor; but no voltage impulse is observed as a result ofmotion. this 118 Part I 6.3 The Magnetic Potential Difference of a Current. Applications 119

I Part I 1

To the galvanometer

2 To the I galvanometer I

Figure 6.9 A current enclosed twice within the path of a magnetic potentiometer – at left, within a closed loop, and at right within an open curve whose upper and lower ends are one above the other vertically (Video 6.1) of the potentiometer. We have I

Umag D H ds D I : (6.9)

This equation is known as “AMPERE’s law”.

A numerical example: I DR83 A. A deﬂection of the ballistic galvanometer of 12 cm corresponds to Udt D 1:7 104 V s. Multiplication by the apparatus constant of the magnetic potentiometer, i.e. by 5 105 A/V s, 4 5 yields the magnetic potential difference Umag D 1:710 Vs 510 A/V s D 85 A.

5. In Fig. 6.9 (left), the path of the potentiometer encloses the current twice (double loop). The magnetic potential difference is then doubled. Continuing in this manner, we ﬁnd for an N-fold enclosure of the current I that the measured magnetic potential difference becomes Z

Umag D H ds D NI : (6.10)

6. In Fig. 6.9 (left), the current I is enclosed twice within the path of the potentiometer, and its beginning and end points are held together. This is however not necessary; the potentiometer could just as well be in the form of an open helix with N turns enclosing the current N times and with open ends (Fig. 6.9 (right)). i ; a (6.12) (6.11) mag U . ; 4 I a = 0 at the center b). Then we ). The mag- NI . This result is D 6.11 6.11 z 716 : wires carrying the turns in their windings) 0 N N ). (Fig. D (Fig. axis ( i 0 ; . It proves here to have z z 1 : ) H and 6.10 : mag l 4.2 a NI 2 U l 6.10 D (Fig. C : with l 2 r H I r 2 along the 4 = , it is constant to within 10 j I coils H p a or j :::/ is given by D l ) from Sect. 1 l : C NI H 0 4 NI is composed of two additive terms, 4.1 / D ) applies, and at the center of a current-carrying a D jÄ = at a distance r from a current-carrying straight z H z mag 4.1 j / . ELMHOLTZ r Hl U . ) in compact form. 15 C6.4 and length : 0), we find . 1 r . We find that the term due to the region outside the D 6.10 NI l axis for Hl 1 . z 1, 0 D D H D ELMHOLTZ i (‘inside’ and ‘outside’). Within the coil, the magnetic field is , is negligibly small compared to ; is constant. The magnetic potential difference along an arbitrary H o D N ; mag o ; . Therefore, the magnetic potential difference along the whole H mag U I H mag , we see that Eq. ( U mag l U . The magnetic potential difference along one of its circular field U The homogeneous magnetic field of a long coil The magnetic field H r path is wire have current coil, homogeneous, apart from thestrength short pole regions (coil ends), and its field and Thus, 2. To remind us ofapplications can these be important helpful: facts,1. the following three examples of netic potentiometer is threaded throughan the arbitrary coil path. and closed outside Its it path along thus encloses once each This is just the same as Eq. ( been a special case of the general equation ( If the spacing of the two coils (each with radius We mention without derivation that the field strength at the center of a cylindrical and thus, along the Figure 6.10 is equal to their radius, then the field between the coils) is given by coil (solenoid) of radius circular ring ( An often-used arrangementsuch for circular rings, producing called H homogeneous fields consists of two 1 Summary: curve is simply equalenclosed to one the time strengththe within of potential the the difference curve. current becomes if N When the times it the latter is current is enclosed N times, coils expressed by Eq. ( For D B ). The 1800 A/m. 4.4 D ba 0 l in the neighbor- = B T. NI 4 field was computed B 10 T –4 The Relation Between Electric and Magnetic Fields 5 10 : 6 Distance from the center of the coil (cm) –100 100 22 along the axis of twoboth coils, with 20 10 0 The longitudinal component of the C6.4. In order to getpression an of im- the homogeneity of the field of a longfollowing coil, figure the shows the strength of the axial field, i.e. the “field profile” of long coils. The field strength in their interior at the center is Curve a): length/diameter = 100 cm/10 cm; b): 50 cm/10 cm (the same geometry as in Fig. decrease of hood of the ends ofis the practically coil independent of the length of the coil.calculation (The was performed by J. A. Crittenden, Cornell University.) 120 Part I 6.3 The Magnetic Potential Difference of a Current. Applications 121 a b G

G Part I

Figure 6.11 The distribution of the magnetic potential difference in the field of a long coil. The coil has 900 turns in its windings, a length of 0.5 m and a diameter of 0.1 m. A current of 1 A passed through the coil produces Video 6.1: D a magnetic field H 1800 A/m. a): The magnetic potentiometer is passed “The Magnetic Potentiome- through the entire length of the field coil. Switching the current off or on 3 ter” yields a voltage impulse of 1:7 10 Vs,i.e.fromEq.(6.8), Umag D 850 A. The length and position of the ends of the potentiometer coil outside the field http://tiny.cc/obggoy coil are practically unimportant. Thus, the field outside the field coil makes In this experiment, the mag- no significant contribution to the magnetic potential difference. netic potential difference in b): The potentiometer follows an arbritrary path completely outside the field the region outside the coil coil. The voltage impulse which is induced in it is only about 9 105 Vs. is 10 % of that measured Umag thus has a value in the region outside the field coil of only around 45 A, within the coil. The data of and can therefore be neglected relative to the magnetic potentialR difference the field coil are: Windings: of 850 A measured within the field coil. The path integral H ds for the N D 4300, length l D 40 cm, outside region is indeed practically negligible, even for this field coil which diameter D 11 cm, current is not really very long. (Video 6.1) (Exercise 6.2) I D 0:15 A. 5 UB D 3:2 10 V s/scale division. lines (Fig. 1.4) with a radius r can be found from Eq. (6.10), taking the symmetry of the problem into account:C6.5 C6.5. “taking the symmetry into account” means here Umag D 2rH.r/ D I ; that one can find no reason and thus why the tangential compo- 1 I H.r/ D : (6.13) nent of the magnetic field 2 r on a concentric circle (of ra- The field is directed tangentially, and its strength is determined only by r dius r) around the current (for its sign,seeFig.4.10). path (wire) should not be 3. Magnetic potential difference measurements in the magnetic fields of constant. It must therefore be permanent magnets. Our treatment has emphasized the essential similarity of the magnetic fields of current-carrying conductors and those of perma- constant. The fact that the ra- nent magnets. This can be reinforced once more by measurements with dial and axial components are the magnetic potentiometer. In Fig. 6.12, the magnetic potential differ- both zero can however not be ence between the poles of a horseshoe magnet is being determined. For concluded from symmetry these measurements, the magnet is pulled away from the potentiometer alone; it is instead an exper- rapidly. The potential difference is once more found to be completely in- imental finding (the absence dependent of the path which is investigated with the potentiometer (i.e. the of a radial component can be shape of the curve which the potentiometer represents). On a closed loop, it is always zero. The potentiometer can of course not enclose individual clearly seen in Fig. 1.4,for molecular currents; it would have to be threaded through single molecules example). for that! Every hole that might be bored through a permanent magnet however passes not through the individual molecules, but rather between them.

In a few cases with simple geometry in which the strength of the magnetic field along the magnetic potentiometer is constant, we could use flows is de- (6.14) I E ;or,more A d = N S I can be given : The current d j ) was general- D 5.5 j and 6.9 . Experiments have s H d (Eq. E I H ’s law in differential form. in the wires is surrounded D I ; . A condenser is discharged j s and the magnetic potential dif- ’s laws, which will be treated in d P D B MPERE 6.13 . A H H d maintained, on the contrary, that the ’s Third Law H j curl R AXWELL . It then follows, in analogy to the deriva- j D I , that the relation between AXWELL corresponds to the one in Fig. To the galvanometer in an audacious manner. His train of thought can AXWELL is the current per cross-sectional area, 6.1 j 6.8 corresponds to the voltage s . This equation is A b. d corresponds to the field AXWELL 6.11 H 4.10 and M j H 2 The current density tube of magnetic field lines has no ends; it forms a closed torus: the following section. generally, in vector notation, shown that the magnetic potential differencefield is of equal the current to density the enclosed tion given in Sect. where the sign onwith the Fig. right side is to be determined by comparison in the form of a differential expression as: ference It is a part of the third of M be explained by referring to Fig. through an external circuit. During its discharging, a current through its leads, and within the condenser, its electric field by concentric circular magnetic fieldure lines. could contain We field imagine lines thatwe around this all can fig- the say, wires in roughlyrounded the by but circuit. a unmistakably, Then “tube” that ofsides the magnetic of whole field the lines. wire condenser, whereof is This the the tube sur- wires condenser. ends meet at the M both plates (electrodes) caying at the same time. The current Figure 6.12 density 2 The experimental result that 6.4 The Displacement Current the potentiometer to determinecase the of field a strength, single for current-carryingiment wire. example in The is importance the however of much thisasshowninFig. more exper- far-reaching. The measurement setup A magnetic potentiometer in the field of a permanent magnet. The experimental setup corresponds to the one in Fig. (Video 6.1) ized by M The Relation Between Electric and Magnetic Fields 6 Video 6.1: “Magnetic Potentiometer” http://tiny.cc/obggoy 122 Part I 6.4 The Displacement Current and MAXWELL’s Third Law 123

I Part I

H H

Electrons

Figure 6.13 Schematic drawing of the magnetic ﬁeld from a conduction current and a displacement current (I corresponds to the conventional current direction from C to )

Figure 6.14 Schematic draw- E∙ ing of the magnetic field of a displacement current.The vector field EP is the time derivative of the vector field A E (the arrows show the direc- H tion of a displacement current Iv which is pointing upwards).

The varying electric field within the condenser is also surrounded by circular magnetic field lines. Such circular magnetic field lines are however one of the principal characteristics of an electric current. The corresponding current is referred to – somewhat strangely –asthedisplacement current. All of the usual meanings of the word current, ‘flowing’ or ‘streaming’ in analogy to a current of water no longer apply to this ‘current’. The word displacement current here indeed refers only to the fact that an electric field is changing with time (Fig. 6.14). Following the introduction of this new type of current, we can say that: In nature, there are only closed current loops. In a conductor, these are conduction currents, and in an electric field (e.g. of a condenser), they are displacement currents. Electric currents have no spatial ends or beginnings. At the end of the conduction current, the displacement current begins, and vice versa. Like every current, the displacement current can be measured in ampere. On the other hand, it is supposed to be a quantity which is determined by the time derivative of an electric field. The latter field would therefore have to have the unit ampere second.Thisisthecase for the product

Displacement density D

Cross-sectional area A of the ﬁeld D DAD "0EA

2 2 (for example: D in A s=m , A in m , E in V=m, DA in A s, "0 D 8:854 1012 As=Vm).

The displacement density D (Sect. 2.13) is related to the electric field E via D D "0E (Eq. 2.5). We denote the rate of change of D and E ) - D M dis- 6.9 P ), is ( E (6.19) (6.15) (6.16) y 6.3 d (6.18) (6.17) x and t onduction d = D d D : P along the perimeter of : P D E ; like those of a conduc- 0 z H P : " E P : EA 0 0 " P E C 6.13 " I P 0 j EA D " 0 z D " D D P field which is produced by the D D s ’s complete third law: must be added as a second term. P P D d P D D D DA y ) can be obtained as a differential equa- d I x d x D y D C d P H H @ DA j s @ C6.6 H 6.16 D d I ; its derivation is similar to that of Eq. ( D j D magnetic ). However, for technical reasons, in elec- AXWELL curl v y I x H H 6.15 is also called the “displacement current den- H @ @ ; we thus obtain y 6.19 A I field. extended this law by a term containing the d = curl v x I , written as D P electric D AXWELL passes through the surface element, then on the right side of the I . Then we obtain the displacement current ’s law t d equation, its current density In words: At every pointtime within produces an a electric magnetic field, aof field. change this of The magnetic the latter field field iscurrent is with a density. equal vortex to We field, have the and assumed time the here derivative of curl that the the displacement surface element d penetrated by only acurrent displacement current. If in addition a c or, after including the other components in vector notation, tion by referring to Fig. The relation described by Eq. ( a surface element d above. We have to compute the path integral of = E PÈRE Unfortunately, we cannot simply make thethe magnetic displacement field current lines visible from in Fig. tion current by using ironexpedient, direct filings. demonstration That of the would validity bethe of right a the side didactically second of very term Eq. on ( tric fields with long field lines, we cannot produce a displacement The quantity This equation yields the change in an placement current We have thus obtained M again with a dot above the field symbol, i.e. sity”. So much for the measurement of the displacement current. A was originally discovered through experimentscurrent. with M the conduction d in ) A 6.16 is positive s : A d D s E . d A encloses the area A Z s H t The Relation Between Electric and Magnetic Fields d d / 6 A 0 I . " s such a way that d Here, the path of integration C6.6. Written completely in vector notation, Eq. ( becomes (i.e. it curves clockwise) if one is looking in thetion direc- of the surface-normal vector 124 Part I 6.5 MAXWELL’s Fourth Law 125

y @H H + x dy x @y @H Part I dy H H + y dx y y @x

dxx

Figure 6.15 Taking the path integral of the magnetic field H along the perimeter of a surface element dx dy;thez axis points out of the plane of the page (right-hand coordinate system) and the sense of the integration (curved arrow) is clockwise as seen in the z direction (as in Fig. 6.4) current of sufficient strength. But such a demonstration would in any case not prove that the origin of the magnetic field was to be found in the displacement current; one could always claim that the magnetic field observed in Fig. 6.13 was due to the conduction current in the leads to the condenser plates. A real proof of the origin of the magnetic field from the displacement current can be obtained only by referring to circular, closed electric field lines. We will do this in Chap. 12, by the detection of freely propagating electromagnetic waves. Until then, the magnetic field of the displacement current remains only a plausible hypothesis.

6.5 MAXWELL’s Fourth Law

At this point, we want to introduce the last of the four MAXWELL equations. In integral form, it is given by: I B dA D 0 ; (6.20) i.e. the magnetic ﬂux (Eq. (5.3)) through a closed surface, integrated over the whole surface area, vanishes. In differential form, this becomes

div B D 0 :

These equations are analogues of Eqns. (2.9)and(2.8) in Chap. 2, which related electric fields to electric charges. The observation that magnetic field lines have no beginnings and no ends, as shown by the experiments in Chap. 4, means that there is no magnetic analogue of electric charges, i.e. there are no magnetic charges. This explains the zero on the right-hand side of the two equations above. for and (6.22) (6.21) (6.23) (6.24) E ). 0 (its data 2.14 " Maxwell )are: 6.24 D B 6.11 D and ; was measured out- E . One coil is pulled for Eq. ( 6.8 a I turns, their length is ; 4 ; N mag P E of the permanent magnet U 0 s " d 0 H ). R 6.2 ; C ; j P B 6.3 )), and Chap. 0 : 0 % in the long coil of Fig. " 0 H 6.22 D D D D ’s four laws (also known as the E B E B ) (Sect. o ; div div . Their windings have curl curl ). mag for Eq. ( U , the magnetic potential 4.6 6.3 AXWELL 6.1 in the vertical conductor in Fig. I 6.12 when the ends of the potentiometer are touching each (Sect. mag . The four laws in differential form (for ); they describe electric and magnetic fields in vacuum (and ), Sect. U H We imagine a bundle of identical long coils with square cross- In Fig. Referring to Video 6.1: Determine using the description of 6.12 0 ), or, as shown in Fig. 6.11, it forms a curve completely i ; 6.21 mag D U , and all of them are carrying the same current B side a permanent magnet betweenine its that ends a (Video thin 6.1). channel has Now been imag- bored through the magnet along its out of the interior of the bundle. a) In the ‘tunnel’ which isa left magnetic where the potentiometer. coil wasference How removed, large we insert is the magnetic potential dif- l sections as in Fig. 6.2 in Fig. equations thus practically also in air). Note that in vacuum, 6.3 The experimental justificationtion of of these the equations displacementbeen (with given current the in density the excep- in preceding sections the (for third example, equation) in has Sect. Eq. ( outside the bundle ( the magnetic potentiometer in Sect. 6.1 a) The current Exercises We summarize M b) the average magnetic field are given in the figure caption); and c) the magnetic potential difference other outside the bundle? b) Compare the magnetic potential differencessured which would if be mea- the( potentiometer is either completely within the tunnel The Relation Between Electric and Magnetic Fields 6 126 Part I Exercises 127 length. Compare the magnetic potential Umag;i that would be measured along this channel between its ends, with Umag;a. (Sect. 6.3,

14.5) Part I

Electronic supplementary material The online version of this chapter (https:// doi.org/10.1007/978-3-319-50269-4_6) contains supplementary material, which is available to authorized users. atwl edu oadee nesadn fidcini moving in induction of understanding Chap. deeper in described a already as to conductors, us lead will fact h rm fReference of Frame the on Depend Fields the How nR In Evaluation Quantitative A 7.2 de- fields magnetic and electric that the show upon to pend want we chapter, this In Remarks Preliminary 7.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © n 3. and snttetu infiac fteeprmn;ta isi logical Fig. a to in return lies it. We from that directly experiment; drawn the be can of which significance conclusion true the not is current duction h edln atr ssece nFig. in sketched as pattern field-line the ota hyaemvn iha ria velocity orbital an with moving axis, common are a they around direction that same so the in rotating are which rings spae nti ktha etbtentetopae,ta si the in is that plates, two the the between of rest strength greatest at of sketch region this in placed is h ieto ftervelocity their of consider direction and regions the long comparatively over curvature their neglect ftern-hpdcag areshv eylreradius large very a have carriers charge ring-shaped with the deal If now we fact; significant very formulation. quantitative of a perpendicular direction its is the are This to perpendicular lines motion. also relative field and electric the field Its its electric to the produced. addition of those is in to condenser, field the magnetic of a motion field, the of result a As whichhaveanareaof n ( and eta ifrneis difference tential n st odapoiainhmgnostee nacoe path closed a 2 On length there. homogeneous of approximation rings good closely-spaced a two to the is between and only magnitude non-zero a has etdensity ment OWLAND 2.5 ) h oaigcagdrnspoueamgei field magnetic a produce rings charged rotating The )). fR of b hc assaon n fterns h antcpo- magnetic the rings, the of one around passes which D seprmn (Sect. experiment ’s rm freference of frame OWLAND ol eiiae ymvn hre ln.Btthis But alone. charges moving by imitated be could fmagnitude of 4.12 U mag A n mgn htacnesrcnit ftwo of consists condenser a that imagine and D olsItouto oPhysics to Introduction Pohl’s D bl H sExperiment ’s ( D z u l b D ob osat ewe h plates, the Between constant. be to D electric rmwihte r bevd This observed. are they which from ngetn h edi h exterior the in field the (neglecting 4.3 2 Q ,temgei fet fa of effects magnetic the ), = r A ,teei neeti displace- electric an is there ), field seSect. (see 7.1 5 Sect. , h magnetometer The . E . u , hnw observe we Then . 5.2 https://doi.org/10.1007/978-3-319-50269-4_7 2.13 ntegop 2 groups the in qs ( Eqns. , r ecan we , H con- 2.4 .It M ) 7 129

Part I : c C7.1 Left (7.1) (7.2) (7.3) .The . which u u u and the ), 0 S 7.1 contraction, . This (small) is at rest relative B 0 /: S . E ; and 2 ). This increases the 7.1 ORENTZ . Above and below the enclosed by the path. c ; u E u is moving at the veloc- l M = y . with the velocity 7.4 0 2 = 0 u S ) u u 1 relative to the observation . ,or uD Qu 0 (a compass needle with a light l ) the equation ( S , appear to be foreshortened ). " 7.1 is the L = 0 1 D M D S

b 7.2 D 7.1 /; z (the ‘laboratory system’) is at rest Qu p I 0 A Q H B S E u D moving D in Fig. at the velocity b D z S u or M . (see also Sect. H y to the electric ﬁeld when the condenser, 0 lb ; uQ S D' / 0 0 ) with the aid of the theory of relativity, can be seen from Fig. D E in D u

" ﬁnds: z 7.1 u H S u D D z z' . x' x m/s is the velocity of light in vacuum. For E B S' D in addition 8 10 H Then the theory states that: The origin of the factor yS y' 1), we obtain from Eq. ( 3 is moving relative to Magnetic ﬁeld lines from positive and negative electric charges (the magnetometer 0 C7.2 C7.3 . . S D S c

in u In order to obtain Eq. ( effect was neglected in deriving Eq. ( the carrier of the electric field, is definition of the vector apparatus charge density on its plates, and thereby also pointer). This field occurs magnetometer to be at rest in the system and, in general form and vector notation, which makes the condenser, at rest in along the direction of relative to the room and the magnetometer ity to the plates, while the lower system (i.e. can thus be considered torelativity be experimental evidence for the theory of where which are moving parallelof to the each page) with other their (perpendicular plate-shaped carriers, to both at and the into same velocity the plane Two right-handed coordinate systems. The upper system let us consider the condenser to be at rest in the system Figure 7.1 region). This is equal to the current pair of plates, the magnetic fields essentially cancel each other. Then the observer in , ). force (1848– in ” (Dover 5.4 http://www. ,wherethe ORENTZ 0 S Einstein’s The- OWLAND ), was too diffi- . R 7.3 5.2 Lectures on Physics How the Fields Depend on the Frame of Reference ENRY 7 H (Sect. cult even for a scientist like charges are at rest, soonly that an electric field, butmagnetic no field is present, would find that the magnetometer (compass needle) is subject to a torque.is This certainly the case, but the demonstration that a magnetometer which is moving relative to an electric field experiences a torque, analogous to the L Sect. C7.3. The principle of relativity requires that even an observer in C7.2. See Fig. C7.1. Introductions to the special theory of relativity can be found intextbooks. many See for example M. Born, “ ory of Relativity Publications, 1965) (available online for download at https://archive.org/details/ einsteinstheoryo00born Other introductions with examples include R.P. Feyn- man, Vol. I, Chaps. 15–17 and 34, and Vol. II, Sect. 13.6 for a relativistic treatment of moving charges. This can be read online at feynmanlectures.caltech.edu/ See also “Electricity and Magnetism” by E.M. Pur- cell and D.J Morin, 3rdCambridge ed., University Press (2013), Sec. 5.9. 1901)! 130 Part I 7.3 Induction in Moving Conductors 131 7.3 Induction in Moving Conductors Part I The earlier experiments (Figs. 5.10 and 5.11) are summarized in a schematic sketch in Fig. 7.2. There, the points where the magnetic field lines of the homogeneous field perpendicular to the paper pass through the plane of the page are shown as black dots. The slider of length L is moving relative to the frame of the magnetic field (the field coil) at the velocity u perpendicular to the direction of the magnetic field lines. The observer finds negative charges at the point C, and positive charges at A; and, using the arrangement shown in Fig. 5.10, the magnitude of the induced voltage is found to be: U D BuL : (5.9)

This experiment can demonstrate the validity of the principle of relativity convincingly, by moving either the conductor (slider) or the field coil which produces the magnetic field. In both cases, the same induced voltage is observed. The interpretation of this result depends on the frame of reference of the observer. Initially, the observer is at rest next to the field coil in the reference frame S, and describes the induction effect as follows: ‘Like every object, the slider contains electric charges, an equal number of each sign. These charges participate in the motion of the slider at the velocity u. During the motion, they collect at the points A and C (Fig. 7.2); the slider can serve as a current source. Therefore, within it, there must be forces F which separate the charges,asin any current source. Here, they are a result of the motion. They pull positive charges downwards and negative charges upwards’. While these forces F are piling up charges at the points A and C, according to Eq. (5.9), an electric voltage is produced between A and C. It, in turn, causes a restoring force F to act on the two separated charges (Eq. (3.1)). In vector form, this restoring force is given by: F D Q.B u/: (7.4)

For positive charges, it acts upwards, and for negative charges it acts downwards. If only these forces were acting, the charges would again be pulled together. The experimentally-observed stationary state is

Figure 7.2 Schematic drawing of in- Current I Negative charges duction in moving conductors. The magnetic ﬁeld points perpendicular C to the plane of the page towards the reader. The circular cross-section of u L the ﬁeld coil seen in Fig. 5.10 has been replaced here by a rectangular A cross-section. The rod A-C corre- S sponds to the slider in Fig. 5.10. Current I Negative charges Op- dur- (7.5) (7.6) to the in the .Then cannot

u F force ), and induc- hyperphysics. D 5.3 F in addition 0 between two types field. This field, as (in the frame of ref- ORENTZ ), i.e. : 7.4 to the magnetic field, , act perpendicular to both B ). electric /: that separate the charges are B u F C7.4 ). in Eq. ( , Point 2b), with negative charges D experimentally u

field instead of moving a slider rod. . 5.5 ORENTZ 0 F ). For this observer, the charges are Q 0 Q in addition F 3.10 relative to the observation apparatus S D effect, which is a splitting of the spectral u D ’s experiment – only the roles of the elec- which are moving at the velocity 0 F force is a new experimental fact. E magnetic Q TARK ). Induction in moving conductors is therefore would be observable in principle without a con- 0 ”, Chap. 14, Sect. 47; or see e.g. S 7.2 OWLAND passed fluorescent molecules at a high velocity . For the observer who is at rest in the frame of the ,wehave ORENTZ B , when the field coil, the source of the magnetic field, F field which then acts on the electrons in the molecules IEN , we had to distinguish 5 on the charges field. This field occurs forces B electric moving at the velocity and These forces, named for H.A. L be acting on them, but instead, only an shown by the experiment, is given by magnetic field is (the slider in Fig. a counterpart to R for the From this point of view, an electric field occurs in S tric and the magnetic fields have beenAn reversed. electric fieldframe which of reference appears during the relative motion in the ducting slider.artificial Imagine resin. that During the the motion,would slider it be would is “frozen cool a in”. andwould its rod act When interior as made state it an was of electret removed (Sect. a from the heated field, the rod at the top end and positive charges atWilhelm the bottom. W field coil, the L magnetic field u at rest. Therefore, the magnetic field with its L through a homogeneous tik und Atomphysik lines into several components in an electric field (13th edition of “ phy-astr.gsu.edu/hbase/atomic/stark.html erence which we have called Now, we let the observer move with the slider of induction, namely induction in a coil at rest (Sect. tion in moving conductors (Sect. In the former case, the explanationa was coil as follows: at With rest, induction the in electric forces that separate the charges are produced by an 7.4 Fields and the PrincipleIn Chap. of Relativity therefore possible only if the forces equal and opposite to the forces The was detected via the S , , 0 S S How the Fields Depend on the Frame of Reference 7 insofar as they are different from those measured in will likewise be written with aprime. C7.4. All of the quantities measured in the system 132 Part I 7.4 Fields and the Principle of Relativity 133 ing the variation of the magnetic field. It surrounds the magnetic field with closed, circular electric field lines (Sect. 6.1,Fig.6.2;themath- ematical formulation is given as the MAXWELL equation (6.22)). Part I According to Sect. 7.3 also, in the second case, that is induction in moving conductors, the forces which separate the charges are due to an electric field (which is seen by an observer who is at rest relative to the moving conductor). However, it remained completely unex- plained just how the motion was able to produce an electric field. The answer is given only by the theory of relativity:C7.1 It states that the windings of the field coil are no longer exactly electrically neutral during its motion. They acquire an excess of charges of each sign, and between the excess charges, an electric field acts. We can already show this here; from the theory of relativity, we need only the LORENTZ contraction: An observer whose frame of reference (the S system) is at rest relative to some object measures its length to be l. An observer who is moving parallel to this length relative to the object with a velocity u measures in his/her frame of reference S0 a reduced (‘contracted’) length p l 0 D l 1 u2=c2 (7.7)

.c D light velocity in vacuum D 3 108 m=s/:

Following this brief but sufficient introduction to the LORENTZ contraction, we repeat in Fig. 7.3 the content of Fig. 7.2. The terminology needed is given in the caption of this figure. – Initially, an observer is at rest relative to a field coil through which the current I is flowing (S system). In the windings of this coil, the positive charges (the lattice ions) are fixed, and the negative charges (the electrons) 1 can move at a rather small velocity ue,i.e.wehaveue c.The magnitudes of the charges q and their densities are equal within the windings of the field coil; the conductor is electrically neutral, so that q %C D % D D % (7.8) V .V D ld2 is the volume of a wire of length l/:

For an observer at rest relative to the slider (S0 system), the positive charges which are fixed within the wires are moving at the velocity u (Fig. 7.3); for this observer, in the windings of the field coil just below (3) and just above (1) in the figure, the positive charges have the same charge density % %0 D q D p q D p ; C 0 d2l d2l 1 u2=c2 1 u2=c2 or, for u c (i.e. .1 C u2=2c2/) Â Ã 2 0 1 u %C D % 1 C : (7.9) 2 c2

1 Compare the footnote in Sect. 8.3. , )

l ), l x' e u u )isat 7.11 axis is (7.10) (7.12) (7.13) (7.11) L ). This u S' z e ,asisthe : 0 u parallel to : S ; are all mea- u y' C L Ã e Ã / u / u z' e charges in the e , whose S uu uu excess of positive 2 2 wires (3), combin- C : : 2 2 N u (4) negative u u u e . . is moving to the left (at the e 2 . This ﬁeld is perpendicular ) are observed in 2 2 2 c

S c l c c 1 1 = 2 2 Nqu Nqu ) yields an NI e e and section of windings just below C C u u D 7.8 D 1 1 D C 0 C Â Â 0 excess of negative charges H q system, the negative charges (elec- q % % y 0 upper , the system l 0 S E' D D (3) (1) S N N ) , which is moving at the velocity 2 2 0 D c c S is passed through this coil, it produces a homo- D = = ) with Eq. ( I I I 0 C 2 2 0 )inthe / / A Q ) yields an C e e u Q 7.9

u u L 7.8 C % Current Current % section of windings, just above (1), e u u Exercise 5.6 . u ( . )and( . As observed in (2) 0 0 y turns in its windings, made of wire with a square cross-section 1 and the velocity of the electrons (drift velocity) 1 C7.5 E lower 7.10 Induction in a moving conductor. The sketch shows the rectan- x N ). I p p u . When a current . The excess charge densities ( ) with Eq. ( 2 : S d D S D axis (as is sketched to the right in the figure for clarity). The length 7.9 0 0 x % % y system (with 0 z observer thus finds the charge density of the S and for the and ( For the lower section of windings (1), the combination of Eqns. ( sured in charges ing Eqns. ( For the upper section of the coil containing and in the lower section (1) a slightly greater velocity ( electric field trons) in the upper section of the windings (3) have a velocity ( velocity rest in the frame of reference to the plane of thefield page lines and through points the towards page the is reader, marked and by the dots. passage of The its slider (of length also perpendicular to the plane of the page. The coil has along its length gular cross-section of athe plane long of field the coil, page. It whose is long at axis rest in is the perpendicular frame of to reference the the current For the same observer in the Figure 7.3 all together of area (3), to be geneous magnetic field of magnitude is . E’ B : )tobe and u 2 e ,wherethe u c u 7.14 S ) which is ob- 0 ), the field

=" Nq ll 0 0 7.16 D

" How the Fields Depend on the Frame of Reference D D , moving with the velocity of the slider. The field is 7 0 0 0 field source is at rest.in As Eq seen ( proportional to This field is caused by the relative motion of the slider and the magnetic field source. It is a relativistic effectdoes and not appear in theoratory lab- frame served in the moving system S E u shown in Eq. ( C7.5. The goal of theing follow- calculations is to derive the electric field (in the form E 134 Part I 7.5 Summary: The Electromagnetic Field 135

0 0 Therefore, between QC and Q, there is an electric field which is 0 produced by the LORENTZ contraction. Dividing QC by the cross- 0 0 sectional area l l of the windings gives its displacement density D D C7.6. The lengths l and d Part I 0 C7.6 have no components along "0E , and thus the electric field: the direction of u,sotheyare 1 Nqu u 1 Nqu u Nqu u not affected by the LORENTZ E0 D e D p e D e (7.14) 0 2 2 contraction. "0 l l c "0 ll c2 1 u2=c2 "0 ll c . was defined in Sect. 7.2/: C7.7. This field can also be derived directly from the

We know that que D Il and 0NI=l D B. Inserting these quantities theory of relativity. For this into (7.14) gives derivation, we consider that the field coil is at rest in the u E0 D B : (7.15) frame S (Fig. 5.4 in the foot- 2 "00c note in Sect. 5.2)andthe charge Q is at rest in S0.Then = 2 D " Applying the relation 1 c 0 0 (see Sect. 8.2) and taking the we have directions into account, we finally obtain the electric field in vector 0 D . /; notation:C7.7 E u B and E 0 D .u B/; (7.16) B0 D B : 0 and for u c,Eq.(7.6) follows from this. The field E is directed In S0, then, the force F0 D downwards in Fig. 7.3: The electric field which results from the mo- . / 0 Q u B will be observed. tion of the field coil relative to the slider in S thus has the magnitude Using a relativistic transfor- and direction that were determined empirically in Sect. 7.3. Result: mation, we can compute from The observer who is at rest relative to the slider can explain the ob- C7.8 it the force F which will be served charge separation in terms of the LORENTZ contraction. observed in S. The transfor- Now, finally, we are in a position to understand the experiments of the mation equation for forces in second group in Sect. 5.2 in detail: As seen from the reference frame this case (where the geometry is particularly simple) gives S of the field coil, the LORENTZ force produces the induced voltage 0 in the induction coil, which is at rest in S . As seen from the induction 1 0 0 F D F D Q.u B/: coil in the reference frame S ,theLORENTZ contraction of the charge densities in the field coil produce the same voltage in the induction This is precisely the coil. In the general case of induction in moving conductors, one must LORENTZ force! It thus of course also take into account (in addition to the LORENTZ force follows directly from the or the LORENTZ contraction) the possible time dependencies of the theory of relativity. (Note magnetic field, as described in Sect. 5.6. again: The field E measured in S is zero.) More details may be found for example in 7.5 Summary: The Electromagnetic P. Lorrain, D. R. Corson, and F. Lorrain, “Electromagnetic Field Phenomena”, Freeman, New York (2000), Chap. 13.

Because of their importance, the results of this chapter will be briefly C7.8. The explanation given summarized here: here can also be applied to 1. An observer can place a measuring apparatus (the magnetometer the experiment shown in in Fig. 7.1 or the moving conductor in Fig. 7.3) at rest within a frame Fig. 5.11. The reader should of reference which is moving at a velocity u relative to the source also try to apply it to BAR- of the field (condenser or field coil). Then she or he will observe LOW’s wheel! in ). For )or ). In the ORENTZ 7.6 7.1 L relativistic 7.3 electromagnetic Induction in moving conduc- C7.9 ) applies, and in the second case, Eq. ( 7.1 to the electric field also a magnetic field (Fig. . contraction. Or, conversely: to the magnetic field also an electric field (Fig. 0, the additional fields vanish. They are thus purely . Their presence or absence depends upon the frame of reference D ORENTZ L tors is an experimental verification of the existence of the field chosen for the observations. effects. 2. Electric fields andpendent magnetic of fields one are another. not autonomous They and are inde- both parts of an u first case, Eq. ( 3. Induction in moving conductors can occur ascontraction a result of the in addition addition )! 5.11 following Eq. ( 5 How the Fields Depend on the Frame of Reference 7 C7.9. Making use of this newly-found information, the reader should reconsider the remarks made at the end of Chap. 136 Part I ocsi antcFields Magnetic in Forces .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © 8.1 Figure Sect. In ( Eq. Fig. test can we However, velocity the at density moving flux is of which field Q charge magnetic a a on acts force This L the of existence the conductors, of moving aware in became induction we of consideration detailed our From Forces the of Demonstration 8.1 iua obt h velocity the both to dicular aeopst in u h aedrcino oin(seg nFigs. in e.g. (as motion of direction same the but signs opposite field) have induction magnetic the as hrecrirwsfudt eeuvln oteivsbemto of motion invisible the to equivalent be to found was carrier charge a are,freapeacagdsa ube ecno aethe make cannot We bubble. soap charged product a example for carrier, charge macroscopic ( mechanically-moved a equation using experiment this stration verify cannot we to Unfortunately, and field (Fig. magnetic charge the the to of velocity both the perpendicular is force the product, h eea case, general the 5.12 BB BB ( As). F ) ntergthn mg,te aeopst ietoso oin In motion. of directions opposite have they image, right-hand the In b). 8.1 F o xml nnwo ( newton in example for 4.3 xeietlyi te ways. other in experimentally Qu nMvn Charges Moving on Forces teR (the ufiinl ag o uhalrecriro charge. of carrier large a such for large sufficiently uuu u and F OWLAND D B r o epniua oec other. each to perpendicular not are Q olsItouto oPhysics to Introduction Pohl’s . u u F D F n otemgei u est as known (also density flux magnetic the to and xeiet,temcocpcmto of motion macroscopic the experiment), 7.5 B D 1N), B ntetolf-adiae,techarges the images, left-hand two the In . 8.1 / n t osqecsa hw in shown as consequences its and ) Q B nmvn hre.Te r perpen- are They charges. moving on . B scnb enfo h vector the from seen be can As . ): u nVs/m V in B / ORENTZ 2 ( D 1T), 7.5 , https://doi.org/10.1007/978-3-319-50269-4_8 ihademon- a with ) force u F nm/s, in Q 5.10 in ( uI u 7.5 C8.1 in to ) velocity orbital its that so V, 200 ca. of voltage a by is accelerated beam The page. the of plane the to perpendicular points that field magnetic a through moving is beam electron an Cologne), Co., W Göttingen): IWF Lechner, K. (Photo: the figure in here shown as beam, electron an using by avoided is difficulty This C8.1. yrgngsa rsueof p pressure a at gas hydrogen containing bulb glass a In ftecrua orbit. circular the of center the towards acting ity, veloc- orbital momentary the to perpendicular is and the orbit along point every at stant fH pair of a by produced is field es frtto n radius a and rotation of sense counter-clockwise a with force), (Lorentz field the by orbit circular a into deflected is beam The reader). the a 8 ca. L ORENTZ D a(rmteLe the (from Pa 1 ELMHOLTZ r 10 D u 4 6 sc.10 ca. is ,drce towards directed T, oc stu con- thus is force 8 10 ol ( coils 2 7 .The m. /.The m/s. ybold B 137

Part I ) is m D as = l l 4.4 B (8.1) ( ), lead- 13 W s 15 A, : 7.5 1 D D which carries I l m .Thefield 2 8.2 10 5 15 A 2 : : : m B The observer can use the charge Il = IB l C8.2 D , within a conductor of length D 5Vs Q : 1 F Qu D or F , Q 2 m = 13 N. . The conductor is : . I 1 5Vs : 8.2 1 8.3 D D . Quantitatively, we found m A horizontal current-carrying conductor perpendicular to the perpendicular to the field The field-line pattern corre- = u B I m, 2 13 N m : 1 10 a current To verify this equation, wetor make use in of a a homogeneous horizontal,produced magnetic linear by field conduc- an as electromagnet inconductor. and Fig. is The directed conductorwires perpendicular forms and to a is the hanging trapezeample from with is a given its force in two the metershowninFig. legend (balance). rigid of lead the A figure. numerical The ex- pattern of field lines is This equation for the magnitudesing can to be inserted the into force Eq. on ( a section of conductor of length a ‘rest’ frame of reference.a In velocity that frame, only the other charges have D homogeneous magnetic fieldforeshortened of due an to electromagnet. the5 perspective. The conductor Numerical appears example: carriers of either sign, sponding to Fig. perpendicular to the plane of the page. electric charges within a conductor. Figure 8.2 Figure 8.3 m (See 2 u = with the kg is the QB 31 8.3 . In perfect D 10 ). )). 3.9

8.1 11 : 9 Forces in Magnetic Fields D 8 m lines in Fig. C8.2. Compare the field-line patterns in Fig. patterns of the electric field of the electrons, the so- called cyclotron frequency, is given by The frequency of rotation mass of the electron), and is independent of the (non- relativistic!) velocity Exercise 8.1 (see Eq. ( the magnetic field of one current distribution (here: of the electromagnet) and the current in another conductor analogy to the electric case, the force is determined by ( 138 Part I 8.2 Forces Between Two Parallel Currents 139 8.2 Forces Between Two Parallel

Currents Part I

As an application of Eq. (8.1), we compute the forces between two parallel, straight conductors of length l at a distance r from each other, which are carrying the currents I1 and I2 (Fig. 1.9). The current I1 produces

1 I the ﬁeld strength H D 1 (6.13) 2 r at a distance r,thatis

I the ﬂux density B D 0 1 : (8.2) 2 r Equations (8.1)and(8.2) together yield the attractive force (when the currents in both conductors are ﬂowing in the same direction) or the repulsive force (when they are opposite):

I I l F D 0 1 2 : (8.3) 2 r

Numerical example: I D 100 A, l D 0:5m, r D 1cm, 0 D 4 107 Vs/Am,F D 101 N.

We now apply Eq. (8.3)toaspecial case: We imagine that the two currents are carried by two identical strings of charges which ﬂow beside each other through space (Fig. 8.4) (these could be electrically- charged particle beams with a linear charge density Q=l). Thus, in contrast to the usual conductivity currents in metals, etc., the ‘background charges’ of opposite sign are missing here. As a result, in addition to the attractive magnetic force Fmagn between the two strings, there is a repulsive electrical force Fel perpendicular to the direction in which the charges are moving. For the magnetic attractive force, we obtain from the combination of Eqns. (8.3)and(4.4):

Q2u2 F D 0 : (8.4) magn 2 rl

Figure 8.4 Two parallel strings of charges of the same sign, moving in the same direction 2 c = 2 (8.7) (8.8) (8.5) (8.6) u forces : s m in vacuum! 8 c must refer to 10 0 magnetic 0 " 998 : . This acts according 2 p : rl Q = 0 : D 8.5 1 : " 2 : : rl 2 2 Q : 2 u 2 2 0 Vs rl ) yields 0 c D Am c u Q 1 must be a pure number; it rep- 7 " el 0 E 0 D 2 8.5 2 F " waves (at the time experimentally 1 u 0 " 0 D 10 2 D in 1856 described this velocity simply as D ), we obtain 0 1 0 " D " QE , in 1862: He explained light waves as )and( el 4 8.6 p magn el magn D F F 8.4 F F el F 1 As Vm OHLRAUSCH 12 AXWELL electromagnetic ) into Eq. ( forces between the same charges at rest. We want to 10 on a cylindrical surface of is exactly the same as the velocity of light 8.7 r 1 and R. K ) on the charge string at the right, exerting a force 859 : 3.5 8 electric r EBER to Eq. ( Derivation: The charge string at the leftat a produces distance an electric field strength W. W produced by moving electric charges are smaller by the factor a velocity. A calculation gives This velocity This is not simply anrelation accidental between agreement, but the rather velocity aena. of fundamental The light first to and recognize electromagneticwas this fact James phenom- in Clerk its M wide-ranging consequences elucidate this statement by referring to Fig. In words: Under similar geometric conditions, the In this equation, the product resents the ratio of two forces. Therefore, 1 still unobserved!). Inserting Eq. ( short-wavelength a “critical” value whichforces. could make magnetic forces just as strong as electrical than the 1 Taking the ratio of Eqns. ( For the electrical repulsion, we find We thus have the experimental result that Forces in Magnetic Fields 8 140 Part I 8.2 Forces Between Two Parallel Currents 141

r I u I u Part I l

Attractive force

Fmagn

Figure 8.5 A numerical example to elucidate Eq. (8.8): Two parallel copper wires of length l D 1m and1mm2 cross-sectional area are a distance r D 0:1 m apart, and carry a current of I D 6 A. According to the footnote below, the current I is due to negative charges of Q D 1:36 104 As whichare moving at a velocity u D 0:44 mm/s. We thus find .u=c/2 D 2:16 1024. The current flowing on the right produces a magnetic flux density (induction field) of B D 1:2 105 Vs/m2 at the position of the left wire (Eq. 8.2). 5 This causes a force Fmagn D QuB D 7:23 10 N to act on the charge Q there. From Eq. (8.5), the mobile negative charges Q alone (i.e. without the presence of the equally strong positive background charges) would repel each 19 other with a force Fel D 3:34 10 N. Then the ratio of forces is Fmagn=Fel D 2:16 1024 D .u=c/2.

Fig. 8.5 shows schematically two current-carrying wires (not to scale!). The mobile negative charges (electrons) within them move as gray clouds through the lattice of ﬁxed positive charges (lattice ions), at velocities which are normally less than 0.5 mm/s.2 Thus, .u=c/2 is only a tiny fraction, on the order of 1024! If the positive ions were lacking, the two clouds of negative charge would repel each other with the force Fel. The magnetic ﬁeld generated by their motion produces an attractive force between them of only 24 Fmagn 10 Fel. For this reason, Eq. (8.8) leads to the following conclusion: The production of magnetic forces through electric currents is certainly of

2 The copper wires used in everyday house wiring are normally limited to current densities of less than 6 A/mm2. A copper wire with a cross-sectional area of 1 mm2 and a length of 1 m has a mass m D 8:95 g, and thus the amount of substance n D 0:14 mol (Vol. 1, Sect. 13.1). It contains copper ions, each carrying one positive electrical elementary charge. For these ions, the FARADAYconstant is the quotient

Charge Q As D 9:65 104 : Amount of substance n mol

The copper wire therefore contains a positive ionic charge of Q D 1:36 104 As. The mobile negative charge Q between the lattice ions is just as large. Inserting this quantity of charge into Eq. (4.4)gives

Il 6A 1m m mm u D D D 4:4 104 D 0:44 : Q 1:36 104 As s s are C8.3 ”. induction ’s law, the current ENZ relativistic effects ): . ”or“ Video 8.1 eddy currents second-order effects ’s Law and Eddy Currents (1834): , an aluminum ring as “induction coil” is hanging be- , the induction was caused by the increase of the mag- ENZ 8.6 A ring-shaped “induction coil” 5.5 ENZ . This results from the conservation of energy. If the sign want; finally, it becomes just ainduced disk in of that metal. disk The are currents called which are netic field. Itcurrent in must the therefore field flow coil. in the2. opposite In direction Fig. totween the the conical polemagnet pieces away on of its a guidewhich rail. horseshoe is the magnet. The cause ring of follows the We it. induction, pull is Their3. the opposed. separation, Now we reverse thethe experiment, i.e., ring we and push try thenet magnet to pole towards force pieces the where ringfrom the the into approaching field the magnet; its region is approach,induction, between which strongest. is is the again the avoided. mag- The cause of ring the moves away 4. In cases 2 and 3, we could make the hole in the ring as small as we induced in the induction coil must impede this increase of the mag- The electric fields,always currents directed and to forces oppose producedduction the by processes which givewere rise opposite, the to process that causing in- thea positive induction feedback would and be would subject increase without to from limit, nothing. creating energy Examples (see also is hanging like a pendulumpoles between of the a horseshoe magnetslid which along can a be guide rail (an optical bench) Figure 8.6 Induction processes give riseThe to sign electric of their fields, direction currents isH. F. determined and E. from L forces. the rule formulated by 1. In Fig. netic field from the field coil. According to L eminent importance for electricallongs technology; among but the physically, it “ be- 8.3 L ’s law exhib- ).) ENZ have been de- r illations and separated by ’s Law” v Forces in Magnetic Fields ENZ 8 yields Eq. (8.8). rived (in Eq. (5.29) there; the first term on the left andterm the on the right). Their ratio the distance wobble osc from the collection of physics demonstrations at Cornell University. (Note as well the Video 8.1: “L two point charges (protons) moving with the same velocity point charges, the repulsive electric force and the attractive magnetic force between and D.J. Morin, 3rd edition, Cambridge University Press (2013), Sect. 5.9, p. 264. Here, instead of two rows of of relativity can be found in “Electricity and Mag- netism” by E.M. Purcell C8.3. The derivation of this equation based on the theory http://tiny.cc/mcggoy The video shows an impressive demonstration experiment of L http://tiny.cc/qcgvjy ited by the aluminum ringthe at end of the video (seeVideo also 11.12 in Vol. 1, 142 Part I 8.3 LENZ’s Law and Eddy Currents 143

Figure 8.7 Eddy currents brake the fall of a silver coin in an inhomogeneous magnetic ﬁeld NS

(Video 8.2) Video 8.2: Part I “The Eddy-Current Brake” ab c http://tiny.cc/1bggoy N A round aluminum disk is released as close as possible to the center in the region of NS strongest field (and therefore also the greatest field gradient). It has been previously cooled to the temperature S of liquid nitrogen (77 K), d which strongly enhances its N electrical conductivity and therefore the braking effect. Then, a long aluminum plate S is pulled through the field to demonstrate the astonishingly Figure 8.8 A rotating magnetic field with various “induction rotors”. Part b strong forces, which increase shows a schematic drawing of the rotating magnetic field produced by the with the velocity of the plate. apparatus in a, at two different positions separated by 60ı. The small circles mark the axis of rotation for an observer who is looking directly down from above. The magnetic field lines between its rotating poles S-N are indicated. Parts c and d show two rotors which can be used above the rotating magnet in place of the rectangular frame. The application of rotor d is a reversal of Video 8.3: the experiment shown in Fig. 8.9. (Video 8.3) “Induction Rotor” http://tiny.cc/5bggoy The principle of an induction If we drop a silver coin through the inhomogeneous magnetic field motor is demonstrated with of a large electromagnet (Fig. 8.7), then it doesn’t fall freely with the a setup as shown in a shadow usual acceleration in the air. Instead, it sinks slowly as if it were in projection in Fig. 8.8 a. a sticky fluid. Here, again, the induction process impedes its origin, in this case the falling motion of the coin.C8.4 C8.4. Modern coins which are made of alloys have lower electrical conductivities. This 5. We replace the linear motion by a rotation; in Fig. 8.8,werotate makes the induced currents a horseshoe magnet around its long axis and thereby obtain a “rotat- smaller, and thus also the ing magnetic field”. Into this rotating field we then bring an “induc- forces which slow the coin’s tion coil” which is mounted on bearings so that it can also rotate; it fall. takes the simple form of a rectangular metal frame. The frame follows the rotation of the field. The angular motion between the field and the frame, which is the origin of the induction in the frame, is impeded. Soon, the frame is rotating almost as fast as the magnet. It cannot move exactly as fast; in that case, the field change within the area of the frame would no longer be present, so there would be no induction at all. The fractional velocity difference between the coil and the rotating field is called the slip. For the technological application of this experiment, the simple rectangular frame is replaced by a metal cage (Fig. 8.8c); one then refers to it as an induction rotor or ‘shorted rotor’ (see Fig. 9.21). / ,we B .They ,and 8.9 1 u F . rotation Axis of Q of the field D B inhomogeneous F Cross-sectional view of the field coils eddy currents forces magnetic field through 2 F is thus larger than 3 3 1 F F F 2 ORENTZ F , we have sketched a cross section shift the electron orbits to the right. 2 8.10 F inhomogeneous ) Video 8.2 is perpen- B ( The production ) act, as indicated by the arrows. The flux density 8.9 Eddy currents slow the 8.1 is greater below than above in the figure. (Sect. The production of these eddy currentstion is in best moving understood conductors. in termsof In of the Fig. induc- magnetic fieldcle indicates and a a closed segment pathelectrons for of participate the the electrons in within disk. the theing rotation metal perpendicular The disk. of to small, All the the the disk. field dashed lines, cir- Therefore, and they L are mov- this gives rise to adirection. circular Furthermore, motion the of forces theThese electrons two motions in are a superposed counter-clockwise orbits and of result the in eddy a currents. cycloidal motion for the ing process. (See also Figure 8.9 rotation of a circular diskaluminum. made Its of axle lies farplane behind of the the page. Theof front the surfaces magnet poles couldparallel also to be each other, butthe then inhomogeneous only regions at their edges would play a role in the brak- of eddy currents in thedisk moving of Fig. dicular to the plane ofpage the and points along theof line sight of the reader) Figure 8.10 a tough resistance to motionof which eddy is currents again surprisingly opposes strong. its Induction cause, the rotation of the disk. 6. In thewere fourth produced by experiment, moving we an a encountered metal plate of limitedplate size. changed with The time. magnetic flux passing throughEddy the currents can howeverin also the be geometrical produced arrangementsee of without a the any circular components. changes aluminum disk Inmagnetic which field Fig. extends of into the an electromagnet.hind the The axle plane of of the the disk drawing. is well The be- disk is hard to rotate; it shows Forces in Magnetic Fields 8 144 Part I 8.4 Damping of Rotating-Coil Instruments. The Creeping Galvanometer. Magnetic Flux Circuits 145 8.4 Damping

of Rotating-Coil Instruments. Part I The Creeping Galvanometer. Magnetic Flux Circuits

We refer to the second experiment in the previous section. There, we saw in Fig. 8.6 a metal ring mounted as a pendulum in a magnetic field. When it was set into swinging motion, it was seen to come to rest after only a few swings – it was strongly damped. The forces due to induction impede the oscillations of the pendulum (LENZ’s law). This induction damping is often used to prevent unwanted vibrations. It is also called “eddy-current damping” or “eddy-current braking”. Imagine the ring in Fig. 8.6 to be replaced by a metal disk. Induction damping is in particular indispensable in the construction of various kinds of measurement instruments (including modern electronic analytical balances). It is used to prevent the annoying and time-consuming oscillations of the pointer before it settles down to its final position. As a rule, the aperiodic limit3 can be reached, so that the pointer comes to rest in its final position after a minimal time. We offer as one example the induction damping of a rotating-coil ammeter (Fig. 1.19). As a rule, it consists of two parts: First, one uses a rectangular metal frame as support for the coil windings. It acts like the ring in Fig. 8.6, and analogously for rotational oscillations. Second, the rotating coil itself can act as a closed-loop induction coil. The instrument is part of an electrical circuit; thus the ends of the rotating coil can be connected in some manner by a conducting link (“short circuit” or “shunt”) as needed. The resistance of this shunt (for example in Fig. 1.33, around 106 ) is called the “external resistance”. A suitable choice of its magnitude allows experienced observers to attain the aperiodic limit for the pointer oscillations. C8.5. This is also the oper- When the damping is too strong, the pointer creeps to its equilibrium ating principle of the first position, without oscillating, but very slowly. This makes the instru- telegraphic systems for ment useless for measurements of the momentary values of currents use over long distances. C. F. GAUSS and W. WEBER or voltages. However, in contrast, a creeping galvanometerR is very set up a telegraphic connec- useful for the measurementR of “current impulses” . I dt/ and “voltage impulses . U dt/:Itsums a series of impulses over a longer time tion in 1833 over a distance of observation automatically.C8.5 It is the limiting case of the ballistic of ca. 1 km between the Göt- galvanometer treated in Sects. 1.10 and 2.11. A mechanical analogy tingen Observatory and the may be helpful: Physical Academy. Positive and negative voltage impulses In Fig. 8.11, a gravity pendulum is shown with its lower end dipping were used to cause a light into a very viscous liquid, for example honey. This causes its motion pointer to deflect to the left or to be strongly damped. We applyR a hammer blow to the pendulum, the right by a small amount. giving it a mechanical impulse . F dt/. The pendulum is deflected The apparatus can still be with a jerk and then practically stands still: due to its strong damping, seen in the Historical Collec- tion of the Physics Institute at 3 This is not yet the “creeping” pointer setting; see below! the University of Göttingen. ). Now to the 8.12 . They can be calibrated as of a current-carrying coil (see of the empty field coil. V s ˚ ˚ –3 5 ∙ 10 , we can see that the coil is framed by an 8.12 , for example in volt second. As an example of the etc., but now strongly damped by the small “external re- 5.3 ). In Fig. . To amplify the deflections of the galvanometer, one could use 2 The effect of an “iron core” on the magnetic flux of a coil. The The operation of a creeping 2.38 abc d 010 4.14 , 2.36 V s. That is the magnetic flux 4 also Fig. improvised induction loop. Thethe pointer zero of point the of galvanometer is the near scale (bottom image in Fig. experiments: 1. The current invanometer the pointer field begins coil to (about move10 to 3 A) position is a. switched This on. corresponds to The gal- sistance” of the inductionaround loop. 25 cm The cross-sectional area of the iron core is an iron core on the magnetic flux application of a creeping galvanometer, we investigate the effect of flux is measured usingFigs. a creeping galvanometer, the same instrument as in Figure 8.11 several turns instead of only a single induction loop. galvanometer Figure 8.12 it can return toutes. its A rest second impulse position (another only hammerpendulum blow) at very would the thus slowly, end strike after point the oftion several its would min- first add deflection, so to the(hammer the second blow deflec- first. from the An left) impulse would be from correspondingly the subtracted. Creeping opposite direction galvanometers weremainly for formerly registering voltage usedshown impulses in for Fig. measurements, Forces in Magnetic Fields 8 146 Part I 8.5 The Magnetic Dipole Moment m 147

2. We put the field coil over one leg of the U-shaped iron core. The pointer moves to position b, since the flux ˚ has increased to 1:3 3 10 Vs. Part I 3. We bring an iron closure bar slowly into position on the U-shaped iron core and finally lay it firmly onto the core, closing the magnetic circuit. The pointer moves stepwise to position d, and the flux ˚ has now reached a value of 9:4 103 Vs. 4. We switch off the current to the field coil, and the pointer of the galvanometer moves back to position c on the scale; the “remanent” magnetization of the iron core has a magnetic flux of 2:2 103 Vs. Finally, we remove the closure bar and the iron core. The pointer returns to the zero point. The field coil with no current and no iron core is once again free of magnetic flux. A qualitative evaluation can be made readily using the simple model of molecular currents (Sect. 4.4). The magnetic field of the field coil aligns the magnetic fields of the molecular currents in the iron so that they are all parallel to each other and to the external field. Then the invisible ampere turns within the iron add to the visible ones in the field coil, greatly increasing the strength of the magnetic flux density B. We will treat this topic quantitatively later, in Chap. 14 (Sect. 14.11, ferromagnetism). For the next chapters, the experience gained from these measurements will suffice: The magnetic flux ˚ of a current-carrying coil can be increased nearly 100-fold by inserting an iron core. Furthermore, it can be conveniently varied by changing the magnetic circuit of the core.

8.5 The Magnetic Dipole Moment m

The simplest and most convenient indicator of a magnetic field is no doubt a compass needle. The magnetic field exerts a torque Mmech on a suitably-mounted bar magnet. The latter can also be replaced by a current-carrying coil, for example as in Fig. 1.10. How does this torque come about, how can we describe it quantitatively? We give the answer first for the case of a current-carrying coil. The planes of its windings are parallel to the direction of the magnetic field, as C8.6. To avoid confusion shown in Fig. 8.13. We show only one turn of the windings instead with the magnetization M of the whole coil; for simplicity, it has a rectangular cross section. Of (Chap. 14), the symbol for the four sides of the coil, two (the vertical sides) are perpendicular the mechanical torque M and the two others are parallel to the field. Therefore, a force F D BIl is denoted by a subscript acts on the first two. The two forces act as a force couple on the lever “mech”, as already used in arms r and produce the torque:C8.6 Sect. 3.9. For many of the equations in the following

Mmech D BIl2r D BIA (8.9) paragraphs, just the magnitudes are sufficient. The A is the area of the windings, independently of the shape of the coil (rect- directions can be read off angular, circular etc.). from the figures. E m ,we (8.11) (8.12) (8.10) and the (a force 3.17 A F : .Wedefine sees the cur- Á , 2 m F . The bar was F A m R = I rr F in V s l : B B ; 2 : corresponds to the vector Á B : i magnetic moment B in A m A 2 A m I m plays no role here. ; X the torques that act on each of the R D D ) for an electric dipole in an electric I m add in N m is unit: 1 A m rectangular winding, one has coils with D mech 3.25 I ), the vector is a vector quantity. Its direction is perpen- M mech produced by a pair of forces . For the overall torque, we obtain M mech m F 3.9 turns have practically the same area single 8.11 M ) attached to the end of a spoke N ). Then the torque is given by S S m as in cylindrical coils (solenoids), or varying as in which denotes the orientation and magnitude of the A A and .InEq.( For example: The origin of the A B flowing in a clockwise direction (or: the curved fingers of the I In the case of well-definedtheir cylindrical windings, coils all with only a few layers in Figure 8.13 magnetic moment (the current many turns of arbitrarysectional shapes area (long or flat, with a constant cross- relative to The vector product describes the torque for every orientation of Now we introduce a new quantity, the The magnetic moment in the analogous equation ( field. Usually, instead of a dicular to the areasurface vector enclosed by the current path, i.e. parallel to the it by the equation flowing in the conventional direction) Analogously, we can simply couple). The length of the spoke multilayered coils, especially in flatfor coils). the For second these time cases, an we experiment recall from mechanics. In Fig. right hand point inparallel the to direction of current flow, when the thumb is rent current path. An observer looking in the direction of individual windings of a coil,the independently common of axis; their this distancesment, is from treated just in the Sect. same as for an electric dipole mo- subject to a torque saw a bar (vector Forces in Magnetic Fields 8 148 Part I 8.5 The Magnetic Dipole Moment m 149

N Part I I S

Figure 8.14 A bar magnet and two iron-free coils, all with the same magnetic moment, of magnitude m 34 A m2. The long coil has a diameter of 10.6 cm and 4300 turns; the flat coil is 25.4 cm in diameter and has 730 turns. The current in both coils is 0:9 A. The straight arrows show the directions of the three equal magnetic moments m and also the direction of the magnetic flux density B at the center of the flat coil. Looking in the direction of m,we would see the current I circling clockwise (curved arrow, right-hand rule!). The north pole N of a compass needle would point to the geographic north pole of the earth. same orientation. Therefore, their combined magnetic moment is m D INA : (8.13) Two examples of the magnetic moments of coils are shown in Fig. 8.14. Permanent magnets of all kinds, and magnetized pieces of iron or other magnetic materials, are no different (outside their own volumes) from current-carrying coils or bundles of coils (Sect. 4.1); but the orbits of the circulating charges within these materials are invisible. As a result, we cannot compute the magnetic moment m of a permanent bar magnet or a similar object in the same way as that of a current-carrying coil (Eq. (8.12)). However, it can be measured using Eq. (8.11), in the simplest case with m perpendicular to B, M m D mech : B For this measurement, we mount the permanent magnet on bearings (like a compass needle) with minimal friction, in a horizontal plane. In equilibrium, that is Mmech D 0, its magnetic moment m orients itself parallel to B (see Eq. (8.11)). Then we use a calibrated torque (a spring balance F at the end of a lever arm r as in Fig. 8.15)torotate the axis between the poles of the magnet until it is perpendicular to a homogeneous magnetic field of known flux density B (e.g. from a field coil). Figure 8.15 shows a measurement of this type using a bar magnet in the magnetic field of the earth.

Small torques Mmech cannot be measured with great precision as the product of force times force arm; it is better to compute them from the oscillation period T of torsional oscillations. From Vol. 1, Eq. (6.13), we ﬁnd the . B mech 2 1m, : 0 M ,to (8.17) ough (8.15) (8.16) (8.14) D r (length) , i.e. an anti- . .Webringit 2 ı m is parallel to mass 180 / m ˛/ : . This requires that 12 39 A m D = B torsion coefﬁcient 1 cos D ˛ . )of h D ˛: B 1 = 8.11 . mB 2

: T mech N with the lever arm mB 2 h 3 relative to M

˛ B B F magnetic moment 2 2 D 4 ˛ North D T 10 sin 4 ˛ D m , the work has its maximum value, d 8 D mB B : ˛ ). 7 , for a bar magnet ˛ mech 2 m 2 D S N M sin D r and ) combined yield ˛ mech F 0 Z m . Then the object will orient itself, presuming by an angle M in kg m 8.15 in V s/m B mB m

B . Then we ﬁnd 2 , a freely suspended, horizontal bar magnet (compass) is D ˛ )and( in s, T W 8.14 Vs/m Measurement of the magnetic moment of a bar magnet, mounted angle , for example 5 moment of inertia). If it is rotated out of its rest position thr rF 10 D D small 2 , we can rotate .

subject to a restoring torque according to Eq. ( Equations ( the (For example (Vol. 1, Eq. (6.11)), ratio of the torque to the rotation angle, termed the ( be B D mech h mB parallel orientation of 2 horizontally and free to rotateM in the earth’s magnetic field. (A counter-torque is produced by the spring, used as a force meter). The torque vector the torque perform work: An arbitrary magnetic object (current-carryinga coil, paramagnetic molecule compass etc.) needle, has a Figure 8.15 from the springThe points magnetic perpendicular flux density toB of the the horizontal plane component of of the the earth’s page field and is into it. It is stored in the form of potential energy. For Applying an external torque thatby is opposed to the torque produced that it is free to move, so that its magnetic moment into a magnetic field Forces in Magnetic Fields 8 150 Part I 8.5 The Magnetic Dipole Moment m 151

abc Current in the windings of the

small coil and field coil Part I anti-parallel parallel

F = 0 FF

NNNSS S

Figure 8.16 Part a: In a homogeneous ﬁeld, a current-carrying coil, that is an object with a magnetic moment m, is not acted on by a force. Parts b and c: In an inhomogeneous ﬁeld, in contrast, forces act. This is at the same time a model of a diamagnetic substance (Part b) or a paramagnetic substance (Part c).

In an inhomogeneous magnetic field, in addition to the torque Mmech, there is also a force F. It pulls or pushes the object in the direction of the field gradient, e.g. @B=@x. This important difference between homogeneous and inhomogeneous fields is elaborated in Fig. 8.16. We will elucidate the origin and the magnitude of this force by referring to Fig. 8.17. We imagine the magnetic field to be perpendicular to the plane of the page and pointing towards the reader. The points where the field lines pass through the page are marked with dots. The field strength increases on going from top to bottom in the figure. Our ‘object with a magnetic moment m’ is a rectangular wire loop carrying a current I (its area is A D lx). Its magnetic moment m therefore points in the same direction as B. The forces directed to the left and to the right, Fl and Fr, cancel each other. The forces pulling upwards and downwards, Fu and Fd, however, have different magnitudes. These are given by Eq. (8.1): Â Ã @B F D IlB and F D Il B C x : u d @x

Therefore, the net force F D Fd Fu is pulling downwards; i.e.

@B @B F D Il x D IA ; @x @x

Figure 8.17 The derivation of I Fu Eq. (8.18). The current I flows in the conventional direction and the vector field B points perpendicular to the plane of the page towards the reader. ∆x Fl F When m and B are parallel, the resul- r tant net force pulls downwards; when they are anti-parallel, it pulls upwards. Fd l . ) ) D D po- 0 l 3.5 2 8.18 = ( 8.19 H by the m (8.18) (8.19) 2N. : NI Q 0 acting at ;theresult . When the into regions 116 A AB F : / 0 5.8 l . With m , it is acted on ). This torque = B 8.20 D and it is carrying NI 8.19 could be localized . A m 8.12 and the charge . ˚ D H F ) e.g. to determine an un- . The sign is found as shown : 4 8.18 : x at the “centers” of the shaded areas. B E H @ @ S Q . For these two magnets, the ˚ b and c, we had m 5 is a scalar quantity. Equation ( is replaced by ) carrying a current of 29 A); D and D 2 E D . ˚ 8.16 according to Eq. ( F , electric charge Q 3 N F , it then follows that (for simplicity we F using a test coil with a known magnetic 8.18 0 x 3.18 and is located in a homogeneous magnetic ˚ and ): l 0 , the rectangular current path shown is one of =@ NIAB ˚ D B ˚ 72 V s/m . Its cross-sectional area is :InFig. : @ 1 B D A 8.13 0 D B ), x ), one can no longer speak of poles at all. are vectors and mech =@ , as explained in the following section. ) also holds when the field gradient is parallel to the direction of and B 8.10 M @ H . This current produces the magnetic field strength ˚ 0 can be schematically defined as in Fig. 8.14 . We can make use of Eq. ( I . , the pole regions of a long current-carrying coil and a per- 8.18 magnetic flux = m and 0 8.17 4.7 ound today, the flux distribution is shown by the lower image in Fig. B turns of a solenoid whose axis is perpendicular to the plane of F numerical example inside the solenoid. In the homogeneous field l N D A (i.e. 2 turns with an area of 20 cm Therefore, = 0 Equation ( For bar magnets made of steel, which were formerly used extensively and can moment the current with a stronger or weaker field strength This force pulls the object with the magnetic moment H each end of the coil as shown schematically in Fig. by a torque NI in Fig. known field gradient Here, leave off the prime on the page. It has a length field of flux density magnetic flux is so strongly localized,between we can apply a formal analogy lar regions corresponds to a force couple, with a force There, one can localize the poles In the case ofimage magnetic in fields Fig. from flat current-carrying coils (as in the left-hand the field. Suppose that in Fig. still be f magnetic flux 4 5 8.6 The Localization ofIn Magnetic Fig. Flux or, with Eq. ( manent magnet of similarwere shape compared. made of For a both, ceramic the oxide magnetic material flux using a wire induction loop (measured e.g. as in Fig. is shown at the top of Fig. corresponds formally to the equation Suppose that in Fig. the Forces in Magnetic Fields 8 152 Part I 8.6 The Localization of Magnetic Flux 153

V s cm 5 10–7 Part I l Φ S N Solenoid or ceramic magnet

1 10–5 Steel bar magnet S N Magnetic flux ∆ Length element ∆

10 20 30 40 50 60 cm

Figure 8.18 The distribution ˚=l of the magnetic flux ˚. In the upper part of the figure, a long current-carrying coil (solenoid) or a long bar magnet made of a magnetic ceramic oxide. Lower part: a bar magnet made of steel. An induction loop (as in Fig. 5.8) is slid stepwise along the length elements l and measures their contributions ˚ to the magnetic flux ˚.(Exercise 8.3)

–F = –ΦB/μ0 l +Φ –Φ

F = ΦB/μ0

Figure 8.19 Schematic of a magnetic dipole in a homogeneous magnetic field. The field H or B lies in the plane of the page and points downwards, perpendicular to the line l. for a charge Q in an electric field E. By analogy, the magnetic flux ˚ was previously termed the “magnetic charge”. It was presumed to correspond in the magnetic case to a quantity of electricity or charge Q in an electric field E. The implementation of Eq. (3.5) with an electric field was treated in detail in Sect. 3.4. What was said there can be applied correspondingly to the implementation of Eq. (8.19) with a magnetic field, i.e. in particular: For H,inEq. (8.19), one must use the undisturbed magnitude, as measured before ˚ was present. From Eq. (8.19), it follows that the magnitude of the torque which acts in Fig. 8.19 is given by 1 Mmech D Fl D ˚Hl D ˚Bl 0 and that of the magnetic moment is M 1 m D mech D ˚l : (8.20) B 0 We list some additional formulations which follow from the analogy between the magnetic flux and the electric charge: 1. The magnetic field at a large distance from a magnetic pole which has the magnetic flux ˚. Figure 8.20 shows a schematic drawing of . of even ˚ (8.22) (8.21) m C8.9 ). The longer ; 8.20 : 2 ). Division by the symmetrically over r 0 r A ˚ ˚ 0 8.18 1 4 2 ). For simplicity, we have D D 4.4 r , this step corresponds to the C8.8 s H H H 8.20 or and and B 2(cf.alsoFig. 2 ˚= C8.7 A ˚ r . Therefore, at a sufficiently large distance, 1 2 2 ˚ of the coil gives the magnitudes of the fields D r 4 s A D s ˚ D B r B . Current-carrying coils (with or without an iron core) m The left end of a long, thin , we show the measurement of the magnetic flux directly at the face of the coil; we find the values: . s 5.8 H 8.14 and The magnetic field at a large distance R from an object with a mag- The magnetic field directly in front of the flat face of a polar region s i.e. half of theface total. of a That coil yields end for the magnetic flux through the a long coil.before The it measurement was loop pulled wasloop out; starting near from in the far Fig. center offlines to of in the the the right. process coil of It being has withdrawn cutIn from through contrast the all coil. to the this field step,in we front now of place the theaway, measurement end only loop of the directly the field coil, lines above to the the arrow. left When of it the is arrow pulled pass through it, In Fig. 3. netic moment and permanent magnets can have the sameif magnetic moments their shapes andFig. compositions are quite different; we saw this in once again completelycharge. analogous to the electric2. field of a point In the neighborhood oftern of such the coils field lines or certainly permanent depends on magnets, the the shape of pat- the magnetic Figure 8.20 solenoid whose field lines emergeproximately with radial ap- symmetry the field lines of adrawn only long its solenoid left-hand (Fig. end in thisAt figure. some distance from the polar region,to the a pattern of good the approximation field lines radiallythe is symmetric bar (Fig. magnet ormagnetic coil, flux the distributes itself more at precise largea distances is spherical this surface approximation. 4 we The find for the magnitudes of cross-sectional area B plays .Ifwe B ˚ , from sym- ˚ ) can be exper- ): In the interior 8.21 8.22 Forces in Magnetic Fields 8 metry. Note that only the axial component of each of the newly-formed ends will make the same contribution to of a long field coil,the we magnetic find flux now split the coil in halfa (in thought experiment), then arolehere. C8.9. A different derivation of Eq. ( coil). by Eq. ( imentally determined with quantitative precision by using an induction coil (probe C8.8. The behavior described C8.7. See also the calculated field distribution in Com- ment C4.4. 154 Part I 8.6 The Localization of Magnetic Flux 155

Figure 8.21 The flux NS μ m B = 0 density B at a large R 27R3 First principal orientation distance R from the Part I center of a bar magnet or a coil with the S μ m R B = 0 magnetic moment m 47R3 N Second principal orientation object. But, at a sufficiently large distance, the fields B and H are determined only by the magnetic moment m of the object.Thisis shown for the two principal orientations in Fig. 8.21. Here, the carrier of the magnetic moment is a small bar magnet with its north and south poles marked as N and S, often called a magnetic dipole.

Derivation: Each end of the bar (or coil) produces a flux density Br D ˚ at the point of observation according to Eq. (8.21). Only the differ- 4R2 ence of the two values is important, so that in the first principal orientation Â Ã C8.10. In order to determine ˚ 1 1 B D : (8.23) all the magnetic quantities by . = /2 . C = /2 4 R l 2 R l 2 means of mechanical meas- When the distance R is sufficiently large compared to the length l of the urements, one would first bar or coil, we can neglect l2 relative to R2, and for the magnitude of B,we have to measure the prod- then obtain uct mBh (see Eq. (8.16)) by means of the experiment de- 1 ˚l m B D D 0 : (8.24) scribedinFig.8.15.Then 2 R3 2 R3 one would find, as shown Correspondingly, for the second principal orientation, we find here, the angle ˛ which the compass needle makes m B D 0 : (8.25) with B (independently of 4 R3 h the magnetic moment of 4. The measurement of unknown magnetic moments using one of the compass needle). From the principal orientations. Equations (8.24)and(8.25) are important Eq. (8.25), we then obtain m for measurements, in particular for the experimental determination tan ˛ D 0 : 3 of unknown magnetic moments m. For this purpose, one measures 4 R Bh B in one of the principal orientations, either directly with a probe Combining this with coil (Sect. 5.4) or my making some sort of comparison with the Eq. (8.16), it follows that known horizontal component of the flux density of the earth’s field B2 D 0 : 4 2 h T2R3 tan ˛ (e.g. Bh D 0:2 10 Vs/m in Göttingen). For example, one orients This permits us to measure the directions of B and Bh perpendicular to each other and finds the Bh without knowing the mag- angle ˛ between the directions of Bh and the vector sum of the two fields (Fig. 8.22) with the help of a compass needle. Then the field netic moment of either the bar magnet (coil) or of the sought is given by B D Bh tan ˛. Using this value of B, one then computes the moment m from Eq. (8.25).C8.10 compass needle; and from it, the other magnetic quan- Compensation methods are often favored. In these, one allows a sec- tities follow (C. F. GAUSS ond, known magnetic moment to act on the compass needle in ad- and W. WEBER, 1832; see dition to the unknown moment (Fig. 8.23). The known moment is Wilfred Dudley Parkinson produced by a current-carrying coil of well-known dimensions. The (1983), “Introduction to magnetic moment of this “compensation coil” is computed using Geomagnetism” (Elsevier), Eq. (8.13). p. 353 ff). . ) 8.22 (8.26) ( (8.28) (8.27) : must be large com- V 2 R B of a dipole field in the 0 : B 1 A h 2 2 : B α B ) with the force V D 0 2 1 N x B 2 is excluded. At the same time, (null method). This is a schematic 8.19 0 A ˚ B S x 1 A m D 2 1 2 2 B 2 A 0 D A SN R D ˚ s 1 D 0 2 B 1 V RR magn 2 D ); in reality, the distances contains the energy W x D S N V 8.22 F C8.12 C8.11 F and can then lift a heavy load. In this process, S N x D ). When connected to a flashlight battery, it can pick W 8.24 The determination of the flux density Measurement of an unknown magnetic moment by comparison , the two faces of the magnetic poles approach each other and the field coil). . One pole by itself produces the flux density and volume N - B 8.25 S of the other pole according to Eq. ( The energy content of a homogeneous magnetic field of volume V Forces between the planar, parallel faces of two neighboring mag- up more than 100 kg This equation canelectromagnet be (a verified “pot-type electromagnet”) quite with5.5 impressively cm a (Fig. by diameter of using only a small directly in front of its˚ pole face. This field acts on the magnetic flux was performed. Therefore,density a homogeneous magnetic field of flux pared to themagnet dimensions of the carriers of the magnetic moments (the bar drawing (as is also Fig. with a coil of known magnetic moment 6. Figure 8.23 5. netic poles Figure 8.22 In Fig. a magnetic field of volume over a distance the mechanical work second principal orientation, byhorizontal making component use of of the the earth’s known field flux density of the D B are 0 500, B ,more . 1cm, 2 D depends 14 a value of . and D N B 5000 A/m. 9 B . More details l H Vs/m D 3 in the solenoid, H H . With in the presence of Vs/Am,wefindfor l 10 = 7 0 1Aand : NI 0 10 (Video 8.4) Forces in Magnetic Fields D D 8 not simply related by the factor 4 the flux density about 6 than two orders of magnitude smaller than the value measured in the induction experiment. This is a clear indication that one can use the expression for the field in the interior of a long coil orH solenoid: C8.12. In this thought experiment, we have to take care that the flux density is held constant in the space between the pole faces (see Comment C8.11). This can be fulfilled approximately by two permanent magnets at a sufficiently small spacing, so that stray fieldsbe can neglected. A still more convincing demonstration of magnetic field energy will follow in Chap. C8.11. To estimate the magnetic field are given in Chap. magnetic material (here: the iron core). Also, strongly on the geometry, i.e. here on the width ofgap the I we obtain From this value, with 156 Part I Exercises 157

4 V Part I G

Figure 8.24 An electromagnet (“pot-type magnet”) with a closed iron core. The field coil is at the center, and above it, an induction loop for measuring the 2 3 2 flux density B. (The cross-sectional area A of the iron is 10 cm D 10 m , Video 8.4: D 2 : 3 B 2Vs/m , F calculated from Eq. (8.26)is16 10 N). Using a flashlight “Electromagnet” battery as current source, the windings of the field coil should have around http://tiny.cc/xbggoy 500 turns. (Video 8.4) The strong attractive force of the electromagnet is demon- Figure 8.25 The calculation of the magnetic ∆x field energy strated. It amounts to 530 N. In addition, it is shown how NS sensitively this force depends upon the width of the gap. A few sheets of paper are shoved between the two halves of the magnet. At a gap width of 0.4 mm, the force is only 15 N ! (see A numerical example: The highest flux densities B which can be obtained Sect. 14.6). in iron cores are around 2.5 V s/m2. Then in the field region between the 3 C8.13 poles, a magnetic field energy of ca. 2.5 W s/cm is stored. C8.13. Using superconducting magnet coils with flux densities of 5–15 V s/m2, energy densities of 10– 90 W s/cm3 can be obtained (for comparison: the stored energy density in Ni/Cd stor- Exercises age batteries is in the range of 180–300 W s/cm3;see also the example in Vol. 1, 8.1 For the experiment described in Comment C8.1, find the field Sect. 19.8, Point 1). Such su- B in which electrons that have been accelerated by a voltage of U D D perconducting magnetic en- 100 V will follow a circular orbit of radius r 10 cm. (Sect. 8.1) ergy storage devices (SMES) have been under develop- 8.2 ment for some years. See A flat coil carrying a current consists of 10 turns of wire with a cross- e.g. P.J. Hall and E.J. Bain sectional area A D 100 cm2; it is in a magnetic field, as shown in (2008), “Energy-storage Fig. 8.13. The current through the coil is I D 10 A and the flux technologies and electricity density of the magnetic field is B D 0:1 T. How large is the torque generation”, Energy Policy 36 Mmech which acts on the flat coil? (Sect. 8.5) (12), pp. 4302–4309.

8.3 Estimate the magnetic flux ˚ in the long coil used for the measurement results shown in Fig. 8.18, and compare it with the measurements given in Video 5.1 (Fig. 5.8), which were carried out on a different long coil. (Sect. 8.6) 0, in D https:// magn I W when 1 2cm D l = N 5 cm, and the current strength D ) contains supplementary material, which The online version of this chapter ( r would be necessary to keep it from contracting? ) F 8.6 14 A. Start by determining the magnetic field energy D I the helix. (Sect. the diameter of the helix is 2 Electronic supplementary material 8.4 Current is flowingloosely through and vertically. a Due helical toaxis. the conductor Which current, force which it contracts is along its hanging long doi.org/10.1007/978-3-319-50269-4_8 is available to authorized users. is The number density of its windings is Forces in Magnetic Fields 8 158 Part I hrcagn fteelectrodes the of charging ther noeulbimwt the with equilibrium into hrb nraetevlaebetween and voltage electrodes two the can the to forces increase charges more charge-separation thereby transfer the to continue Then time a plates. for the between flow can obeuae eie ht ti uhtoln,s ti sal brvae as abbreviated usually is it charge-separation the so between long, this clearly too by distinguish to much devalued forces has was is one it it case, any and that, In ‘E.M.F.’ source, Besides current usage. the double of voltage load-independent the n tcno eecee.Teeeti edbtentepae it- between plates charges the between the field on electric forces The produces exceeded. self be cannot it and etsource rent otg,otncle the called often voltage, fteotrcrutbetween circuit outer the If mechanical, of reservoir a energy. from chemical taken charge-separation or is the thermal It charges, the the work. indicates move perform to must ammeter order forces the In current. electrodes!), a the of flow reach charges the when n Motors and Generators to Particular in Induction, of Applications 1 Figure Remarks General Note. Preliminary 9.1 at oee,ta emwsas sdfrthe for used also was term that However, past. and ln oiotlln ewe h lcrds a eicesdby increased be can electrodes, the of sort between some line horizontal measured charges, a negative the along and positive the between distance The .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © separation. ares w fte,acag-are ar r kthdi Fig. in sketched are pair, charge-carrier charge a on localized them, as of them Two of think carriers. can we signs; both of charges are rclmcie a esmaie nafwsnecs h un- The sentences. always few elec- are innovations a of decisive attributes in the and essential summarized principles the indepen- physical be two derlying manner, of can similar use machines the a and trical In needle, the of threads. tip innovations: dent two the by at eye characterized needle’s be the can machine sewing modern A l fteecag-eaainfre eecle eetooiefre”i the in forces” “electromotive called were forces charge-separation these of All C and r once hog namtr ewe hs electrodes these Between ammeter. an through connected are 9.1 nCretSources Current on lcrclquan electrical evst lutaetegnrldfiiino h term the of definition general the illustrate to serves or hresprto forces charge-separation generator tities aro odne ltso “electrodes” or plates condenser of pair A . olsItouto oPhysics to Introduction Pohl’s odidpnetvoltage load-independent hresprto forces charge-separation uha the as such C 1 and . A voltage sitrutd omr charges more no interrupted, is uigtermto ntonly (not motion their During . C voltages and hc eut rmtecharge the from results which A htte rdc,ie for i.e. produce, they that u onalimiting a soon but ; C , and https://doi.org/10.1007/978-3-319-50269-4_9 ilb reached, be will , rvnigfur- preventing , A n comes and cur- 9.1 A . 9 159

Part I ) I 7.5 gen- ( charge- Electrons F A u B A with two additions: F CA 9.2 CA charge-separation forces Current Electrons / B are connected by a conducting .The A u . upwards in the direction of the arrow force Q and D C ) F induction and move them in opposite directions. to ORENTZ Q C , using it as a ‘slider’. This will cause a ,theL u force to electric generators and motors. move the conductor The definition of an “in- The definition of the term “cur- . We repeat that picture here in Fig. Generators which operate by 9.1 cular to Generators and Motors ORENTZ to act on the charges separation force with a velocity block (shaded). We can now separatetor the charges in within two this ways conduc- andthe give two them electrodes: velocities which propel them1. towards We could ductive” current source (the magnetic field points out of thereader page and towards is the perpendicular toof the the page; plane the current isconventional flowing direction, in the rent source”. For demonstrationwe experiments, use two ‘charge spoons’as (cf. charge Fig. carriers 2.11) (they areinfluence charged machine). by an Figure 9.1 We adopt a viewpoint from within thecontains ‘black a box’ magnetic and field, suppose perpendicular that toalso it the that plane of the the two page, electrodes and erators Figure 9.2 We begin with the most important current sources in use today, simple. Theare great not achievements a of matterscriptions modern of of these physics, electrical technical but applications engineering shouldoverview. rather In be of this limited chapter, technology. to we discuss a thethe Physical brief application L de- of induction and 9.2 Inductive Current Sources, which they employ are produced bythe induction terms processes. ‘charge-separation forces’ We and defined ‘current source’of on Fig. the basis Applications of Induction, in Parti 9 160 Part I 9.2 Inductive Current Sources, Generators 161

Figure 9.3 An N alternating- A J current generator Part I with external a b magnetic poles S

To the voltmeter

2. We could change the magnetic flux density B of the magnetic field which acts within the box. That would lead to an electric field around the closed current loop in Fig. 9.2 (cf. Fig. 6.2 and Fig. 5.5), and would move the charges between C and A by means of the forces FC D QCE and F D QE towards the electrodes. As a rule, both of these processes are applied simultaneously in order to use this device as a generator (current source) which produces a voltage between C and A. We will explain this by taking as examples several types of generators: a) The alternating-current generator with external magnetic poles (Fig. 9.3). A coil J is rotated around an axle A in a magnetic field produced in some way. The ends of the coil are attached to two slip rings, which are electrically connected by two spring-loaded slip contacts or “brushes” a and b to the output terminals of the machine. The rotation of the coil represents the periodic repetition of a simple induction experiment. The induced voltage is “alternating”. Its time dependence can easily be registered using a voltmeter with a short response time (ca. 1 s) if the rotation of the coil is not too fast. In the special case of a homogeneous magnetic field and uniform rotation (Fig. 9.4a), this voltage curve is sinusoidal. Its frequency is equal to the rotational frequency.C9.1 C9.1. A quantitative treatment is given in Sect. 5.6,in For practical applications, the coil has an iron core (Fig. 9.5). The particular Eq. (5.10). More coil and the iron core together form the rotor. During their rotation, details on alternating current not only does the magnetic flux ˚ through the rotor coil change, but can be found in Sect. 10.3.

a in volt Voltage

b in volt Voltage Time

Figure 9.4 a): The sinusoidal voltage curve of an alternating-current or AC generator. b): The voltage curve of a direct-current generator with a simple coil rotor and a commutator (Fig. 9.6). The signs refer to the direction of the electric ﬁeld between the output terminals. . 1 2 b C a a b 9.4 coil rotor and the ﬂux . The arch- ˚ C To the voltmeter To shows a demonstration rather than a Time J 9.6

N (field coils or permanent magnet) S volt in Voltage - N b can be “smoothed”. Instead of .Fig. 9.4 ) b cylindrical rotor ) ). 2 , the latter due to the effective variation of the a B 8.12 and 1 a NS S , one uses several, spaced at fixed angles relative to one ab J The voltage curve ( Cylindrical rotor The iron cores of the stator ,and through the rotor are large, and at b, they are small. direct-current generator a over to positive values; the voltage curve shown in Fig. C B cular to Generators and Motors ,com- direct-current generator with a cylindrical rotor J 9.4 and of the rotor coils of a generator. At a, the magnetic flux with two pairs of coils andmutator com- a single coil another. This results in a Figure 9.8 also the flux density gap width (cf. Fig. b) The Figure 9.6 A direct-current generator with asimplecoil rotor mutator a permanent- magnet stator Figure 9.7 Figure 9.5 density of a cylindrical rotor with twoof pairs coils, and how it(superposition is of produced c) A shaped voltage curve in Fig. model, again as a shadow projection. Thecurrent slip rings generator of the are alternating- now(the replaced “commutator”). by a Itof simple reverses the switching the rotor device connections coilhalf between and rotation. the the output This ends folds terminalsFig. the of negative the portions generator of after theresults. each curve Its shown voltage in varies betweensign zero always and remains a the maximum same. value, but its Applications of Induction, in Parti 9 162 Part I 9.2 Inductive Current Sources, Generators 163

I A Part I A

V U C

Figure 9.9 An old-fashioned direct-current generator with 2 25 permanent ﬁeld magnets and a cylindrical rotor with 9 pairs of coils. At 8 A and 12 V, it can bring a 100-watt incandescent lamp to full brightness. One has to provide a muscle power of 8 A12 V 100 W. The machine is “hard to crank”. If the current is interrupted, however, it rotates freely, with hardly any resistance. This experiment demonstrates clearly that we should appreciate the energy content of a kilowatt hour and its commercial price ( 30 Eurocent; see Comment C1.16.). As a demonstration object, the generator from an automobile engine is also suitable. However, as shown in the schematic in Fig. 9.10, it will require external excitation in the form of a current source for its ﬁeld coils FC.

Figure 9.7 shows a schematic, with two pairs of coils and a fourfold commutator. In this example, two of the ‘arch’ curves from Fig. 9.4 (a1, a2) are superposed in an evident way. The result is the smoother direct current shown in curve 9.8b. Figure 9.9 shows a model of a direct-current generator with a cylindrical rotor which is suitable for use in lecture demonstrations. d) The direct-current dynamo. The generators mentioned so far obtained their stator magnetic fields from permanent magnets. These permanent magnets can be replaced by current-carrying coils, so- called field coils (FC in Fig. 9.10). The current in the field coils may be supplied by some sort of auxiliary current source. Figure 9.10 shows a schematic of this external excitation. However, the genera-

Figure 9.10 A direct-current V generator with external excitation

C J

FC ,which .Thero- E .Inthe . In all the 8.12 b . The essential J by means of a rotor J E a ame shaft as the main generator. . shows a machine of this type. It 9.11 ) induces an initial voltage in the rotor 9.11 14.7 is an oscillating iron or steel diaphragm in M is mounted on the stator. In practice, there . Their principle is based on the presence of J ). 9.12 dynamos An alternating-current generator with internal poles telephone as an alternating-current generator without windings. cular to Generators and Motors . In the case of the internal-pole generator, the opposite is true: E J Alternating-current generators with coil-free rotors generators we have consideredgenerator so which far, rotates, the carried rotor,the coils. i.e. magnetic that It flux part is however within of possible the the to induction vary coils without windings. Rotors ofare this mechanically type very have stable therotational frequencies. and advantage that can Figure they therefore be operated at high point in the design ofrotors the (Fig. alternating-current 9.11) generator with wasmagnetic coil-free the circuit. periodic variation Theoscillation of rotation (Fig. the can iron-containing be replaced by a back-and-forth is derived in a readily understandable manner from Fig. tor consists in this model of a narrow, rectangular iron bar causes the magnetic flux throughorientation. the coils to vary, depending onFor its technical applications,magnets one by electromagnets, often i.e. replaces coilsiron carrying the cores. a permanent direct Furthermore, current, field metry all with and the repeat parts themselves areof many arranged the times with rotor around and radial stator. the sym- circumference g) The alternating-current generator with a bar-magnet rotor It is the ‘closure bar’magnetic of circuit the formed by the horseshoe magnet. are many coils whichis are arranged often with in radialcircumference. the symmetry. The form direct The current of rotor forauxiliary a the generator field mounted flywheel, coils on is the with provided s the by an field coils around its f) external-pole generator as described inproduces a), induction was the fixed, magnetic and fieldcoil the which rotor contained the induction The rotor consists offixed induction windings coil which carry a direct current. The place of the moving rotor.the This experiment again sketched is in only Fig. a technical variation on windings. e) The iron in the coils. Whennetic they field begin of to the rotate, iron the (Fig. weak remanent mag- Figure 9.11 tor itself can alsothe supply case the with current for its own field coils. This is Applications of Induction, in Parti 9 164 Part I 9.2 Inductive Current Sources, Generators 165

Figure 9.12 Schematic of the tele- a phone of Alexander GRAHAM BELL

(1876) M Part I J

Figure 9.13 An antiquated telephone as an alternating-current generator (a rotating-coil ammeter is connected through a rectiﬁer D)C9.2 C9.2. The telephone shown D here is indeed quite antiquated, but even today, in the age of mobile telephones (“smartphones”), the same principle is still used in telephone receivers, loudspeakers and even the little sound Figure 9.12 shows the schematic of a telephone transmitter. Here, it converters in hearing aids. is interesting only as an alternating-current generator. Its function is Microphones, in contrast, to convert the mechanical energy of sound waves into electrical en- are increasingly based on ergy. To demonstrate this, we connect a telephone (Fig. 9.13)toan capacitance or piezoelectric alternating-current ammmeter. When we sing into the diaphragm, we elements (see Sect. 3.10). can observe weak currents of the order of 104 A. These alternating currents have the rhythm of the human voice. In earlier times, the audio currents were sent over long-distance telephone lines to the receiver telephone and converted there back to mechanical oscillations (sound waves). Figure 9.14 shows a sketch of a typical arrangement. Today, this setup is completely outmoded; human vocal cords are no longer used as a motor to drive an alternating-current generator. In- stead, the pressure of the sound waves from the voice simply controls currents with the rhythm of speech (microphone)2.

S N JJN S Speaking Listening

Figure 9.14 An old-fashioned connection of two telephones for long- distance calls (bar magnets instead of the horseshoe magnet as in Fig. 9.12)

2 This kind of “control” was already utilized by the inventor of electric telephony, the teacher PHILIPP REIS (1861). The transmitter used by REIS was a microphone in today’s terminology, with a vibrating contact made of platinum (instead of carbon particles as introduced by D. E. HUGHES in 1878). The telephone receiver used by REIS would today be termed a “magnetostriction receiver”. In the ﬁrst publication by REIS (1861), he closes with the words: “Until a practical application of the telephone becomes possible, there is still much work to be done. However, for physics, it is already interesting, in that it opens up a whole new ﬁeld of research.” ;

9.16 .This

)). They ). Then 2 u– , carrying U a 7.5 F– 2 B U a (Eq. ( / B u+ perpendicular to the F+ N B . This motor is in prin- u CA . Electrons Current Q . Their velocities are indi- is brought into this field. By Q D A - F C The magnetic field acts on these s a simple ‘slider’). In practice, 1 2 C9.3 . forces u and C ORENTZ is perpendicu- u S NS B . We imagine that within the box outlined in black, 9.15 An alternating-current synchronous motor connected to an A schematic definition of Generator Motor . alternating-current synchronous motor cular to Generators and Motors torque cated by the arrows cause the charges to move in the direction of the arrow the conductor contains moving charges ciple similar to anshows the alternating-current same (AC) machine generator. ona the Figure motor. left The as rotor a coils generator of and the on generator are the turning right at as a frequency thus it delivers alternating current with the same frequency current passes through theof connecting leads the 1, motor. 2 intocoils. There, the The it rotor direction coils produces ofrent. a rotation torque depends Therefore, which on theposition acts the of torque direction on the must of the rotor the have to rotor guaranteed cur- ensure in the a that right simple it way: direction continues to at rotate. every This can be the conductor along with them (it i a current-carrying coil is mountednetic as a field “rotor” of within the thea fixed “stator”. mag- The forces acting on the rotor produce charges by exerting L alternating-current generator with external poles lar to the plane ofoutwards the towards page, the pointing reader) an “electric motor” ( We give here two examples: a) The Figure 9.16 Figure 9.15 9.3 Electric Motors All electric motors canshown in in Fig. the end be reduced to the simple scheme there is a magnetic field with the flux density plane of the page, and the conductor some means, we cause aexample current from to a flow current through source this conductor operating (for at the voltage ,see 9.15 , in particular the 8.2 Applications of Induction, in Parti 9 Sect. footnote near the end ofsection. that C9.3. For details about the velocities in Fig. 166 Part I 9.3 Electric Motors 167

Figure 9.17 Schematic of a direct-current motor Part I S

In the rotor coils of the motor, the current produces a torque in the direction of the arrow shown in Fig. 9.16 at the moment represented there. After the time T D 1= (the rotational period), the current has again exactly the same direction and strength. If the rotor is then again at exactly the same position, then the torque will again act in the required direction. We have only to start up the rotor with the correct rotational frequency; thereafter, it will continue to rotate syn- chronously with the alternating current from the generator. For a demonstration experiment, we wind a string around the axle of the motor in Fig. 9.16 and then pull it off, causing the rotor to begin turning like a child’s spinning top. The alternating current from the generator has a frequency of D 50 Hz and comes from the power grid (i.e. it is produced by a large AC generator and passed through the power transmission lines to our wall socket). In practice, there are several convenient ways of synchronizing the rotor’s motion with the current when the motor is switched on. Alternating-current synchronous motors are widely used. b) The direct-current motor is superficially similar to a DC generator. The simplified schematic of this motor is shown in Fig. 9.17. Its torque turns the rotor around its axle and pulls its coils until the plane of their windings is perpendicular to the page; then the direction of the current in the rotor coils is reversed, and so on after each half rotation. The reversals are accomplished automatically by the commutator C, which is rigidly attached to the rotor axle and acts as a switching device through its slip contacts or “brushes”. In this simple design, which is still used today, often in toys, the motor has a ‘dead point’. It will not start running if the plane of its rotor coils is perpendicular to the magnetic field. In addition, its torque is not constant during a rotation. These problems are avoided by the cylindrical rotor. We already know its principle from DC generators (Fig. 9.7). Modern DC motors practically all use this design. The fields of the stator are always produced by current-carrying coils (electromagnets). What determines the rotational frequency of the rotor? We repeat the schematic of a motor from Fig. 9.15 here in Fig. 9.18, but with two changes: First, for clarity, only the negative charges (electrons) a u B of one 2 (the also con- i U In the A A' - . U i external C U in circuit b, u a 2 R force acts on 2 a U B U V . How could we are fused into a c A i - . As a result of this 0 U C ORENTZ A . During its motion, the - C' CA 0 Electrons Current A - C and Is Is 0 C the magnetic flux density A - 0 C (which represents the rotor), there ). (that is, opposite to their velocity A c - of equal length. The two conductors which subtracts from the applied volt- is switched on, the conductor decreasing 7.3 i 0 C 2 A U - U 0 of the coils of the rotor is not large, and the i C R tor!). This causes the voltmeter to register an are connected to a voltmeter. ,i.e.by ), preferably a typical motor with a power rating i (cf. Sect. 0 , the current source can no longer deliver current i A U 2 U U 9.19 and D )). This imparts a velocity in the direction of the arrow . Then the acceleration due to the electromagnetic forces 0 The induction process in i C (Fig. 8.1 ). This variable resistor is gradually switched off during the run-up to conductor. Then we can see that the induced voltage cular to Generators and Motors . The resistance limiting speed in the direction of the arrow 3 9.19 are insulators, and the direction of , a strong ‘short-circuit current’ of many ampere flows through the is perpendicular to the page, pointing 2 In large electric motors, the windings of the coils and their power leads are in Is net voltage acting on the electrons in this conductor is additional velocity in the magnetic field, a L single occurs in the current-carrying conductor the electrons in thein direction the lower conduc induced voltage limit that U the current source attached toopposing the voltage ‘rotor’, or by reducing the induced are attached rigidly toelectrodes each other, but are electrically insulated. The of the stator. Both of these changes canexcitation be demonstrated on a motorof with 1 kilowatt. When the currentU source is switched on with its voltage “rotor of the motor”) begins(see to Eq. move ( in theto direction the of electrons the arrow in the parallel conductor When the current source Figure 9.18 the moving rotor of an( electric motor. B towards the reader). induced opposing voltage Now we suppose that the conductors rotor danger of overheating.Fig. This is prevented by using a “starter” ( 3 in the ‘rotor’ arecurrent-carrying conductor shown. Second,is we a imagine second that conductor parallel to the increase this limiting speed? Either by increasing the voltage no longer occurs, and the conductor (‘motorstant rotor’) moves with a to the ‘rotor’ Applications of Induction, in Parti 9 168 Part I 9.3 Electric Motors 169

U1 N I2 A

FC a Part I U2

S I R A I 2 1 A b

Figure 9.19 The induction process in the rotor of a direct-current motor with external excitation, known in electrical engineering as a “LEONARD circuit”. The voltage U2 of the current source can be varied in sign and magnitude in order to change the rotational frequency and direction of the motor; e.g. for driving a conveyor belt (the commutator is not shown here).

age U2 is still very small. It grows only as the rotor gains speed; then the current through the rotor windings is controlled by the reduced voltage U2 Ui, and it approaches zero with increasing rotor speed. The limit Ui D U2, when the rotor current drops to zero, can in practice never be reached; without current, the rotor can draw no more energy from the current source. It would have to continue rotating with only its stored kinetic energy. In reality, even when the rotor is not moving an external load, there are unavoidable frictional losses (and in addition JOULE heating in the windings). Therefore, even without a load, the rotor requires a certain energy input to maintain its rotational frequency. At least a small current must ﬂow through its windings. Loading the motor, e.g. having it lift a weight, or braking the output shaft by manual friction, causes the current I2 in the rotor windings to increase.

To conclude these experiments, we reduce the applied voltage U2 to a very small value; it could be supplied for example by a 2-volt storage battery. Then the rotor reaches its constant, limiting rotational frequency at a very slow rotation rate. If we now increase its frequency manually, the ammeter will show a reversal of the direction of the current I2. The voltage Ui induced in the rotor windings has then become greater than the voltage U2 of the current source. The work performed by our hand is ﬂowing as electrical energy back into the storage battery; the motor, now serving as a generator, is recharg- ing its battery! This experiment is very striking. It shows us that the technically so enormously important machines for converting electrical energy into mechanical energy are based physically solely on the LORENTZ force. In a generator, this force accelerates the electrons, producing an electric current and thereby converting mechanical work into electrical energy. In an electric motor, the same force converts the

speed of the motor, thus always keeping the current strength limited to acceptable values. ough. relative to . ı ) collect the .Sucharo- 0 b .Theyaremu- 8.8 2 9.20 J and 0 b and a (Video 9.1) 1 J (and b and a mechanical vibrations by superposition a vibrations of the same frequency. Two long leaf circular . The left-hand coil, which is just horizontal ı linear carry metal cards with slits parallel to the long axes of the ) after appropriate excitation. The diagonal of this circular b shows a shadow projection of an AC generator on the ı Production of 90 Current and O 9.21 D a cular to Generators and Motors two alternating currents. These have a phase shift of 90 one another. They are carried tocoils the right that in are the figure, mountedand to on two are magnet an split iron in yokemagnetic the field perpendicular center. is to generated. In each To their detect other common this field, center an space, “induction a rotor” rotating in the figure, appearsperspective. to be The a ends circularThe of disk spring the due contacts to two (“brushes”) the coils foreshortened are connected to slip rings. tually displaced by 90 orbit rotates like the spoke ofof a the wheel. magnetic In field a is rotating-field represented motor, the by direction the diagonal. of two perpendicular When projected, the spotare vibrating of with light the followsperiod same a ( amplitude circular and orbit with when a the phase springs lag of one-quarter springs. The overlapping opening of the two slits allows light to pass thr springs Figure left. Its rotor consists of two iron-core coils, Figure 9.20 A magnetic fieldfield’, whose was direction already is treated in rotating, some for detail short using Fig. a ‘rotating electrical energy intoelectrons mechanical and thus work. limits the current In in the addition, rotor it windings. brakes the 9.4 Three-Phase Motors for Alternating tating field can be producednating be currents. We superposing make two use phase-shifted ofSect. alter- the 4.5. general Here, scheme shown we in recall Vol. 1, the essential points in Fig. Applications of Induction, in Parti 9 Video 9.1: “Circular Vibrations” http://tiny.cc/qcggoy 170 Part I 9.4 Three-Phase Motors for Alternating Current 171

J1 J2 Part I

J1 ab a' b'

T J1

Figure 9.21 A demonstration model of a two-phase rotating-field generator and a rotating-field motor with an iron disk as rotor (compare Fig. 8.8) as shown in Fig. 8.8, e.g. in the form of a metal disk, is mounted on an axle. Its axis of rotation is perpendicular to the plane of the page. In the figure, T is the mount for the bearings of this axle. The crossed magnet coils and the induction rotor together form a rotating-field motor. Rotating-field motors are extremely important for practical applications. They can be constructed with a nearly ideal simplicity for power outputs of up to several kilowatt. They start up with a strong torque, without a starter resistor or capacitor (initially, they have a large slip (Sect. 8.3, Point 5)). Their rotational frequency is to a great extent independent of their loading. Apart from the slip, this rotational frequency is equal to the frequency of the alternating current or, with suitable modifications, to an integral fraction of its frequency. We can distinguish between single-, two- and three-phase rotating- field motors. Figure 9.21 shows a two-phase motor. It requires four lead wires and is seldom used today. A three-phase motor with so-called “three-phase current”: Imagine that in Fig. 9.21, there were three rotor winding coils J, spaced at 120ı intervals around the rotor axle. Correspondingly, in the right- hand part of Fig. 9.21, there would be three stator coils oriented at 120ı intervals around the yoke. Then with three alternating currents, shifted by 120ı on the time axis relative to each other, we would likewise obtain a rotating field or a circularly-polarized magnetic field. This would require six lead wires, but with a clever arrangement, two each can be connected in pairs so that only three wires are necessary. One can see these three wires in long-distance transmission lines.C9.4 C9.4. For a discussion of three-phase electrical power, The single-phase motor requires only two lead wires; it is fed with see for example https://en. normal alternating current. The second, phase-shifted AC current wikipedia.org/wiki/Three- which is required to generate a rotating field is produced within the phase_electric_power. motor itself by making use of some technical tricks. It must be phase shifted by 90ı relative to the first AC current. The principle of this design will be explained later in Fig. 10.13. https:// ) contains supplementary material, which The online version of this chapter ( cular to Generators and Motors Electronic supplementary material doi.org/10.1007/978-3-319-50269-4_9 is available to authorized users. Applications of Induction, in Parti 9 172 Part I .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © University). Princeton at physics of lentn Current Alternating Field. Magnetic the of Inertia The 1 10.1 Figure Self-inductance and Self-Inductance 10.1 periment o,temgei edpntae o nyteidcincoil induction the only not penetrates field magnetic the Now, ml yitrutn h urn hog h ol nue voltage a coil induces induction coil, the the in through impulse current the interrupting by ample FC ndmntaigidcinpeoea eue mn testhe others Fig. in among sketched used setup we experimental phenomena, induction demonstrating In modern a for importance electromagnetism. great of of understanding is phenomenon this of Knowledge lotefil coil field the also ntevr ol hc rdc it. produce which coils voltage very a the induces in field magnetic time-varying coil the self-inductance, field In the in voltages induce also icvrdb J by Discovered togteprmn’ h w ol ol eoeoe (Compare in one! winding become after would together coils explanation). fused two its be and The 9.18 dimensions. could Fig. same experiment’. the wires with ‘thought parallel i.e. a form, two coil Fig. the same in the Then coils on induction parallel and wound field the are that Suppose derivation: Another rdcsamgei ed hnei hsfil,cue o ex- for caused field, this in change A field. magnetic a produces h Inductance the ceai fa nuto ex- induction an of Schematic OSEPH 1 eest pca omo nuto processes. induction of form special a to refers FC H ENRY itself. olsItouto oPhysics to Introduction Pohl’s 13 awthae h ae b later who watchmaker (a ,1832 J hrfr,aycag ntefil will field the in change any Therefore, esrdeg nvl second. volt in e.g. measured , L hsi aldself-inductance. called is This . 10.1 h urn-arigcoil current-carrying The . F C , https://doi.org/10.1007/978-3-319-50269-4_10 J cm professor ecame G 10.1 J ,but S 10 173

Part I

, : 2 V 2 3 s –4 10.2 10.2 10 ). We 10.3 02 )), the voltage impulse ). 5.6 , the change in the current 250 V I 10.3 ), and second, the dimensions B Self-inductance releases energy, nductance; at left, using a voltmeter; The ends of the coil are connected (or . C10.1 H depends on H of the coil is a few tenths of a henry (V s/A)). The time L A voltage impulse due to induced in a coil depends on two factors: First, the change in t d ield. Alternating Current U right). Its filamentas just only glows the weakly battery voltage withsee is a it present; dull but flash on red brightly openingvisible light the to throughout as the switch, white we lecture long heat: hall,in and the it magnetic can field only of have the been coil stored According to the law of induction (e.g. Eq. ( R could also replace the voltmeter by a 6-volt light bulb (Fig. to a storage batteryvoltmeter initially and, indicates the in output voltage parallel,battery. of 2 a V When from rotating-coil the we voltmeter. storage magnetic interrupt field the The decays current rapidly. At bya the opening strong same impulse time, the deflection the of switch, voltmeter upbriefly shows the to a ca. much 20 higher V. value The than voltageto the thus original the attains voltage coil, that as was a applied result of its self-inductance (compare Fig. self-inductance the magnetic field, that is and form of the coil. a wire coil withmagnitude of around the 300 voltage impulses, turns the coilrectangular in is iron core wound its (‘yoke’). around a windings. closed, To increase the Figure 10.3 To demonstrate self-inductance, we use the setup shown in Fig. V . 8.12 Demonstration of the voltage impulse resulting from self-i . 14 The Inertia of the Magnetic F 10 C10.1. See also Fig. The effects of ferromagnetic materials on magnetic fields will be treated in more detail in Chap. dependence of the voltage impulse can be displayed with an oscilloscope (Fig. at the right, with a small light bulb (the inductance Figure 10.2 174 Part I 10.2 The Inertia of the Magnetic Field as a Result of Self-Inductance 175 strength during the process. We can thus writeC10.2 C10.2. As we did in Chap. 5 Z in our first discussion of in-

duction, we here initially ig- Part I U dt D L I : (10.1) nore the question of the sign in treating self-inductance. The proportionality constant L is called the inductance. We thus de- Thus, the equations in this fine the section are all to be understood as giving magnitudes Induced voltage impulse Inductance L D : (10.2) only. The signs will be dealt Current change I with in the following section. The unit of this quantity is found to be 1 V s/A, termed 1 henry (H) or, equivalently, 1 Wb/A (weber per ampere). The inductance is easy to calculate for a long, empty coil (solenoid) with a homogeneous magnetic field: We first consider the solenoid to be a field coil; it produces the field strength

NI H D : (4.1) l The change H in this ﬁeld then causes the induced voltage impulse: Z N I U dt D N A : (5.1) 0 J l

NJ is the number of turns in the windings of the inductance coil; here, it is the same as N, the number of turns in the ﬁeld coil. Then we ﬁnd Z N2A U dt D 0 I : (10.3) l

Comparison to Eq. (10.1) yields the inductance L of the solenoid, which we were seeking:

N2A L D 0 : (10.4) l

10.2 The Inertia of the Magnetic Field as a Result of Self-Inductance

In demonstrating self-inductance, we have thus far ignored the sign of the induced voltage impulse. We now take it into account, which will lead us to a deeper understanding of the phenomenon of self- inductance. We repeat the experiment as shown in Fig. 10.4. In the left-hand image, the voltmeter first indicates the output voltage of the battery (2 V) as a deflection to the left. The small fraction of the electrons which flows through the voltmeter is moving in the direction of the , H ,atleft, indicates b 10.5 ONTGOLFIER and a a . Due to its inertia, the H VV H . The current and its resulting magnetic field ex- a . The Hg manometer between the points P P . They behave in an analogous manner to a massive body The inertia of a current of water in a pipe circuit The inertia ab inertia ield. Alternating Current a pressure drop to the left,sistance corresponding (friction) to in the the direction pipes. ofswitched flow In and out the the right-hand re- of image, the the circuit pump has by been closing the valve water continues to flow for amanometer time now in shows the a direction strong ofused the deflection to arrow, to so construct the that a the right.historically technical This as application, principle the the was water “hydraulic lifting ram” device (or known “hydram”) (J. M. M 1796). we see a current ofby water the moving pump around a closed circuit of piping, driven We briefly recall an example of mechanical inertia: In Fig. Figure 10.4 of the electric current in a coil (the arrows indicate the direction of flow of the electrons) Figure 10.5 A moving body or amotion flywheel is show slowed their inertia bymotion. not braking, only but when This also their process when also they requires are initially a set finite in time. The same is true curved arrow. Inswitched off. the right-hand The large image,the impulse right the deflection on of battery its the scale.flowing has voltmeter in The just goes the current to opposite been through direction.coil the Therefore, must voltmeter also the it continue current to thus through flow now forwithout the a an time in external the current original source, directionto even and collect this at causes point negative charges in motion, or a rotating flywheel. hibit The Inertia of the Magnetic F 10 176 Part I 10.2 The Inertia of the Magnetic Field as a Result of Self-Inductance 177 of a current and its magnetic field. This can be shown by an important and striking experiment (Fig. 10.6). The voltage U is again supplied by a storage battery (2 V). The ammeter A is a rotating-coil Part I instrument with a fast response time (less than 1 s). The large coil, wound with thick wires, surrounds a closed iron yoke (cf. the scale drawing). When the switch 1 is closed, the pointer of the ammeter immediately starts to move; but it advances only slowly. After a minute, it is still creeping visibly upwards. Only after 1–1/2 minutes have the current and its magnetic field reached their full values; this demonstrates their considerable inertia. After the maximum values of the current and the field have been attained, we short-circuit the battery with switch 2 and immediately take it out of the circuit by opening switch 1. We again can observe the inertia of the current and its magnetic field. Even after a minute, “These experiments are al- the ammeter is still showing a clear deflection from zero. These ways rather surprising. We experiments are always rather surprising. We normally associate normally associate electri- electrical processes in everyday life with instantaneous, momentary cal processes in everyday or timeless phenomena. life with instantaneous, We have described this fundamental fact, the inertia of currents and momentary or timeless phe- their magnetic fields, intentionally here from a purely empirical point nomena.” of view. Retrospectively, we can see that it is a simple result of LENZ’s law: Let us take the second case as an example. There, C10.3. An exception to this we short-circuited the current source and then removed it from the rule is exhibited by supercon- circuit. In an ideal conductor, without electrical resistance, the cur- ductors. In superconducting rent would simply continue to flow indefinitely. In fact, however, the wires, induced currents best commercially-available wires have a finite resistance R,sothat can be maintained and ob- the current is consumed by quasi-frictional forces (JOULE heating, served for many years. The Sect. 1.12).C10.3 phenomenon of superconductivity is found (at sufficiently This gradual reduction of the current is the cause of the induction low temperatures) in most process. The induced voltage must therefore oppose the decrease of metals and a large number the current, according to LENZ’s law. A portion of the kinetic energy of compounds. It is charac- lost by the electrons through “friction” is replaced at the cost of the terized by the fact that below magnetic field energy, thus delaying the reduction of the current. a certain temperature TC, the electrical resistance of the material becomes zero. Figure 10.6 Switching on and 40 cm 11 cm POHL described this in earlier off of a magnetic field in a coil editions. For a basic intro- requires some time (Video 10.1) duction to the subject, see W. Buckel and R. Kleiner, “Superconductivity, Funda- 60 cm mentals and Applications”, 2nd edition, Wiley-VHC, Weinheim (2004).

Video 10.1: U “The Inertia of a Magnetic 1 Field” A http://tiny.cc/wcggoy 2 I See Comment C10.5. ) L and 10.8 .The L (10.9) (10.6) (10.5) (10.8) (10.7) ): R negative) is t 10.3 d drops across = I 0 U S : relaxation time or d 1 ); i.e. when the current RI I ; /; 1 C 8.3 0 < I Ã L U t t I 2 : I d : d L R t t 2 I L L R I ) becomes e d d . is called the D e L L he process, taking the sign found R R 1 10.7 max = U I Â L positive), the induced voltage opposes D ’s law (Sect. D 0 C t t D R D U U d d L I r = ENZ I U U D I D Z or d is the saturation value reached only after several 0 C10.4 1 is closed, the applied voltage U I max S as if they were spatially separated (although in fact I > 2 R I D ). The time . , with the usual trick that we draw the inductance The derivation of R 0, the solution of Eq. ( = 5.6 0 10.7 ) D U 10.7 0 ; we thus have U 10.7 R ield. Alternating Current is increasing ( and The signs correspond to L into Eq. ( or, in differential form, as one can readily convince oneself by substituting the solution ( current The solution of this differential equation is given by the resistance they are both properties of the same coil). After the switch relaxation times. If we now short-circuit thethat battery and remove it from the circuit, so Figure 10.7 Eq. ( For a quantitative descriptionexperimentally of correctly into t account, we rewrite Eq. ( accompanied by a voltagealso in Sect. the same direction as the current. See For further considerations, we makein use of Fig. the series circuit sketched the current, while a decreasing current ( com- in- L U U t I d d ). L D 10.4 U The Inertia of the Magnetic F D 10 L duced in the coil: U pensates the voltage (see Sect. C10.4. The voltage 178 Part I 10.2 The Inertia of the Magnetic Field as a Result of Self-Inductance 179

I I Imax

⎛ 1⎞ Part I ⎝1– e⎠ ≈ 0.63 1 e ≈ 0.37

τr τr –3 –2 0 5∙10 t0 10 s Time

Figure 10.8 The time dependence of the current accompanying the increase and decrease of a magnetic ﬁeld (Example: L D 101 Vs/A, R D 102 , 3 r D 10 s (see also Fig. 2.51)).

Equations (10.8)and(10.9) are shown graphically in Fig. 10.8.C10.5 C10.5. For the derivation of Eqns. (10.8)and(10.9), we The energy W stored in the magnetic field can be calculated from the have assumed that the coil JOULE heating within the resistor R after switching off the battery (at was empty (no iron core!). the time t ). We find (from Sect. 1.12): 0 In the experiment described Â Ã in Fig. 10.6 (Video 10.1), 2 2R dW 2 U0 t D I R D Re L : (10.10) this is however not the case. dt R In that experiment, the coil was mounted on a closed Integration gives iron yoke. As a result, the Z1 rise and subsequent fall of 2 2R U0 t 1 2 the current were delayed W D e L dt D LI I (10.11) R 2 max considerably. The strong t0 effect of the iron core in the coil is especially clear at the that is, in a coil of inductance L carrying a current I, the stored mag- end of the video, when after netic energy is reversing the current, its rise is much slower than at the 1 W D LI2 (10.12) beginning of the experiment. 2 The reason for this is the .e.g. W in W s; L in V s/A; I in A/: reversal of the magnetization of the iron (see Chap. 14, Making use of Eqns. (10.4)and(4.1), we obtain from this the expres- especially Fig. 14.7). sion for the energy stored in a magnetic field of volume V that we had previously found in Chap. 8: 0 2 1 2 Wmagn D H V D B V : (8.28) 2 20

The inertia of the magnetic ﬁeld plays a decisive role in all applications of electric currents where their strengths and directions vary. Qualitatively, we can demonstrate two signiﬁcant effects by making use of a periodically interrupted direct current (DC). In Fig. 10.9,the current from a 2-volt storage battery divides along two branches of the circuit, each with a light bulb. The left-hand branch contains in addition a coil with an iron core, while the right-hand branch has only S S ). R ):

(10.13) frequency R D : t T ! = analysis, discussed in sin ,1 . Alternating currents of 0 T U , and it increases with in- L D OURIER U and t ! , when the switch is opened, the “sluggish” sin inductive resistance 0 I 10.9 C10.6 D I inductive reactance A demonstration of induc- A Quantitative Discussion ield. Alternating Current Vol. 1, Sect. 11.3, can bealternating applied currents. quite generally to the descriptionFor of sinusoidal alternating currents and voltages, we have Figure 10.9 tive resistance or using a periodically interrupted directcurrent 1. With a lowbut frequency, the both left-hand lamps lamp still is showright-hand delayed one. the by same Its about current one brightness, lags seconda behind relative second the to voltage. to the build It up requires nearly its magnetic2. field. As the frequencythe is magnetic increased, field the to time reach is itsdimmer full no and value. longer dimmer. The sufficient At left-hand for frequencies lampThus, above becomes the 1 Hz, coil it has stays an quite dark. is periodically opened and closed, theswitch is situation is briefly quite opened at different (the time intervals of With a constant currentboth strength, light both bulbs branches glow are with the equivalent and same brightness. When the switch 10.3 Alternating Current: For a quantitative treatment ofthe alternating simplest form, current i.e. (AC), a we sinusoidalmore waveform choose complex forms can always besoidal constructed waveforms. by superposing The sinu- formalism of F In these experiments with periodically-interruptedof direct the current, energy all provided by thefield current source is to lost. build up In the Fig. magnetic a short piece of wire with the same resistance as the coil (ca. 0.3 current in the coilnetic flows field through energy both there lamps intoand and heat. the converts right-hand the (Without lamp, the mag- thiselectric wire would arc of occur between resistance instead the inturn switch the to sinusoidal form contacts (“alternating”) of (switch currents. an arcing)!). We now creasing frequency. , low-pass filter The Inertia of the Magnetic F 10 C10.6. This is a modela of so-called which allows only low frequencies to pass through. 180 Part I 10.3 Alternating Current: A Quantitative Discussion 181

Figure 10.10 The deﬁnition of the effective I I current of a sinusoidal alternating current 0 Part I

T Area A= I2dt 0 2 2 Ieff Ieff T

Ieff 0.7 I0 0 Time tT

Here, I and U are the momentary values (at time t)ofthecurrentand the voltage, I0 and U0 are their amplitudes, that is their maximum values, and ! D 2 is their circular frequency (cf. Vol. 1, Sect. 4.3). The momentary values of the current and the voltage can be observed and measured using instruments with a sufﬁciently short response time, for example oscilloscopes. In general, one does not measure or quote these momentary values, but rather the effective values, as their time-averaged values are called. The time-averaged value over a sine function would of course simply be equal to zero. Therefore, one calculates the average values of the squares of the time functions and deﬁnes their effective values by means of the equations

2 3 = 2 3 = ZT 1 2 ZT 1 2 1 1 I D 4 I2 dt5 and U D 4 U2 dt5 (10.14) eff T eff T 0 0 .T D period D 1= /:

They are illustrated in Fig. 10.10. These effective values are also called “root-mean-square” (rms) values after their calculation procedure (Eq. (10.14)). For sinusoidal currents and voltages, one ﬁnds

I0 U0 Ieff D p and Ueff D p : (10.15) 2 2

The effective values of the current and the voltage are thus proportional to their amplitudes. This definition corresponds to the values of the direct current and voltage which would give the same average JOULE heating in an OHMic resistor. ic ic HM leads HM ’s law 0 (10.18) (10.17) (10.16) ; L HM U =+0° =+90° =+71° φ φ φ /: ), we find is required to com- ı L ) takes the form 90 U L 10.6 and voltage would have : C t I t d d 10.16 2 ) is sufficient to maintain the L .! 0 I IR voltage D sin s). The top curve shows the overall D 0 3 D I L ind on the time axis. The two voltage R ! .FromEq.( ! U 10 U ı L 0 L R I ind U U U 2 : D U : In the upper part of the circuit, only the D D 9 has the same phase as the current. (Numerical t inductive 0 R L by 90 ! ; V V D L U U 0 I R 10.11 U cos = L 0 ic voltage” nce-free cable”, current I ! HM L V 50 Hz, D A series circuit consisting of a conductor with only O . The voltage L , an additional Circuits . D U / R t U along the conductors.

. U I U 10.12 ic resistance determines the voltage (middle curve), while in the ield. Alternating Current HM O (for short the “O resistance (above and middle curve)(below and and a bottom coil curve). with Theto only amplitude Fig. inductive ratios resistance and phase angles are similar pensate the induced voltage the current amplitude voltage example: with the signiﬁcant new result that the voltage amplitude voltage Then the amplitude of the inductive voltage is given by lower part (bottom curve), only the self-inductance acts. For a sinusoidal alternating current, Eq. ( In this case,for one example draws as in the Fig. coil in a circuit diagram in two parts, However, it is frequently notof possible the to conductor in neglect alternating-current the circuits,contain self-inductance especially coils. when Then, they to maintain the current, in addition to the O current Figure 10.11 10.4 Coils in Alternating-Current The effects of self-inductance in a single cableIn can such often be an neglected. “inducta the same phase. At every moment, the voltage as given by O The Inertia of the Magnetic F 10 182 Part I 10.4 Coils in Alternating-Current Circuits 183 amplitudes required to maintain the current amplitude I0,i.e.UR;0 and UL;0, must thus be combined to give a resultant voltage amplitude C10.7 U0. C10.7. For this “com- Part I bination” of the voltage This can be represented graphically in a so-called phasor diagram or amplitudes, the func- vector diagram as in Fig. 10.12. For the amplitude of the resultant tions U D U ; sin !t voltage, we ﬁnd R R 0 and U D U ; cos !t p L L 0 (Eq. (10.17)) must be added. U D I R2 C .!L/2 : (10.19) 0 0 The result is again a harmonic function of the same It leads the current amplitude I0 in time by the phase angle '.The phase angle is given by period, but shifted by a phase angle ', i.e. it leads the func- !L tion UR (Eq. (10.20)). The tan ' D : (10.20) R resulting amplitude can be found in Eq. (10.19). These The quotient results can be most clearly shown in a so-called phasor U p 0 D Z D R2 C .!L/2 (10.21) diagram (or vector diagram) I0 as in Fig. 10.12.Thishas the advantage that other is called the overall AC resistance or impedance Z of the circuit. It is formulas for AC resistances not a constant for alternating currents, but rather it increases with the can be most simply derived AC frequency . mathematically. When the product !L is large, the impedance Z can be orders of magnitude greater than the constant OHMic DC resistance R,asmay be seen from Eq. (10.21). In such cases, R can be neglected in Eq. (10.21) relative to !L. Then, only the inductive or AC resistance (also referred to as the inductive reactance) remains:

U ; L 0 D !L : (10.22) I0 To summarize: The voltage C10.8. In a phasor diagram, U D U sin !t (10.23) 0 the amplitudes of the sinusoidal voltages (or currents) are represented by the lengths

Figure 10.12 The calculation of the AC resistance or 2 of the lines (vectors). ' is impedance from a “phasor diagram”C10.8 ) ωL the phase angle relative to

+( 2 a quantity which is the same R

√ in all parts of the AC cir- 0 I

= cuit. In Fig. 10.12,thisis

2 L,0 the current, and ' is the an- U ωL

0 + I gle deﬁned by Eq. (10.24); 2 = R,0 U

L,0 here, it thus indicates by how √ U = much the voltage is leading 0 U the current. For calculations, ωL the lines or ‘phasors’ in the tanφ= R φ diagrams can be treated like vectors. See also Comment UR,0=I0R C10.10. , ) Al- 9.4 50 Hz) (10.25) (10.24) D /:

, the phase angle rotating magnetic , but now, the current is produced (Sect. 10.11 ı '/ t , we exchange the coil for of 90 mechanical frequency ! D t cular to each other. One of these sin ; 10.13 is an AC current source, .! 0 ). Second: Between the current and U sin 0 6.4 D I phase shift U are incandescent lamps used as ballast resistors D F). We again observe a 2 . I 5 R rotating magnetic field ) (Sect. . The current from an AC current source ( 10 and circular frequency C10.9 1 between the momentary values of alternating cur- is not blocked by a condenser; it passes through as D R . We can draw two conclusions from this. First: 10.13 . In the circuit shown in Fig. ' Z A. C = ) and Fig. 10.12) is positive. 2 1 Demonstration of a phase shift by producing a rotating field 0 1 2 R R D Circuits U 10 ). A metal disk rotates in this field as rotor. (A practi- .! 10.20 D 0 I 9.21 , but its sense(Video 10.2) of rotation is opposite to that observed with the ield. Alternating Current (Eq. ( displacement current leads the voltage. For a quantitative treatment, weage assume a sinusoidal alternating volt- Fig. cal application of this principle is“split-pole found motors” in the widely-used so-called field coil ternating current a the voltage, we again find a instead of rheostats and ammeters, (Video 10.2) with rent and voltage isexperiments. a A favorite convenient subject setupis for for shown fascinating in demonstrating the Fig. demonstration is phase divided shift equally betweenof two coils; branches these are which mounted eachcurrent perpendi branches contain also contains a a coil pair sult with of a the large phase inductance. shift, As a a re- ' (current In the experimenta shown condenser in ( Fig. 10.5 Condensers in Alternating-Current Figure 10.13 produces a current The phase shift B F, see ic resistor 10 HM D C ) is inserted instead 10.5 The Inertia of the Magnetic F 1 H), the disk begins 10 The different impedances explain the different rotational frequencies that are observed in the two experiments. The incandescent lamps used as ballast resistors are of limited usefulness as ammeters. of the coil, the disk rotates, like the magnetic field, in a counter-clockwise sense. “Rotating Magnetic Field” http://tiny.cc/8cggoy First, a “null experiment” is shown: An O Video 10.2: Sect. field rotates in a clockwise sense. If a condenser (capacitance tor is then exchanged for a coil or choke (inductance L to rotate to the right: The in place of the coilduces pro- no motion of the iron disk. When the resis- results. (“shaded poles”). This delays changes in the magnetic flux. A rotating magnetic field C10.9. The split poles of an AC magnet each carry a copper ring on one side 184 Part I 10.6 Coils and Condensers in Series in Alternating-Current Circuits 185

Suppose that the condenser has a capacitance of C (and thus when a voltage U is applied to it, it carries a charge of Q D CU). Then at every time t, we ﬁnd for the charging or discharging current Part I

dQ dU I D D C : (10.26) dt dt ı In this equation, dU=dt D !U0 cos !t D !U0 sin.!t C 90 /,and thus

ı ı I D C!U0 sin.!t C 90 /; .' D90 ; see Eq. (10.24)/: (10.27)

Therefore, for the amplitude of the current which is passing through the condenser (the displacement current), we have

I0 D !CU0 D 2 CU0 ; (10.28) but with an important additional fact: The current leads the voltage by 90ı (compare later to the schematic in Fig. 10.15). The quotient

U 1 0 D (10.29) I0 !C is the capacitive or AC resistance of the condenser, also called the capacitive reactance.

5 4 Numerical example: D 50 Hz, C D 10 F, U0=I0 D 3:2 10 .

10.6 Coils and Condensers in Series in Alternating-Current Circuits

For a conductor (resistor) which has only OHMic resistance, we have:

UR;0 D I0R ; (1.2) and for a conductor with only inductive resistance,

UL;0 D I0!L ; (10.18) while for a condenser through which a displacement current I0 is ﬂowing, we ﬁnd

I0 U ; D (10.29) C 0 !C .the index 0 again denotes the amplitudes/: , 0 .) L ), /

t ! . volt- C U 10.15 10.14 (10.30) (10.32) (10.31) .' Hz. Note ic resistor. 3 10 HM D =+53° =+0° =+90° =–90° φ φ φ φ RC = .Agraphof 2 ' Ã s, 1 across the condenser, it 2 C according to Fig. 1 C ! ' 10 U C : 1 , it is phase shifted by the ! 0 and the capacitive resistance I 44 L : R LC and the inductive resistance 1 ! L 1 , there is a special frequency L p Â ! R D . For this reason, one often speaks C ! 0 C R 2 U = R L C 2 D L and U U U U c). R D . We can ﬁnd ' / 0 L s ı

0 have larger amplitudes and effective values V I tan V V 11.18 90 ic resistor : 50 Hz, 0 C D C U U D HM 0

.' U has the same phase as the current in this series circuit. In b a and ) for the deﬁnition of the phase angle R / V L U ; and relative to the voltage .TheO A series circuit with a condenser, a coil and an O U ' ). Relative to the current 10.24 U resonance frequency C become equal. When we set these two quantities equal, we R is given later in Fig. 10.16 / U / 10.15 C ı . ield. Alternating Current 90 (see Eq. ( =.! (Fig. that the voltage are combined in aages single coil (with an iron core). The partial is phase shifted by than the total (applied) voltage Experimentally, this “resonance of ain series Fig. circuit” is demonstrated phase angle (Numerical example: the example shown, the current lags by 53° behind the applied voltage right side: For each pair ofat values which of the inductive resistance Figure 10.14 When these three components are connected in series (Fig. the three voltages just mentionedgive add the with total their voltage respective phases to 1 obtain the The Inertia of the Magnetic F 10 186 Part I 10.6 Coils and Condensers in Series in Alternating-Current Circuits 187

Figure 10.15 The calculation of the AC resistance of the series circuit in

Fig. 10.14 Part I

⎞ ) ⎠

ωC ) ⎞ ⎠ L

ω ⎛ 1 ( ⎝ ωC +

– 2 ωL R ∙ 0 ωL

I √ ⎛ ⎝ ( 0 0

= I I = =

L,0 0

U U C,0 U – L,0

U δ φ 0 0

I UR,0= I0R I UR,0= I0R ωC ωC = = C,0 C,0 U U

Figure 10.16 An example of voltage resonance EE in a series circuit ( 0 D 500 Hz; the current source (UL + UR)eff UC, eff is an AC power supply = 300 V = 300 V with variable frequency (signal generator); the coil has a closed iron yoke, and L 37 H, R D 1:1 104 . A rotary variable condenser is used to adjust the resonance frequency, with 9 E Cmax 3 10 F. Static voltmeters (electrometers Ueff = 300 V E) are used (Sect. 1.6)). v = 500 Hz

Figure 10.17 The AC resistance Ohm of the series circuit as a function 2∙105 of the AC frequency (Eq. (10.30)). (The experimental data are those in Fig. 10.16,and is the logarithmic Λ ≈ 0.3 1∙105 decrement (Vol. 1, Sect. 11.10); see = AC resistance eff also Sect. 11.7). eff I U DC resistance = 0 0 I U 0 500 1000 Hz Frequency v of a voltage resonance. At resonance in a series circuit, the overall resistance U0=I0 has its minimum value (Fig. 10.17).

In every electrical circuit, due to its OHMic resistances R, electrical energy is converted into heat. The power consumed in this way is WP D I2R. Other losses can also occur, especially in iron-containing coils due to eddy currents and magnetization reversals (Sect. 14.4). All the losses, i.e. the . . C ı V U 90 10.19 (10.33) (10.34) (10.35) a b D ' . Therefore, 0 I and the induc- . In the ideal 2 would be phase : R I , which is larger 0 C ; V V : /= R U R L 2 P LC W 1 / U U RC ), we have C p 1 and P ! ! . V . (its sign is as defined in / 2 R / I D U D U L 0 / L !1 , and at resonance, both would 0 C ) only in the special case that ı ). Then we obtain quite dif- R ic resistance .! and L LC ! .! from that of the series circuit 2 U C U ;! . 0 HM ! C 2 10.32 U 10.18 2 C R 2 L R 1 R p )), by adding together the partial cur- ic . 0, we have q L C R HM flowing through the coil and the displace- D between current resonance ! 1 ! L 10.32 0 ' I ' r I D 0 ! ' D 0 0, the two voltages are again located in a single coil. The two partial D tan U D L ! passing through the condenser can have much greater 0 ! R C10.10 C ) through ( I A coil and an O 1 (which is however often fulfilled). Resonance can be in Alternating-Current Circuits )), , i.e. the current 10.30 L 10.24 = ield. Alternating Current total power lost, is attributed to an effective resistance shifted relative to each other by exactly 180 limiting case than the DC resistance. We thus define increase without limit (resonance catastrophe!). C 2 and for the phase angle Eq. ( At resonance, that is for In the limit of very high frequencies ( This equation is the same as Eq. ( one often speaks here of a ment current amplitudes and effective values than the overall current tive resistance currents (Eqns. ( resistor in series are connectedlel in with paral- a condenser. Themeasured voltages as are effective values, oras else momentary values with an oscilloscope. R Figure 10.18 10.7 Coils and Condensers in Parallel The resistor, the coilgether and in the a condenser parallel can circuit also (Fig. be connected to- ferent results for the amplitude rents. We find Here, as in the series circuit, the O demonstrated experimentally with the circuit shown in Fig. ) .An 10.33 . currents ) can again be 10.34 The Inertia of the Magnetic F 10 and ( carried out with the aid of a phasor diagram, this time however for the amplitudes of the equivalent method is the complex-number formalism. A compact introduction is given for example at http://www.physics.byu.edu/ faculty/peatross/homework/ complex145.pdf C10.10. The mathematical derivation of Eqns. ( 188 Part I 10.8 Power in Alternating-Current Circuits 189

Figure 10.19 An example of a current a resonance in a parallel circuit ( D

50 Hz; a technical paper condenser was Part I used, with C D 3:7 106 F; R D 38 , L D 2:7 H. (The zig-zag symbol used U V here refers to a coil which also has an A A IL IC OHMic resistance.) (Exercises 10.10, I 10.12). A b

Figure 10.20 The AC resistance 2∙104 of a parallel circuit as a function Ohm of the AC frequency (Eq. (10.33)). (The experimental data are from Fig. 10.19; is the logarithmic 1∙104 decrement (Vol. 1, Sect. 11.10); see Λ ≈ 0.14 = AC resistance

also Sect. 11.7)(Exercise 10.11). eff eff I U = 0 0 I U 0255075Hz Frequency v

As a result of the losses mentioned in the ﬁne print in Sect. 10.6, the current amplitude I0 may become very small, but never zero. The phase difference ı between IL and IC can approach the value 180 arbitrarily closely, but never reaches it.

At resonance, the overall resistance U0=I0 D Ueff=Ieff of the parallel circuit has its maximum value (Fig. 10.20) (leading to the name ‘rejection circuit’).

10.8 Power in Alternating-Current Circuits

For the power WP of any electric current, we have found WP D IU.In the case of alternating current, both I and U are periodic functions of the time. Furthermore, in general there is a phase difference ' between them. In the simplest case, i.e. with sinusoidal alternating currents and voltages, Eqns. (10.23)and(10.24) apply, that is, the power is given by P W D IU D I0U0 sin !t sin.!t '/; (10.36) or, after an elementary rearrangement, 1 WP D I U Œcos ' cos.2!t '/: (10.37) 2 0 0 In words: The power WP of an AC circuit consists of two parts: the 1 ' D ' first, 2 I0U0 cos IeffUeff cos , is constant over time. The second varies periodically with time, at the circular frequency 2!. ” The lend (10.38) )). The coil, has or current C10.11 . , and the idle ' 10.18 ' passes not only sin I primary 0 cos I 0 I ' transformers 1 tan D . During the first quarter of an turns of that coil’s windings, the L s ld is produced in the coil. During N ), with the amplitude , with the amplitude of this circuit. An AC generator must be ic resistance is negligible. Then the inductive is induced. With the same magnetic field in Active current acts between its terminals (Eq. ( ). One of them, the field or Reactive current s L HM U ! coil, and in the 0 I 10.21 a constant active power and simultaneously to “ active current loss factor D A current transformer for is given back to the AC current source. During the third reactive current 0 2 (Chokes) 0 ; L LI U 1 2 produce secondary turns in its windings. Its ends are connected to the AC current ield. Alternating Current p is called the able to a reactive power during every seconddoes quarter-period. not produce Reactive losses power in this“operating view, capital”. but it requires a large amount of ratio current (or producing large currents part leads to zero power summed over theIt full is oscillation period. therefore usualknown as to the distinguish two components of the current, and fourth quarter-periods, the sameopposite cycle signs repeats for itself the but with current the and the magnetic field. This second both coils and no-load operation, according to the law of induction voltage secondary voltage Figure 10.21 10.9 Transformers and Inductances Knowledge of the self-inductance asus an a understanding of form the of importantand inertia topic voltage of opens converters up for alternating for current. A transformer consists of two coilsnetic which field encompass the (Fig. same mag- N Here is an examplecuit which contains should a already coiloscillation of be period, inductance familiar: a magnetic Anthe fie AC second cir- quarter-period, theenergy magnetic field again decays, and its source. Its DC or O magnetic field which is associatedthrough with the the primary current coil, buttion or also through the other coil, the induc- . 13.11 The Inertia of the Magnetic F relative to the voltage; ı 10 cation is described also in Sect. the active current is in phase with the voltage. An appli- 90 C10.11. The reactive (idle) current has a phase shift of 190 Part I 10.9 Transformers and Inductances (Chokes) 191

Figure 10.22 An SS induction coil Part I

A C

the two voltages have the same ratio as the numbers of turns of the coils, i.e. the transformer equation holds (Exercise 10.13):

Us;0=Up;0 D Ns=Np : (10.39)

Therefore, by the proper choice of Ns=Np, i.e. by choosing the transfer ratio, one can obtain almost any arbitrary increase or decrease of the voltage amplitudes. Transformers providing many hundred kV are operated today for a variety of research and technical purposes. Their main application, however, is in today’s long-distance power transmission, which would be unthinkable without multiple voltage conversions. The consumer should receive voltages of at most a few hundred volts; apart from gross negligence, such voltages are not dangerous to life. The long-distance transmission lines, in contrast, must transport the electrical energy at high voltages and relatively low currents (e.g. 104 kW at 105 V and 102 A). Otherwise, the cross-sections of the wires would have to be much too large and the transmission lines would therefore be too heavy and too expensive. Voltage reduction in the secondary circuit is accompanied by an increase in current strength. Low-voltage transformers are constructed with only a few turns in their secondary windings (e.g. 2 as shown in Fig. 10.21). They can be used to routinely provide currents of several thousand ampere for demonstration experiments. Technologically, this principle is used to construct induction furnaces for melting steel etc. The secondary circuit in this case consists of only a single loop. It can take the form of a circular trough lined with high-temperature resistant stone. The metal to be melted is placed in the trough. The induced currents may reach several tens of kA. A special type of transformers are called “induction coils”. Their primary coils (terminals A and C in Fig. 10.22) and secondary coils (terminals S) are coaxial and contain an iron core (not as a closed magnetic circuit); an example is shown in Fig. 10.22. Figure 10.23 shows a spark discharge from such a coil.

In the usual AC transformers, the necessary periodic variation of the magnetic field is produced by the AC current in the primary circuit. Induction coils instead operate with a periodically interrupted direct current.The periodic interruption can be provided by one of the numerous automatic switches developed for the purpose. The simplest types make use of toggle 40 cm . Deter- ) 1 D L l 50 H. How 10.1 D L . (Sect. I , which destroys the gas Co 4300, length Co D N of one of the coils used in ) so that the current reaches one- R L Pt 10.1 100 V 100 has an inductance of Pb similar rings were collected into a bundle N R 10.7 . A positive electrode made of a 1 mm thick and ): if in Fig. 10.22). The coil is equipped with a mechan- N S ’s hammer or vibrator, using the principle of a door- L 11 cm. (Sect. 10.24 D EHNELT r AGNER The spark discharge from an induction coil, about 40 cm long The production of toggle switching using an electrolytic inter- . As an example, we mention the arrangement described in Vol. 1, A wire is bent into a circular ring. Its inductance is The coil in Fig. Calculate the inductance ield. Alternating Current bell). A notable development ismoving the parts production (Fig. of toggle switching without action Fig. 11.5 (W . layer, etc. Thetance) frequency can be of adjusted the overR toggle a wide switching frequency (with range a by varying fixed the coil resistance induc- and diameter 2 10.2 and all of them were carrying the same current large must one make the resistance 10.3 mine the inductance rupter, after A. W and surrounds itself withflow. an This insulating induces layer a voltage of impulse gas, in interrupting the the coil current 10 mm long platinum wiresulfuric is acid mounted solution. at When the a end current of flows, a the glass wire nozzle is in heated a until dilute it glows ical interrupter, and the exposure time for this photo was 1 second. Exercises 10.1 Video 6.1 asare: a Number of “magnetic turns in voltage the meter”. windings Their characteristics Figure 10.24 Figure 10.23 between the terminals ( The Inertia of the Magnetic F 10 192 Part I Exercises 193 half of its maximum value in a time of 10 s after closing the switch? (Sect. 10.2) Part I 10.4 An AC voltage with a frequency of D 100 Hz and an effective value of Ueff D 150 V is connected to a coil with the inductance L D 0:3 H. Find the effective value of the resulting current Ieff. (Sect. 10.4)

10.5 An AC generator is connected to a series circuit with a resistor of R D 13:7 and a coil of inductance L D 50 mH. The effective value of the current is found to be Ieff D 12 A. Determine the frequency of the alternating current. (Sect. 10.4)

10.6 An AC generator operating at a frequency of D 50 Hz and producing a voltage of effective value Ueff D 80 V is connected to a circuit with a resistance of R D 10 . In order to limit the current to an effective value of Ieff D 2 A, a coil of inductance L is connected in series with the resistor. Find the required value of L. (Sect. 10.4)

10.7 In an AC circuit, a resistor with the value R D 10 and a coil of inductance L D 100 mH are connected in series. What value must the frequency have, in order that an increase in L by 1 % would lead to an increase in the impedance by the same amount as a decrease in R by 50 % would decrease it? (Sect. 10.4)

10.8 An AC current of frequency D 50 Hz is ﬂowing through a copper wire with an OHMic resistance of R D 5 and a negligible inductance. By winding it into a coil, the impedance of the wire is to be increased tenfold. What inductance L must the resulting coil have? (Sect. 10.4)

10.9 A coil of inductance L and an OHMic resistance R is connected in series with a condenser of capacitance C (see Fig. 10.14). What is the relation between L and C that must hold in order for the impedance Z of the circuit to be equal to R, i.e. the current and voltage in the circuit are in phase? The frequency is D 1 kHz. (Sect. 10.6)

10.10 In this and the following two exercises, we investigate the experiment described in Figs. 10.19 and 10.20. The AC current source which is connected to the parallel circuit has an output voltage of U D U0 sin !t with U0 D 73 V. The frequency and the values of C, L and R are given in Fig. 10.19. a) Find the voltages UL;0 and UR;0 in the left-hand branch of the circuit, and use a phasor diagram (Fig. 10.12) to ﬁnd the impedance ZRL and the phase angle '1 between the current IRL and the voltage U. The deﬁnition of ' is given in Eqns. (10.23)and(10.24). b) Determine the current amplitudes IRL;0 and IC;0 in both branches of the parallel circuit. Draw these amplitudes and their phase angles relative to the applied voltage amplitude U0 in a phasor diagram (the vector representing U0 can be drawn in pointing to the right and par- . t ? d R P W turns .The to the https:// 10.20 p p and its R l 2 / N 0 ' I T by making = ) 1 0 ; . p U 10.7 ic resistance D , derive the trans- ) P HM sin 15°, the compo- W 5.5 0 I ) ) 10.9 ). (Sect. , and its length is p 10.9 10.8 A 10.34 (sect. 11.7) of the parallel circuit ic resistance in the primary circuit ) contains supplementary material, which

The online version of this chapter ( )and( HM to the primary voltage . 0 ; 0 s 10.33 U U found in Exercise 10.10, find the active cur- ). An AC current source with output voltage 0 I , and the reactive current is connected to the primary coil, and an AC volt- turns. The O 10.39 U t s relative to ! N 2 in the parallel circuit. (Sect. , and compare the result to the value given in Fig. ' cos ) 0 delivered by the current source and compare it to the loss ; RLC p P P 10.19 W U W cos 15°, that is the component of the current which is in phase 10.7 0 I D ield. Alternating Current p 10.14 a) Find the time-averaged value of the power use of the lawprimary of coil induction and (since knowledge the ofhere two the the voltages ratio inductance have of of their opposite the magnitudes). signs, (Sect. we mean values found using Eqns. ( A qualitative answer will suffice here. (Sect. phase angle in its windings, its cross-sectional area is former equation ( U c) Compare the impedance found above and the phase angle can be neglected, and that ofof the the voltmeter secondary is voltage infinite. Derive the ratio secondary coil, which surrounds thewindings primary with coil on its outside, has power nent which is out ofpower phase with the voltage by 90°. b) Determine the 10.11 Find the logarithmic decrement in Fig. which is delivered by the current source inb) Exercise 10.13. How does thiscircuit power is change replaced if by the a voltmeter resistor in with the a secondary finite O with the voltage rent 10.13 For a pair of coils as shown on the left in Fig. Electronic supplementary material doi.org/10.1007/978-3-319-50269-4_10 is available to authorized users. allel to the x-axis). From this diagram, find the amplitude meter is attached to the secondary coil. The primary coil has 10.12 a) From the value of (Sect. The Inertia of the Magnetic F 10 194 Part I ahfiue,w a eetteeprmnssoni Figs. in shown experiments the repeat can in and right we the (at figure), circuits” “tank each so-called containing setups, these With et hsyed nA otg ewe h terminals the between voltage AC an yields This rent. value ut nbt fteefiue,the figures, these of both In cuit. nA eeao a efbiae rmaD urn oreand source current DC Fig. a from resistor. fabricated variable be a can generator AC An ytesymbol the by soscillators, as uthv au fteodro 100 of order the of value a have must eaon 10 around be aiberssos while resistors, variable apdoclain (for oscillations damped lcrclOscillations Electrical iue11.1 Figure Fig. In Oscillations Electrical Free 11.2 In as circuits circuits”). these (“tank identify coil will includ- we a currents, chapter, and present alternating condenser the with a dealt containing circuits we ing chapter, previous the In Remark Preliminary 11.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © netgt hi properties. their investigate hl ntergt rud10Vi required). is V 100 around right, the on while 10.18 11.1 U 10.14 = RR I eidcvrain faresistance a of variations Periodic : rdc natraigcret uepsdo ietcur- direct a on superposed current, alternating an produce E tvr o rqece ( frequencies low very at eisL ici n aallL ici “akcircuits”) (“tank circuit LC parallel a and circuit LC series A 4 eswasre ici,adi Fig. in and circuit, series a saw we , ahcnetdt nA generator. AC an to connected each b a h urn orea h eti trg atr 2V), (2 battery storage a is left the at source current The . . hsi copihduigteshm hw in shown scheme the using accomplished is This R A

saﬁe eitr h wthsaeue oproduce to used are switches The resistor. ﬁxed a is olsItouto oPhysics to Introduction Pohl’s D z h resistance the Hz, 1 D Cgenerator AC ,and

D fodr1H) erequire We Hz). 1 order of ntergthn mg,must image, right-hand the in , E E E a ipyindicated simply was or and ntelf-adimage left-hand the in 10.18 , https://doi.org/10.1007/978-3-319-50269-4_11 D D rudamean a around oscillators a r periodically are aallcir- parallel a b a and b . A 10.14 and 11 195

Part I ) in ,is / /; L ). con- ) D )), is C ! 10.32 D ( (where 10.20 =.! 3.20 10.35 b and : E and resonance fre- (Eq. ( ). Then the en- a 10.6 see Eq. ( 2 I 1 CU 10.32 1 2 L H) and a condenser with = ), we would have to make D 3 C 2 hout losses at a constant am- e R 10 W opposite directions, depending 11.1 LC 1 p s), or with an oscilloscope. For L onnecting rod driven by a motor. nt to produce damped oscillations 2 F). With the switch closed, a sliding . has a low resistance, while at the right, D E 0

this limiting case, there is a clear-cut mechan- up to 50 of the circuit between the terminals ). Today, it can be found in most school books on ), while it is high for the parallel circuit (Fig. 1Hz. C I = 11.2 U amplitude; or, put differently, we observe damped 10.17

oscillator circuits Therefore, both series circuits and parallel circuits are , at left, the resistor 11.1 much smaller, as can seen from Eq. ( which we have already encountered in Sect. decaying C in the parallel circuit, this holds only when it is attached to therather generator). low The (Fig. resistance of the series circuit may be has a large value.to This the matches resistance the internal resistance of the “generator” physics and needs no further explanation. ical analog (Fig. . The oscillations start withon deflections whether in the switch iselectrical oscillator closed circuit would or oscillate opened. wit plitude In following the an ideal impulse limiting excitation.circuits case, are an In equivalent. this For limit, series and parallel In Fig. and a large capacitance ( stant and simply openalternating or current close in the switch. bothwith the Then a series we circuit can and observe the an parallel circuit, oscillations. oscillators. Theyergy each (the contain condenser) a andor storage for opening device magnetic the for energy switch(i.e. alternating (the electrical is currents coil). sufficie en- with decayingtation amplitudes) in Closing by these impulse exci- contact on the resistorsby is hand moved or periodically using an back eccentric and and forth, c either just as large as the capacitive resistance of the condenser, 1 i.e. the frequency at which the inductive resistance of the coil, The frequency of these electrical oscillations is the The AC current flowinglines), through which the has coil a and constantmeter the amplitude, of can condenser short be (heavy response observed with time an (0.12 am- quency a coil with a very large inductance ( both the series circuit andplitudes the around parallel circuit, we find the largest am- Then we see something new: We leave the resistances the series circuit at resonance. In order to obtainpulse electrical excitation of oscillations a series of circuit (Fig. higher frequency by im- L ergy initially stored in the condenser, ). 11.2 Electrical Oscillations 11 Today, it can be found in most school books on physics and needs no further explanation”. “For electrical oscillations, there is a clear-cut mechanical analog (Fig. 196 Part I 11.2 Free Electrical Oscillations 197

Figure 11.2 The periodic exchange of potential and kinetic 1 Part I energy in mechanical oscillations (left), and of electrical and 2 magnetic energy in electrical oscillations (electrical oscilla- 3 tor circuit, right). The arrows on the right mark the direction of motion of the 4 electrons within the circuit. 5

also rather small. As a result, we would then have to use higher voltages; that however causes an annoying problem: At high voltages, an arc jumps across the gap between the switch contacts even before the switch is closed. One has to live with these disturbing sparks.C11.1 C11.1. Using a modern pulse But they can be utilized in a useful way, namely: generator and a sensitive os- 1. as a periodically-operating automatic switching device, and cilloscope, such sparks can easily be avoided. Neverthe- 2. as an ammeter with a very short response time. less, the example that follows The spark acts as an automatic switch for example in Fig. 11.3.In- here is very useful, since it il- stead of a moving and a ﬁxed contact, we see there a spark gap lustrates the early precursors consisting of two metal balls. Two thin wires serve to charge the con- of “wireless telegraphy”, of denser from some sort of current source, e.g. an inﬂuence machine. enormous importance today When a certain maximum voltage U is reached, the spark jumps over in its modern forms. the gap and starts the oscillator circuit. This operating voltage U can be adjusted by changing the distance between the balls of the spark gap. The spark can also be employed as an ammeter with very short response time, owing to the dependence of the light density that it emits

Figure 11.3 A spark gap as the switch for an electrical series circuit as in Fig. 11.1. The resistors R and E have been removed, and only the switch remains. .As L ! V between D 4 0 I The image is the = 3. At high fre- 0 U C11.2 p N , photographs of sparks C11.3 ). 11.4 spark”, 1859. ALTER of a high-frequency electrical oscillator EDDERSEN rge inductive resistance of the secondary-coil windings is consider- several hundred turns. Therefore, voltages of Co s N s; a “F s 3 N ). Often, one end of the secondary or induction coil coil. 11.5 ,thecoil The detection of electrical oscillations from a spark (its duration V can be readily produced at the ends of the secondary Currents for Demonstration Experiments 5 . These are discussed in the following section. ESLA 11.5 its ends without increasingThe the number current of to turns more than a few ampere. several 10 ably greater, e.g. a result, we can produce voltage differences of several 10 nal is then the source of lively bundles of reddish, branched sparks coil. There, long, reddish-blueminals forked (Fig. sparks jump betweenis the grounded ter- (to a water pipe or something similar). The free termi- quencies, it still has a la 1. The T 11.3 High-Frequency Alternating Figure 11.4 on the current.oscillation The cycle. light density To exhibitsimages make two of these maxima variations the during in sparkbe each light separated which spatially. density follow A visible, each rapidlya rotating simple other polygonal method closely mirror of provides in doing time this. must In Fig. is of the order of 10 In Fig. circuit serves at theformer. same It time consists as of the only primary a winding few of turns, a e.g. trans- negative of a photograph taken by B. W of glowing metal vapor.nize the direction In of the thebrighter early end current of during images, the the spark one individual always maxima. can indicates The theThe also high negative recog- voltages pole. which accompany this excitationelectrical of high-frequency oscillations can beperiments used for impressive demonstration ex- taken with this method arewas shown. 50 kHz. The Initially, frequency the ofbe periodic readily the distinguished; variations oscillations in in later images, the they light are density obscured by can clouds , ’s 21 Annalen EDDERSEN ,4thseries, Electrical Oscillations ”, Dissertation, University 11 Flinke Funken im schnellen discovery. 223 (1906). C11.3. Similar photographs of lightning discharges can be found in: B. Walter, der Physik ument contains a detailed account of F Feddersen und der Nachweis der elektrischen Schwingun- gen of Hamburg, 2000. This doc- C11.2. Martin Henke: “ Spiegel – Berend Wilhelm 198 Part I 11.3 High-Frequency Alternating Currents for Demonstration Experiments 199

J S Part I Co

Figure 11.5 ATESLA transformer. The primary coil Co is the coil of a damped electrical oscillator and carries a high-frequency AC current ( 400 kHz; the arrows indicate leads to the current source, e.g. a resonant transformer). The secondary coil of the resonant transformer also makes up an oscillator circuit together with the condenser C; its frequency is the same as that of the line voltage, usually 50–60 Hz (current resonance, see Sect. 10.7). (Video 11.1) (Exercise 11.3) Video 11.1: “TESLA Coil” Figure 11.6 Snap- http://tiny.cc/adggoy shot image (0.01 s) of The video shows experiments the brush discharge with an historic apparatus from the terminal from the early 20th century, of a TESLA coil which is still used for demon- (Fig. 11.5) strations today at Cornell University.

1 m

(“brush discharges”), which weave around and can be over a meter C11.4. This is explained long (Fig. 11.6). Their physiological harmlessness is surprising.C11.4 in Video 11.1 at 9:40 min. See also e.g. Christo- 2. Demonstration of self-inductance in non-coiled conductors.For pher Gerekos, www.tesla- alternating current, the inductive resistance U =I D !L of a con- 0 0 coildesign.com/docs/ ductor is in general large compared to its OHMic resistance R. With TheTeslaCoil-Gerekos.pdf high-frequency AC, this effect can be demonstrated using a “coil” or also https://en.wikipedia. consisting of a single turn: a wire loop. org/wiki/Tesla_coil. Compare In Fig. 11.7, a thick copper-wire loop is carrying high-frequency al- the last section of the latter ternating current in the primary circuit of a TESLA coil. It is bridged reference. While there are at its center by an incandescent light bulb, which would be practi- no electric shocks and no cally short-circuited for direct current. Nevertheless, the bulb glows electrolysis effects of the brightly. The ratio U=I, deﬁned as the resistance, must therefore be high-frequency current, there much greater for the high-frequency AC than for DC. This simple are still health hazards, as experiment shows the inertia of the magnetic ﬁeld in an obvious way. explained there. It contains nothing essentially new; but it is important, since the be- ginner tends to forget about self-inductance in conductors which are “...thebeginnertendsto not wound into a coil (Exercise 11.2). forget about self-inductance in conductors which are not 3. The skin effect. We can imagine a wire to be composed of a thin wound into a coil”. central axis and concentric, tube-shaped layers surrounding it. Its inductance L is smaller within the outer layers than within the inner layers. H .

H C11.5 point a Diameter of the wire aa H b c a the wire: along the inner axis of . 11.8 point upwards; therefore, the induced within b , we see in part b a straight current-carrying con- 11.8 , parts a and c). In both images, the direction of the 11.8 . At the surface of the wire, these newly-formed electric fields Magnetic field lines A demon- magnetic field lines. Variations of this magnetic field with time magnetic field lines, while below, in part c, we see those of several ). Thus, in part a, above, we see the points of penetration of several C increasing the induced field hampersis the decreasing, increase the opposite of occurs the current. When the current Several of them are sketched inIn Fig. addition, one sectionsection of the (Fig. conductor iscurrent shown is twice indicated as by afield a longitudinal long lines feathered are arrow. shownor Furthermore, where the they magnetic penetrate theexterior plane of theinterior image (as induce an electric fieldlines are with drawn closed-loop in parts fieldis a lines. and c as Pairs rectangles,due of for to the these case self-inductance field that thedownwards, have current while opposite the directions. arrowselectric at fields The cancel each arrows otherof at to the a wire, great in extent.field contrast, is this Along opposite the compensation to central the is axis external lacking. applied There, field (feathered the arrow); induced as a result, Justification: In Fig. ductor in cross-section (shaded). Themanner conductor by is the surrounded ring-shaped, inThese closed-loop the field usual field lines lines however of notbut also a only penetrate magnetic surround into its the field interior. conductorcarrying Each a of on the current its tube-shaped must exterior, layers indeed which is be surrounded by magnetic field lines. Figure 11.7 around and within a wireindicated (the by wire shading), is and theirtive effects induc- (the feathered arrowthe shows conventional direction of the electric current; in part b, itplane points of out the of page the towards the reader) stration of the inductive resistance of a wire loop Figure 11.8 ’s ), which 6.13 AXWELL ). These equa- 6.23 is not ferromag- ). 14 Electrical Oscillations 11.8 11 equation ( Chap. Fig. netic, the difference between the conducting material and vacuum can be neglected (see tions were in fact derived for empty space, but asas long the current conductor in C11.5. See Eq. ( follows from M 200 Part I 11.3 High-Frequency Alternating Currents for Demonstration Experiments 201

Figure 11.9 Induction with high- J frequency alternating currents (coil Co

as in Fig. 11.5) Part I

Figure 11.10 Demonstrating the skin effect

the wire, the induced field and the applied field have the same direction, so that the induced field hinders the decrease of the current. Result: Along the central axis of the wire, in contrast to its surface, its self-inductance is stronger.

This uneven distribution of induction within the cross-section of the wire becomes especially noticeable with alternating currents at high frequencies. To detect this current displacement (“skin effect”), we use the setup sketched in Fig. 11.9.ThecoilCo carries a high- frequency alternating current. It induces currents in the induction coil J, a thick copper-wire ring. An incandescent lamp included in the circuit is used to estimate the current strength. We then surround the copper ring with a concentric copper tube (Fig. 11.10). The walls of the tube have the same cross-sectional area as the wire. A similar lamp is connected between the ends of the tube and of the copper wire loop. These two induction coils, one inside the other, are now brought near the field coil Co in Fig. 11.9. The lamp connected to the ends of the tube shines brightly, while the lamp between the ends of the wire glows only a dull red or not at all (see Video 11.1). 4. The detection of closed-loop electric field lines. According to the detailed explanation of induction processes (Sect. 6.1), there should be closed-loop electric field lines resulting from induction. They could unfortunately not be made visible using insulating powder (e.g. fine gypsum crystals). With the high-frequency AC currents in electric oscillator circuits, we can return to this topic and fill in this missing link in our earlier discussion by making the closed-loop field lines visible in a clear-cut way. The apparatus is shown in Fig. 11.11. The field coil Co, with just one turn, produces a high-frequency AC magnetic field. Its field lines are perpendicular to the plane of the page in the figure. This rapidly varying magnetic field should surround itself, according to Fig. 6.2, with closed-loop electric field lines. , 0

Co 11.1 autocontrol or , the circuit itself S . This can be accomplished s autocontrol R at the rhythm of the resonance . Energy losses caused their am- s task was accomplished long ago . R b , a continuous frequency spectrum, , but it is connected only to a part of ssed so far, which were produced by C s R and damped range a C11.6 ). . Pendulum clocks and all kinds of watches are well- of the oscillator circuit produce an alternating current of Video 11.1 The detection 0 at the left: The “AC generator” again consists of a DC 1 Hz). We use the circuit already described in Fig. electrical oscillations. Electrical Oscillations Using Feedback with Triodes

11.12 but also a broad frequency positive feedback known examples. The controlwill of an be electrical treated oscillator bycies by first feedback ( considering oscillations at very low frequen- occurs within the dampedmany oscillations. physical and technical This purposes. One iscurrents often rather with requires constant alternating disturbing amplitudes over for longer times,defined and frequency. with Therefore, a it well- isundamped very useful to be able toIn produce mechanics, the(Vol. corresponding 1, Sect. 11.2) by making use of the technique of source and a variable resistor (right-hand image); however, with aFig. small modification as shown in the coil between the points frequency constant amplitude. In order to achieve must produce this periodic variation in of closed-loop electric field lines, the “electrodeless ring currents”. Noble gases at low pressure, for example neon, already begin to glow at electric field strengths of ca. 20 V/cm. Figure 11.11 Repeated, uniform variations of impulse excitation, were all The electrical oscillations discu Now we bring up a glassregion bulb of filled with these neon closed-loop at electric lowwithin pressure field the into bulb lines; the glows a brightly, widely ring-shapedsee visible region to an a image, large audience. albeitloop We rough, field of lines, the with no AClines beginning electric is or field essential end. for with the Knowledge itschapters. elucidation of closed- This of these demonstration electromagnetic field is waves therefore inderstanding quite later ( important for our un- 11.4 The Production of Undamped plitudes to decrease withone time. frequency A (Vol. damped 1, oscillation Sect. has 11.4); more than not only its eigenfrequency hough it at 5:10 min. Electrical Oscillations 11 C11.6....Event does not prove the existence of closed-loop electric field lines, as is emphasized in Video 11.1 202 Part I 11.4 The Production of Undamped Electrical Oscillations Using Feedback with Triodes 203

I I A A a a UC U Part I b b C V V RS c

Figure 11.12 The production of undamped electrical oscillations in a parallel tank circuit at very low frequencies ( 1 Hz); at the left using external control, at the right with autocontrol (feedback) (Rs 10 k, C D 50 F, L D 103 H). for example by using a triode (electron tube with three electrodes) as a “switch” or “valve” (Fig. 11.12, right side). In a triode, within the evacuated bulb between the thermal-emission cathode and the anode plate, there is a control grid. The resistance of a vacuum tube of this type is of order of 10 k. It can be varied periodically over a wide range by applying an oscillating electric ﬁeld between the cathode and the grid1. The required voltage can be obtained for example by induction from the section b c of the coil. In this way, the oscillator circuit at the right in Fig. 11.12 produces an undamped AC output at a frequency 1 Hz. The time dependence of the current and the voltage as well as the phase difference between them can be registered by rotating-coil galvanometers with a short response time or with an oscilloscope. This method of autocontrol (often called feedback) can be applied with common electron tubes up to frequencies of around 100 MHz. Figure 11.13 shows an example for frequencies of a few hundred kHz. A small lamp is used as an indicator for the AC current.

Figure 11.13 The production of undamped electrical oscillations with frequencies of the order of 500 kHz. This ﬁgure and some of the following ﬁgures show shadow projections of operable setups. The triode can be seen at the far left. The circuit diagram is drawn on a glass plate (cf. Fig 11.12, right-hand image).

1 In Figs. 11.12, 11.13 and 11.17, imagine that a 1.5 volt battery is inserted into the wire leading to the grid of the triode. POHL gave details in his “Elektrizitäts- lehre”, 21st edition, Section 17.5. English: See e.g. https://zipcon.net/~swhite/ docs/physics/electronics/Valves.html. . I d = U 11.2 ,and D becomes I d characteris- = V U - I β whose properties change interrupter, the alternating of the parallel circuit adds to b ) acts as an N-type diode. In the diodes across the conductor EHNELT U and C11.7 a 10.24 α , in the S type, the current increases up to a volt- (Fig. S type N type Voltage drop Voltage triodes have been almost completely replaced by . Initially characteristic curves, referred to as S and N types. V – ; in the N type, the voltage increases up to a current maximum

represents a diode of type S; in a parallel circuit, a small I

˛ diodes

). Both make use of conductor the interrupter D

triodes (transistors).

: In some range, the differential resistance d through I Current during the periodic control process. In a series circuit, the vacuum-tube 11.15 11.14 EHNELT . W with it, the gaspands or bubble contracts: surrounding The the gas bubble glowing “sings”. platinum In wire both tank ex- circuits, the They are found for conductors with a negative differential resistance d (schematic drawing). age maximum ˇ voltage between the terminals the direct voltage fromvariation the of the current voltage source. between the This electrodes causes of a the periodic diode tic (i.e. the dependence ofcurrent the it voltage drop is across carrying) theFig. exhibits resistor the on form the shown for two examples in Here again, an ACoscillations generator with is a constant used amplitude.bination in of As order a generator, to again DC maintain theis current com- electrical employed. source with The a variationof periodically-variable of a resistor the particular resistance property in of this the case resistor employed. is Its the result solid-state Figure 11.14 Autocontrol with diodes also begins with the discussion in Sect. Today, 11.5 Feedback with Diodes electric arc, the alternating currentdirect from current the from series the circuitthe current adds arc source. to periodically, and the along This withthe it changes arc: (due the The to arc current heating), “sings”. the in volume In of the W negative! Conductors with thisfollowing as characteristic are referred to in the How diodes canis effect explained the using(Fig. autocontrol two of demonstration experiments electrical as oscillations examples audibly electric arc E , ’s , triode OHL 111 Video 1 ”, the oper- solid-state http://tiny.cc/ vacuum-tube ). Electrical Oscillations 11 Elektrizitätslehre ation of a “ C11.7. In Chap. 27, Sect.in 3 the 21st edition of P z8fgoy 23:30 min, was explained in terms ofﬁrst the experimental triode, which had not yet been developed sufﬁciently for technical applications (R. Hilsch and R. W. Pohl, Zeitschrift für Physik 399 (1938); see also 204 Part I 11.6 Forced Electrical Oscillations 205

a a 220 V 100 V Part I

E D

R b R = 100 Ω b

Figure 11.15 Autocontrol (feedback) of electrical oscillator circuits using diodes whose properties change audibly ( 1 kHz). At the left is a series circuit similar to the schematic in Fig. 11.1 (left-hand image); here, E is an electric arc. At the right is a parallel tank circuit similar to the schematic in Fig. 11.1 (right-hand image); here, D is a small WEHNELT interrupter (the coil is similar to Fig. 10.2, right, but without an iron core: L D 3:3mH;the condenser has C 1 F. The electric arc is struck between two pure carbon electrodes of 1 cm diameter). frequency of the tone can be adjusted by varying the capacitance C of the condenser.

In the feedback circuit using a parallel tank circuit, one can leave off the condenser altogether. Then only a toggle oscillation at an audible frequency remains (Vol. 1, Sect. 11.17); it can be varied by changing the value of the resistor R (cf. Fig. 10.24). This once again demonstrates the transition from harmonic oscillations to toggle oscillations.

A different conductor with an N-type characteristic (Fig. 11.14, right side) is the Dynatron (cf. 21st German edition of this book, Sect. 17.5; see also https://en.wikipedia.org/wiki/Dynatron_oscillator), as well as semiconductor diodes (e.g. tunnel diodes). With these, oscillator circuits for frequencies up to 100 GHz can be assembled.

11.6 Forced Electrical Oscillations

Let us consider some sort of mechanical pendulum, which swings following an “impulse excitation” or undamped with “feedback” at its resonance frequency, 0. But every pendulum can also be forced to oscillate at a different frequency by a suitable external excitation source, so that the pendulum oscillates as a resonator. To accomplish this, we let a periodic force or torque act with the desired frequency on the pendulum. This process of forced oscillations was treated in some detail in Vol. 1 (Sect. 11.10) because of its general importance. Similar conclusions hold for forced electrical oscillations. Instead of a torsional pendulum with a ﬂywheel and a spiral spring, we consider an electrical oscillator circuit with a coil and a condenser; instead of a periodically recurring torque, we require a periodically varying voltage. The latter can be produced without a conducting connection , illator 11.17 physical , which oscil- as resonators; . The resonance 11.13 11.13 11.16 illations often lack , while the right-hand circuit is the resonator. . One case of this kind can be seen in Fig. C excitation source Two oscillator circuits (“tank circuits”) with which high- ecessary technical accessories. At the left: An oscillator circuit with autocontrol (feedback) 1 MHz. At the right: A tank circuit in which oscillations can be

serves as the As an indicator for the forcedIn oscillations, we this again way, use a we smallwithout can light unn bulb. produce high-frequency AC currents in an osc 2. High-frequency circuits with undampedclarity. osc Coils anddiscrete condensers devices; often, can the no electrodesnecessary longer of capacitance vacuum be tubes considered already provideat separately the the as left; it isWe an see oscillator only circuit a with coil aright which resonance in is frequency the tapped figure, of in on 1 contrast, MHz. we onea see side, coil a and regular with oscillator a circuit, two containing vacuum turns tube. and At a the rotary variable condenser. The left-hand circuit frequency of the left-hand circuiton can the be parallel-plate adjusted condenser; using the that micrometer of screw the right-hand resonator is fixed. frequency AC currents can beA demonstrated suitable excitation by source means is of the resonant circuit shown excitation. in Fig. Figure 11.17 Figure 11.16 between the excitation sourcequalitative and examples: the resonator. We start1. with two We use thethe excitation two source tank could circuits be the shown circuit in in Fig. Fig. produced by resonance at the same frequency for the excitation source,duce so a that voltage the inthe magnetic the condenser field in windings of the of tank itsfrequency. the circuit, coil we resonator Then can can coil. the readily in- Both obtain lamp resonator By the circuits in resonant adjusting have a thecurves small resonator are damping, so rather circuit that sharp. shines their resonance brightly. lates continuously, without damping. The resonators are set up near Electrical Oscillations 11 206 Part I 11.7 A Quantitative Treatment of Forced Oscillations in a “Tank Circuit” 207 11.7 A Quantitative Treatment

of Forced Oscillations Part I in a “Tank Circuit”

In the preceding chapter, in Figures 10.14 and 10.18,wevariedthe voltage between the terminals a and b of the oscillator circuits by using a generator which produces a sinusoidal alternating current. That is, we input an alternating current into the circuits. But now we say: The circuits are devices which can oscillate; we excite them to oscillations as resonators, and we employ an alternating current source for the excitation. We wish to compare these forced electrical oscillations with forced mechanical vibrations or oscillations like those that were discussed in Vol. 1, Sect. 11.10. The alternating current in Fig. 10.14,thatisina series tank circuit containing a coil and a condenser, was described by Eq. (10.30). With this equation, we could ﬁrst calculate the amplitude I0 of the current for various frequencies (Fig. 11.18b), and then, by referring also to Eq. (10.29), we could obtain the amplitude UC;0 of the voltage on the condenser (Fig. 11.18a). Furthermore, with Eq. (10.31), we could calculate the phase angle ', and with it, the phase difference between the excitation voltage U and the condenser voltage UC (Fig. 11.18c); P D 1 2 and ﬁnally, we computed the average power W 2 I0 R dissipated by the resonator (Fig. 11.19); this curve holds at the same time for the D 1 . 1 2/ average magnetic energy Wm 2 2 LI0 stored in the resonator. Its value can be read off the right-hand ordinate scale.

The values of L, C and R were chosen to be about the same as those for the demonstration experiment in Fig. 10.16.

The formal similarities between the forced oscillations of an electrical series-circuit oscillator and the forced oscillations of a mechanical harmonic oscillator are evident. The content of equations (10.30)and (10.31) is illustrated in a very clear-cut way by Fig. 11.18. In the energy resonance curve (Fig. 11.19), the ratio of the resonance frequency of the resonator to its linewidth H is called the quality factor or Q-value:

Q D 0 : (11.1) H This quantity serves to determine experimentally several important values which characterize the oscillator circuit, namely its logarithmic decrement (Vol. 1, Sect. 11.10) and its damping constant 0. When the logarithmic decrement is Ä 1, then the following relation holds: H 1 D D (11.2) 0 Q

.1=Q is also called the loss factor/: 0

/ C11.9 1, the e . (11.3) 0 = ampli-

1 > 1 . –3 2 of its maximum p = = 100 V = 100 = 500 Hz 0 = 4 nF = 15 kΩ Time (s) Time = 0.6 = 25 H 0 weakly-damped series L C R v Λ U within which the of the resonator

). The linewidth (full width v 0 2 4∙10 for free oscillations free for

C U : voltage Condenser . The angle plotted in Part c is L 10.24 C of the excitation source of the excitation source v 10.6 v H r 500 1000 Hz R D √2 0, max I relaxation time Frequency

is the frequency range at whose limits the cur- a c Resonance frequency b Frequency C11.8 : H 0 0 0.5 1.0 1.5 2.0 2.5 3.0 0 6 4 2 0 A –3 V 0° is the . 500 400 300 200 100 r 2 ∙10 +90°

+180°

C is deﬁned as in Eq. ( illator, nor at the slightly lower resonance frequency of

on the condensor the on and the condenser voltage voltage condenser the and in the circuit the in U

' C the excitation voltage voltage excitation the I current of U U voltage of

. =/

these quantities, in the case of

0 C,0 5 I Phase shift between shift Phase

U : The characterization of forced electrical oscillations in a se-

0 = Amplitudes 1 ), where 1 D ı p 0 0 90 of the forced oscillations approaches the fraction

calculate C D =

' These quantities can alsolowing be relations: obtained experimentally using the fol- ( value. The maximum value of the current (Part b) lies as usual at at half maximum, FWHM) rent (effective value or amplitude) has decreased to 1 tude 63 % of its steady-state value. We and parallel tank circuits ries tank circuit, calculated as in Sect. 1

Figure 11.18 of the undamped osc the freely oscillating damped oscillator circuit, but rather at the frequency With extremely strong damping, i.e.maximum a deﬂection logarithmic decrement in of Part a occurs neither at the resonance frequency ound .The 0 ; C U (Fig. 11.43 0 / . and and the am- t 0 I d C of the torsional ”, John Wiley, ˛ a to the condenser b to the current 0 U ˛= ˛ d . Electrical Oscillations 11.18 11.18 11 The Physics of Vibrations plitude amplitude of the angular velocity “ and Waves 5th ed., 1999 (Chaps. 2 and 3). to the physics of vibrations and oscillations can be f for example in: H.J. Pain, C11.8. A good introduction C11.9. Note the analogy to the mechanical harmonic oscillator: The momentary deﬂection Fig. voltages pendulum (Figs. 11.42a and b in Vol. 1) correspond in in Vol. 1) corresponds in Fig. amplitude, 208 Part I 11.7 A Quantitative Treatment of Forced Oscillations in a “Tank Circuit” 209

Figure 11.19 The energy reso- watt watt second nance curve of forced electrical ∙10–4 oscillations (series tank circuit). 0.3 2.5 Part I Even with extremely strong damping, i.e. >1, its max- 2.0 imum falls at ,thatisatthe 0 0.2 eigenfrequency of the undamped 1.5 resonator. The FWHM H here H is the frequency range at whose 1.0 limits both the power dissipated 0.1 1 ∙ Wmax by all damping mechanisms, as 2 0.5 well as the average stored magnetic energy, have decreased to 0 half their maximum values. power consumed in resonator Average 0 500 Hz

Frequency v of the in the resonator magnetic energy Average excitation source

0 0.5 1.0 1.5 2.0 Frequency v of the excitation source

Resonance frequency v0 of the resonator

For the series circuit,

H U ; for D 0 D D C 0 D reciprocal of the resonant voltage 0 UC;0 for D 0 increase (11.4) and for the parallel circuit, H I for D D D 0 0 D reciprocal of the resonant current 0 IL;0 for D 0 increase : (11.5)

Derivation: For the series circuit, at the frequency D 0, UC;0 is equal to the amplitudep U0 of the power supply. At the resonance frequency, 0 D 1=.2 LC/, we ﬁnd from Eqns. (10.29)and(10.30) for the voltage amplitude UC;0 of the condenser r L 1 U ; D U ; C 0 0 C R and, making use of Eq. (11.3), r U ; for D 0 C C 0 D R D : (11.4) UC;0 for D 0 L

Parallel circuit: The amplitude I0 of the currentp in the leads to the parallel circuit at the circular frequency !0 D 1= LC follows from Eq. (10.33),

U0!0CR I0 D p : R2 C L=C

At the same circular frequency, the current amplitude IL;0 in the coil is givenbyEq.(10.21),

U0 IL;0 D p : 2 2 R C .!0L/ . 7 8 C = ) (see D D ) 0 / " https:// in the Q ,the N 11.5 / ( 75 t : 11.3 D . 1 , I 0 ; is deﬁned ,whichwe C is to be esti- H U b minimum in the parallel 11.2 , only the large 0 , the inductance = 11.5

a a ), two coils with .!/ I 0 the condenser has 11.11 ln 8 11.2 . – b D . Find the amplitude of . (Sects. a 0 ; ”, Vol. 1, 21st ed., 1973, t 1 0 is thus at its C L : 5 cm. The diameter of the I 11.9 : U (which is however differently

12 D ) still holds here. H 12 cm and has metal-foil elec- D ). The coil consists of L D L 11.2 D C a ) contains supplementary material, which nce of their magnetic ﬁelds, is pre- 13.1 d r The online version of this chapter ( R D 18 cm inside and outside. The thickness ) and the current 0 are connected in series. Their “mutual ; 0 L I D , and the width I 2 circuit, at the time , right), and its voltage is thus 52 mm, and we may assume a value of Q L h coil in Video 11.1 and Fig. 10.20 : 1 ,wehave LC ; Theorie der Elektrizität 11.2 D and ) )), where the AC resistance has its maximum at the reso- t ) on the coil of inductance ESLA C11.10 1 0 2. L 0

; (Fig. p 11.2 10.33 11.3 L1 = 0 ), has a diameter of 3 mm. Under the assumption that D U Q

D 2.54 b In an electrical oscillator circuit (Fig. For the T In a loss-free on the condenser and the time-dependent current / t defined in this case) as given in Eq. ( here as the frequencyas range large at as whose at limits thea the minimum factor power (!); of that consumed 1 is, is the twice AC resistance has decreased by nant frequency (see Fig. circuit (Eq. ( Thus, for relationship between In spite of the qualitatively different form of the current . C B.G. Teubner, Stuttgart); English:teractions”, “Electromagnetic Vol. I: Fields Electromagnetic and Theory In- and1964). Relativity (Sect. (Blaisdell, circuit. (Sect. 11.2 e.g. Becker/Sauter, “ mated. In the experiments shown in Figs. the voltage consider to be an oscillator circuit, the frequency 11.3 An electrical oscillationU begins. Find the time-dependent voltage inductances of inductance”, i.e. the mutual influe sumed to be negligible (forapart; example this because is they not are sufficientlyplitude essential of far to the the voltage on principle the of condenser is this exercise). The am- trodes up to a height of primary coil is connected to thejar condenser. (Fig. The condenser, a Leyden Electronic supplementary material doi.org/10.1007/978-3-319-50269-4_11 is available to authorized users. Exercises 11.1 achargeof for its dielectric constant (Table wire is 2 of its glass wall is of a single circular ring is given by circular windings with a radius of ”, Electrical Oscillations 11 Electricity and Magnetism C11.10. See e.g.: B. Kur- relmeyer and W.H. Mais, “ Van Nostrand, Princeton 1967, Chap. 14. 210 Part I h odne ltsi hr ie ota t efidcac is self-inductance its that very so decays wire, field short The a small. is plates condenser the o,i hscatr ecnie hr n nlcase: final A and 3. third a consider we chapter, this in Now, pcnso h edlnsat lines field the of spacings et lopoue antcfil;ti a aepasbeb the by plausible made was this field; magnetic cur- a displacement produces (the also field rent) electric time-varying ex- a demonstrates fundamentally formation that their perimentally yielded First, have physics. for waves results electromagnetic important of forms Both tube. speaking a in waves sound the pnsae res hyare they else or space, open ae rpgt ihri l directions all in either propagate waves ntefil opoaaebetween propagate to field the in h edlnsi h gr.W hsosreavr ihbtsilfi- still propagation but finite for high possible This very it a fields. makes observe electric velocity thus for We velocity figure. propagation the nite in lines field the h eddcycue ytebig a led rgesdmuch progressed already has bridge the at by further caused decay field The once yacnutn rde nFig. In bridge. conducting a by connected ˇ .A 2. asstedcyt cu sol” h eddcyocr at occurs decay field The “slowly”. occur to bu decay decays, the field causes electric The wire. of denser lcrmgei Waves Electromagnetic iue12.1 Figure c ab The 1. field electric the of follows: discussion as the roughly organized we chapters, previous In Remarks Preliminary 12.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © ie have lines M αα βα rcial iutnosy hsi niae nFig. in indicated is This simultaneously. practically AXWELL ail-ayn lcrcfield electric rapidly-varying lwyvrigeeti field electric slowly-varying ttceeti field electric static ˛ lcrccharges electric :Asai lcrcfil;badc eaigfilswti con- a within fields decaying c: and b field; electric static A a: qain,btwsinitia was but equations, hnat than ˇ hti niae ytedfeetsaig of spacings different the by indicated is That . β olsItouto oPhysics to Introduction Pohl’s e h ceai nFig. in schematic the See . tterends. their at conducted ˛ and lcrmgei waves electromagnetic rapidly ˛ h w ltso odne are condenser a of plates two The . .InFig. ˇ and l nyahptei (Sect. hypothesis a only lly . h efidcac ftecoil the of self-inductance the t ..tetm o h change the for time the i.e. : ˇ ln rnmsinlns like lines, transmission along freely a olne eneglected. be longer no can 12.1 12.1 iesudwvsin waves sound like , ,tebig between bridge the c, ,ti saln coil long a is this b, , https://doi.org/10.1007/978-3-319-50269-4_12 12.1 12.1 βα ofr.Such form. to .Tefield The a. yequal by b ˛ 6.4 and ). 12 211

Part I Us- : 11.17 . Its disad- 12.2 . The latter is in- sion, satellite relays, hods for limited visibility condi- in a clear-cut setup hnology, including televi An oscillator Oscillator Circuit 100 MHz) circuit with feedback (frequency Figure 12.2 To demonstrate and investigate electromagnetic wavesroom, in we the ﬁrst lecture need100 AC MHz. current sources These with cancillations. frequencies best of be A around generated suitable using arrangement damped is electrical shown os- in Fig. ern communications tec mobile telephones, navigation met tions, etc. were all developed thanks towaves. knowledge of The electromagnetic methods devised for thesemeating purposes other are areas increasingly of per- technology asof well this as our evolution daily is lives.these The developments not end have in grown out sight.dependent of technological physics disciplines Only in and their have one own become right.itself thing in- Physics to limits is the certain: fundamentals.the All following We sections. should of keep this in mind in reading 12.2 A Simple Electrical Harmonic Second, the discovery of electromagnetic waves hastend allowed us the to spectrum ex- which wasand originally infrared known only radiation for seamlessly visibleup out light to to many waves kilometers, withwavelengths and in wavelengths in the of the regions of other X-rays direction and to gamma radiation. extremelyIn the short realm of technology, bothand types of conducted, electromagnetic waves, have free achieved an extraordinary importance. Mod- vantage is obvious: Thecondenser essential and parts the of coil,invisible the are next oscillator to only circuit, the vestigial,We the trivial and correct accessories they this which in are provide theing practically the well-known an unspeciﬁed manner feedback. circuit shown (“black box”) inproduce to Fig. resonant provide the oscillations excitation, we Electromagnetic Waves 12 212 Part I 12.3 A Rod-Shaped Electric Dipole 213

Figure 12.3 A simple closed-loop electrical oscillator circuit for demon-

strating forced electrical oscillations. Part I The light bulb serves as an indicator of the alternating current in the wire loop. Small light bulb

dicated in Fig. 12.3. We see just one simple circular copper wire loop of ca. 30 cm diameter. At its center, below the wooden handle, it contains a small light bulb as a current indicator. At each of its ends, a condenser plate about the size of a credit card is attached. The two plates are spaced around 5 cm from each other. This circuit serves as a resonator, and we bring it close to the excitation-source circuit shown in Fig. 12.2. By bending the copper-wire loop (and thus changing the spacing of the condenser plates), we can adjust the resonator frequency to be sufﬁciently close to the frequency of the excitation source. The light bulb glows brightly. In the resonator circuit, an AC current of about 0.5 A is ﬂowing at a frequency of around 100 MHz.

Compare the experiments shown in Figs. 2.36 and 12.3.InFig.2.36,the ﬁeld decays once, and the resulting current impulse is around 108 As.In Fig. 12.3, the ﬁeld decay occurs about 108 times per second, so that we can observe currents of the order of 1 A.

12.3 A Rod-Shaped Electric Dipole

With the high-frequency AC now at our disposal, we can proceed to something new and important, namely to a rod-shaped electric dipole or antenna. In mechanics, a simple ball-and-spring pendulum consists of a body with inertial mass and an elastic component (helical springs, etc.). In electromagnetism, an electrical oscillator circuit with a coil and condenser is a correspondingly simple setup. We described the analogy of these two systems in Sect. 11.2, and refer here to Fig. 12.4. In the simple ball-and-spring pendulum of mechanics, we can distinguish clearly between the inertial component and the component with , 12.4 a clear-cut sep- without s to the case of electrical oscillations or vibrations e b c d a illator circuit An example is a column of air in a tube (an organ A simple mechanical ball-and-spring pendulum and the corre- The tran- C12.1 electric field. But in othercillating, electrical localizing circuits these functions which separately are isin capable just the of acoustically-vibrating as os- impossible air as column incase an of organ this pipe. latter kind An isthis extreme a simple rod-shaped system electric in dipole. more We will detail. discuss pipe). Everycomponent length and a element section in of ‘tensed the spring’ (Vol. air 1,A Sect. corresponding column 11.7). description is applie bothtions. In an the typical inertial tank circuit, e.g.we as shown can at the clearly right inthe identify Fig. magnetic the field, and coil the as condenser as the the location location of of the “elastic” the “inertia” of sponding electrical osc a suitable ammeter. It would indicate a current of around 0.5 A. sition from a closed (“lumped”) oscillator circuit to an open, rod-shaped electric dipole (“distributed” circuit). The light bulb could be replaced by Figure 12.5 elastic (restoring) force. As longlarge, as we the can mass of completely the neglect ball the is small sufficiently massHowever, of in the mechanics springs. wewhich could can also support oscilla listaration numerous and arrangements localizationcomponent. of the inertial component and the elastic Figure 12.4 Electromagnetic Waves 12 a “lumped system”. ponents and functions, like a ball-and-spring pendulum or a tank circuit with separate coil and condenser, is called is called a “distributed system”. In contrast, a system with separate, localized com- nents and their functions are spread over the whole system rather than being localized in specific devices, C12.1. This kind of system, in which the compo- 214 Part I 12.3 A Rod-Shaped Electric Dipole 215 Part I

Figure 12.6 A rod-shaped electric dipole, around 1.5 m long

We again turn to the simplest of our oscillator circuits, the one shown in Fig. 12.3. When a current is passing through the wire loop and the light bulb, the electric field in the condenser is changing. We enlarge the region around the condenser, at the same time reducing the size of the condenser plates. We want to carry out the transition sketched in Fig. 12.5. The gradual atrophying of the condenser can be compensated by lengthening the two halves of the wire loop. The light bulb continues to glow, thus an AC current is still flowing. In the limit, we arrive at Fig. 12.5e, a straight rod with a brightly glowing light bulb at its center. Fig. 12.6 shows how the experiment is carried out. The hand can serve as a length scale. The excitation source (Fig. 12.2) can be thought of as being about 0.5 m away. The length of the rod is not very important. 10 cm more or less at each of its ends will play no significant role. The rod is a resonator with strong damping (Sect. 11.7). During its oscillation, the two halves of the rod are alternately positively and negatively charged. The corresponding charges can be thought of as each localized around a “center of gravity”; then we have two electric charge regions of opposite signs, separated by a distance l. An object of this kind was called an electric dipole in Sect. 3.9, and we adopt that name here for our electrically-oscillating rod. The electric charges oscillate back and forth along the rod. They represent conduction currents which alternate in direction, i.e. an alternating current. This is the electrical analog of an air current which alternates in direction within an organ pipe that is closed at both ends, a “stopped pipe”: In the dipole, electric charges are periodically accelerated, while in the organ pipe, air molecules are periodically accelerated. The fundamental oscillation of the pipe is shown in Fig. 12.7 by three “snapshot images”. A grey shading refers to the normal number density of the air molecules, a darker shading is an increased density and a light shading a reduced density. These distributions are sketched graphically in the lower part of the figure. The wave (density or pressure) crests lie at the ends of the pipe and the node is in its center. These variations in the particle density occur because the individual volume elements of the air column in the pipe are flowing back and forth along it. The air currents are also distributed sinusoidally, but their maximum (wave crest) is at the center of the pipe. There, the amplitudes of the velocities of the air molecules, which are directed alternately to the right and to the left, have their greatest magnitudes (Fig. 12.8). 12.9 12.11 corre- 12.7 Change in the particle density of the air and the air pressure Longitudinal air current (and the velocities of the air molecules) for the organ pipe. 12.7 , they are sinusoidally distributed 1 (Vol. 1, Sect. 11.8). For electrical oscilla- Node E Crest conduction currents The sinusoidal distribution of the longitudinal air currents in The distribution of the particle density of the air molecules and , this sinusoidal distribution of the conduction current is Crest Crest a b c Node Node 12.10 The electrons in these currents, due to their enormous numbers, move only The analogy can bedinal carried mechanical still vibrations is further. proportionalmodulus to The of the frequency elasticity square of root longitu- of the along the length of the rod-shaped dipole. This is shown in Fig. using three light bulbs:the The two middle bulb on isFig. each glowing side brightly, while exhibit only a dull reddish-yellow glow. In shown as a graphical representation. The periodically alternating current produces aelectric-charge periodically distribution. reversing The grey shading in Fig. corresponds to the lower part of Fig. sponds to an electrically neutraland state; the the light dark shading is shadingexcess a a makes positive, the negative potential chargeand (the excess. for voltage example between the A a ground,a positive or point negative charge the on charge center the excess of makes rod the the rod) potential positive, while negative. Figure the air pressurebelow, in a a graphical pipe representation.upper closed part Its at of ordinate, both theair like figure, ends. molecules the corresponds and grey at to Above, shading the the same three in particle time “snapshots”; the (number) to the density air of pressure. the a pipe which is closed at both ends through very short distances, on the order of a tenth of an atomic diameter. 1 Figure 12.8 The electrical oscillationsresponding of manner. a rod-shaped Thethe dipole air electrical behave currents in in a the cor- pipe are analogous to Figure 12.7 Electromagnetic Waves 12 216 Part I 12.3 A Rod-Shaped Electric Dipole 217 Part I

Figure 12.9 Demonstration of the sinusoidal distribution of the conduction currents along a rod-shaped dipole

Electrical conduction current NodeCrest Node

Figure 12.10 A graphical representation of the sinusoidal distribution of the conduction current along a rod-shaped dipole

Change in the electric potential NodeCrest Node

Figure 12.11 The distribution of the potential along the axis of a rod-shaped dipole. The abscissa line corresponds to a zero potential when the dipole as a whole is neutral. tions, instead of the modulus of elasticity, we have the reciprocal of the capacitancepC. The frequency of an electrical oscillation is proportional to 1= C. The capacitance C is itself proportional to the dielectric constant " (Sect. 2.17). In ap medium with a dielectric constant of ", a dipole of length lm D l= " has the same frequency as a dipole of length l in air orp vacuum. This is shown in Fig. 12.12 for a dipole in water (" D 81, " D 9, Table 13.3). The correspondence between organ-pipe vibrations and dipole oscillations is still not exhausted. The pipe in Fig. 12.7 is vibrating at its fundamental frequency, while Fig. 12.13 corresponds to a pipe

Figure 12.12 In distilled water, this short dipole has the same resonance frequency as the dipole in air shown in Fig. 12.9,whichis B nine times longer (B is a piece of insulating twine) (Exercise 12.4) . 12.14 illation, and its cur- electrical conduction current (first harmonic, upper curve). : The coil and condenser shown there dark illations by using autocontrol (feedback). 11.13 first overtone bright bright A dipole undergoing its first-harmonic osc A dipole with positive feedback (triode for auto- Just like an organalso pipe be excited in to mechanics undampedThis (acoustics), can osc be a accomplished by dipoleIt using can is e.g. derived the of directly circuit from course sketched Fig. have in Fig. degenerated here intohas straight a segments. refreshingly simpleunfortunately The requires a dipole circuit knowledge with of diagram, dipole feedback oscillations. but understanding it in detail control) So much for theof rod-shaped knowledge: dipole. The It distribution of brought us an an important bit Figure 12.14 Figure 12.13 which is vibrating atIn its Fig. 12.13 below, a dipole aboutgether 3 m from long two is of sketched; it the waswhich dipole put to- segments have used been before. inserted Thedistribution of along light the bulbs its electric length current:remains The allow light dark, us bulb while to at the the visualizebright. two central the node This on dipole each is sidesponding also manner, at we oscillating the could at add current sections its to maxima firstof make dipoles 4.5 overtone. are with m, lengths In 6 m, a etc. corre- rents Electromagnetic Waves 12 218 Part I 12.4 Standing Waves Between Two Parallel Wires: The LECHER Line 219 in the interior of a rod can take the form of a standing wave, in its fundamental as well as its harmonic oscillations. Part I This distribution of currents implies a particular distribution of the electric field. The investigation of this electric field and its time dependence is our next task. It will lead us to travelling electromagnetic waves, both those which are guided along a transmission line and those which propagate through free space.

12.4 Standing Waves Between Two Parallel Wires: The LECHER Line2

The electric field lines of an open, straight dipole must in some way be stretched in long arcs between various points along the length of the dipole. Along their paths, they meet up with the walls of the room, with the experimenter, etc. We will initially not attempt to describe this rather complex situation surrounding a straight, open dipole; instead, we begin by investigating the shape of the field lines in a simpler case. In making the transition from a closed (“lumped”) tank circuit to an open dipole, we passed through several intermediate stages, one of which is shown in Fig. 12.15 (top). We can call it a “folded” dipole for short. We bring it near to the excitation source at a frequency of 100 MHz (Fig. 12.2), and use the light bulbs to observe the distribution of currents in the dipole. The center lamp burns most brightly, and this is the location of the maximum current, the crest of the standing current wave. With this dipole form, there can be no doubt about the shape of the electric field lines between the two “legs”. The distribution of the electric field strength E is shown graphically in Fig. 12.15b. The two curves indicate the maximum values or amplitudes, similar to Fig. 12.11. In the upper curve, the upper half of the dipole has its maximum negative charge, while the lower curve shows the lower half with its maximum positive charge. The two curves follow each other at a time interval of one-half of a complete oscillation (cycle).

Figure 12.15 A“folded” a dipole and the distribution of the electric ﬁeld strength along its two legs b Electric field strength in V/m strength in

2 E. Lecher, Annalen der Physik 41, 850 (1890). system ), or by E loop as a re- ECHER b). . The light bulb at induction 12.16 in the figure. It “short cir- 12.16 E . Wecouldalsousean C12.2 ). We see this in Fig. cage). ), are converted to DC by a rectifier diode (detector) and line) can now be measured in an extremely simple and 0 influence 12.13 E a by short ticks. Below, in part b, the field distribution is 12 3 ). It is permeated by the magnetic field which is directed BB B 0 E 12.16 ARADAY ECHER cuits” the electric field, andthe an wire alternating current by is produced within ceiver ( or L One can read thethe ordinates field either strengths as (displacement field currents),est strengths since field the strength or are regions as also of changes regionsmost high- where rapidly. in the field strength is changing We could extend theends dipole by (Fig. adding one or more segments at its precise way. We mention two different procedures: 1. We can measure the strength(antenna of in the the electric form of field a locally. shortlegs; A dipole) it receiver can is be the located short between the piece two of wire labelled can be measured with galvanometers (G).and We forth move the in receiver the back the directions nodes, of that the is double the arrow, zero and points can of thus the locate electric field, with consider- perpendicular to the plane offlowing in the the page line. and The produced ACinduction currents by ( produced the by currents influence ( the far left continuesbefore. to burn The brightly, ends i.e.Fig. of the the oscillations individual persist dipole as segments are marked in again indicated; the fieldtroughs, has and its drops maximum to strength zero at at the the nodes crests (Fig. and This field distribution in such a system (called a L GG Demonstrating the standing electromagnetic waves between two parallel conductors

EE' line). When the ﬁeld strengths are sufﬁciently high, small incandescent light bulbs can

strength in V/m in strength Electric field Electric ECHER (a L be used to sampleto the the plane field of distribution. thestrength. page) The in Therefore, the nodes detectors standing ofusing waves for a are the sheet-metal the at magnetic box the magnetic field (a positions fields of F (which must the is crests be perpendicular in well the shielded electric field against electric fields c Figure 12.16 b a .) Electromagnetic Waves 12.31 12 C12.2. The standing wave of the magnetic field is thus shifted by 1/4 wavelength relative to the standing wave of the electric field! (Compare these field distributions with those of a travelling wave, Fig. 220 Part I 12.5 Travelling Electromagnetic Waves Between Two Parallel Wires: Their Velocity 221 Part I

Figure 12.17 Visualization of the field distribution of standing electromagnetic waves between two parallel conductors (a LECHER line) (Video 12.1) Video 12.1: “LECHER line” http://tiny.cc/gdggoy able precision. Such methods are always applicable. We could then In the video, the field distri- connect the two lines at the positions of the nodes with a finger or bution between the parallel with a wire “bridge” B (compare Fig. 12.16c). This would not in the lines is made visible in least perturb the standing waves: The light bulb at the left continues a simple way by using a flu- to glow with undiminished brightness. orescent tube which is held adjacent to the LECHER line. We could “cut out” each of the rectangles bordered by neighboring The nodes of the electric field bridges B and let them oscillate alone. To detect their oscillations, the are always to the right of the bridges can be outfitted with small light bulbs. During the oscillation, pe- insulating supports for the riodic maxima of positive and negative charges are localized at the centers lines; the magnetic field has of the long, horizontal sides of the rectangles. Their charge transport back and forth through the two short sides (bridges) causes the light bulbs to its oscillation maxima there. glow.

2. Second method: We could pass the two parallel conductors through a long glass tube filled with neon at low pressure (Fig. 12.17). Then in the regions of high electric field strength (the crests and troughs of the standing waves), a gas discharge occurs in the neon; we can see the light from the positive column of the glow currents. The spatial alternation of dark and bright regions of gas in the tube gives a clear-cut image of the field distribution between the parallel conductors.

This method requires relatively high values of the electric ﬁeld strength. They can be achieved most simply by using a damped excitation source, e.g. the circuit sketched in Fig. 12.17, with a spark gap (cf. Fig. 11.5).

The experiments described in this section lead to a simple but important result: The electric ﬁeld between parallel conductors can reproduce the form of a standing wave.

12.5 Travelling Electromagnetic Waves Between Two Parallel Wires: Their Velocity

Standing waves are formed by superposition or interference of oppositely-directed travelling waves (Vol. 1, Sect. 12.5). There- fore, detection of standing electromagnetic waves proves that there are also travelling waves between the parallel conductors. A momentary image (snapshot) of such a wave is sketched in the upper part of Fig. 12.18. . 3m) line) 12.2 (12.1) ). The D 10.32 ECHER ,theirwave-

are related by the litude refers to the , the travelling wave u at rest when the length of the line is 10 m lling electromagnetic wave ( : D u we see a different, but equivalent representa- . To an observer u alternating currents 12.18 line, the frequency of the excitation system can be is a segment of the oscillator circuit from Fig. S

, Top: A snapshot of a travelling electromagnetic “wire wave” The conduction of a trave

in V/M in S

and their velocities of propagation Electric field strength field Electric ECHER 12.19 At the bottom of Fig. field strength in either case,representation quoted shows for nothing example of in the V/m. shapes But nor this the graphical extent of the field lines. tion. Wave crests correspond totroughs electric to fields electric directed fields upwards, directed and downwards. wave The amp With a L which has a lightfrom bulb travelling at electromagnetic its waves that far propagate end. along theFor This line. all lamp receives types the of energy travelling waves, their frequencies lengths equation short compared to their wavelengths. between two parallelelectric conductors field, (the and arrows theirBottom: spacing indicate A shows the different themagnetic representation direction magnitude wave. of of of the the the snapshot of field a strength). travelling electro- freely chosen. In principle, it can be calculated from Eq. ( In Fig. Figure 12.19 One can imagine that thisrection entire with image the is velocity moving in a horizontal di- Figure 12.18 along a double linewide. which is The mounted on electromagneticwires the are waves edges called which of simply are a plastic conducted band along 10 mm two parallel At two points, it is connected to a long double line (L appears as a periodically-varying electric field. Electromagnetic Waves 12 222 Part I 12.6 The Electric Field of a Dipole. The Emission of Free Electromagnetic Waves 223 wavelength can be measured; it is equal to twice the distance between two neighboring nodes. Inserting these values into Eq. (12.1) 8 yields the velocity u D 3 10 m/s D vacuum velocity of light, c. Part I

This result is rather surprising: Along a LECHER line, every segment l, just as in any pair of conducting wires, has an OHMic resistance, an inductive resistance, and a capacitive resistance. Because of these resistances, the propagation velocity of sinusoidal waves along the line depends on their frequency. Non-periodic signals are thus collected together into wave groups (Vol. 1, Sect. 12.23). Their group velocity at frequencies of the order of several 100 Hz is found to be only around 2 108 m/s. Furthermore, the wave groups are damped along their paths. All of these technically important properties are described quantitatively by the so-called telegraph equation.C12.3 C12.3. The telegraph equation is a second-order partial Why do all these properties of the transmission line play no role at differential equation, whose the high frequencies of the LECHER system? Why do we find the full solution describes the prop- vacuum velocity of light, c D 3 108 m/s, as the propagation velocity agation of electromagnetic of the travelling waves in the limiting case of high frequencies? An- waves along a conducting swer: At high frequencies, the influence of the conduction currents transmission line. In addition in the conductors becomes completely unimportant. The magnetic to the inductance and capac- field due to displacement currents – the large rate of change EP of itance of the line, it takes the electric field – is much stronger than the magnetic field from the the OHMic resistance of the conduction currents. It induces an electric field between the next seg- conductors into account. If ments of the line, and so forth. the latter is negligible, the The essential processes for the propagation of the waves thus do not equation simplifies to the take place within the conductors, but rather in the space between well-known wave equation, them; that is, in the air, or more strictly, in vacuum. Therefore, which can be derived directly at high frequencies, the propagation velocity of the waves becomes from MAXWELL’s equations. independent of the properties of the conductors which make up the See for example B.E. Reb- transmission line. han, “Theoretische Physik”, Spektrum Akademischer Ver- lag, Heidelberg, Berlin 1999, Chap. 16; or R.P. Feynman 12.6 The Electric Field of a Dipole. et al., “Lectures on Physics”, The Emission of Free Addison-Wesley, Reading, Massachusetts, 1964, Vol. II, Electromagnetic Waves Chap. 24 (available online; see Comment C6.1.) Considering the results in the the preceding section, we can see that the parallel-conductor transmission line at high frequencies represents only an incidental accessory. It simply guides the waves so that they propagate along the line, rather than freely through space.It conducts the electromagnetic waves along a fixed direction just like a tube conducts sound waves in acoustics. Since this role is relatively unimportant, we can in the following forget about the transmission line. This will not hinder in any way the essential process, which is the mutual generation of alternating magnetic and electric fields due to their rapid time variations. We thus arrive at travelling electromagnetic waves which propagate freely in three-dimensional space. This brings us to our last and especially interesting question: How A 12.20 to the in terms E . . It can be S A ,twosmall ). This cur- b and 12.10 They are mounted source S a E C12.4 S shows a convenient arrange- 12.21 has about the same length as the source E A snapshot of . The receiver ). The left part of Fig. ) probes the field E 12.16 11.3 (“source”). A small dipole . This receiver is thus somewhat too large for measurements in the with a gap offlexible ca. cable 0.1 with mm two which conductors (avides forms household the the doorbell connection spark cable) to pro- gap.transformer an A operating AC at current long, 50 source thin, Hz). (around At 5 kV, the a terminals small coils (“chokes”) are includedcable in from the the circuit; high-frequency theymakes currents no isolate in noticeable the the noise; dipole. supply weis hear The only mounted spark a on low gap a humming.now The wooden on, dipole column we about will one-half refer meter to it high. as the From “transmitter” or S tipped, rotated, or carried to anothertion. location at will during its opera- The setup for thein detection Fig. of the electric field remains the same as the electric field of aS dipole receiver ( strength Figure 12.20 position that we want toinfluence. probe; We a again current call will thisreceiver flow probe is in the alternating the “receiver”. current wire with Thedipole. due the current A to frequency in small of the rectifier diode oscillation (detector) ofcurrent, converts the the which AC into we a can direct readily measure using the ammeter of its spatial distribution.we We do are this already using the familiar. methods with We which bring a short length of wire are free electromagnetic waves emitted,to how is the their emission accelerated related motionforth (e.g. of in charges an oscillating which dipole)? are swingingOur experimental back starting and point isWe once briefly again the recall rod-shaped dipole. thedipole. distribution It of exhibits a conduction maximum currents at along the the center (Fig. rent distribution corresponds to a particular distribution offield. the electric Thecorresponding field points lines on the must twoshows somehow a halves of rough describe sketch the large for dipole.the two the arcs ends Figure case of between that the there dipole. is maximumWe charge now want on to investigate the electric field of the dipole In order to avoid disturbances,walls we of have the to room, keepof the the the floor distances dipole. etc. to the Thusploy large we damped compared choose oscillations to a for the(Fig. dipole excitation, dimensions about using 10 a cm spark long. gap as We em- switch ment. The dipole consists of twoend identical surfaces thick are brass coated rods. with Their magnesium flat foil. ). 11.4 Electromagnetic Waves 12 C12.4. The magnesium amplifies the sparks through which the alternating currents flow (Fig. 224 Part I 12.6 The Electric Field of a Dipole. The Emission of Free Electromagnetic Waves 225

Figure 12.21 A small dipole SE S as source (at Part I left) and as re- a ceiver E (right) (a and b are “choke’ b coils, and D is a rectiﬁer diode) (Exercise 12.1)

Figure 12.22 Measuring the radial component of the electric ﬁeld (dipole ﬁeld) in the neighborhood S of the source dipole S r φ E

immediate neighborhood of the source; it would smear out the ﬁner details of the ﬁeld distribution there. This disadvantage of the relatively long receiver is compensated by its great sensitivity.

The receiver is also a dipole. It reacts to the oscillating ﬁeld from the source by undergoing forced oscillations. The similarity of the lengths of the two dipoles guarantees that they can oscillate in resonance.

The receiver (Fig. 12.21, right) is connected to an ammeter by a thin, flexible pair of wires so that it is just as readily movable as the source. We can thus use it to sample the whole field distribution around the source. We start by searching for radial components of the electric field in the neighborhood of the source; that is, we orient the source and the receiver as indicated in Fig. 12.22. These observations are carried out under various angles '. In the neighborhood of the source, we find radially-directed electric field components at all values of the angle ', but their magnitudes decrease rapidly with increasing distance r between the source and the receiver. At distances corresponding to two or three times the length of the dipoles, they are already negligible. We continue our investigation, searching for transverse components of the electric field close to the source. We use the orientation illustrated in Fig. 12.23. These transverse components increase strongly with increasing values of the angle '.Butevenfor' D 0, that is along the dipole axis of S, they still have noticeable magnitudes. E is E of the plane, through r r same

r φ r S S . We now rotate either connecting the receiver to section of a snapshot comes exclusively from the r 12.23 , the field strength is at a max- to ı 12.24 0. Such components can be found 90 image look? We can easily carry out D at various sampling points which are D 12.21 ' E ' . The directions of the arrows indicate the .For from the dipole. The number of parallel arrows ' r , the electric field exhibits a rather simple pattern, 12.24 whole snapshot The distribution of transverse Measuring the transverse component from the source, five or six times the length of the dipoles. r (transmitter). It has propagated over the distance S greater distances components of the electric dipoleous field directions in vari- of the dipole field Figure 12.23 the center of the source dipole. Thus far, we have kept the source and the receiver in the imum; the field is transverse to the line the plane of the page of Figs. Figure 12.24 We then look for transverse componentsdistances of the electric field at larger Here, we find notion measurable of transverse the component dipole along axis, the i.e. direc- at only at larger angles the source or the receiverfield slowly strength out decreases. of It this vanishes plane: whenand then the the the long axes measured receiver of are the source mutually perpendicular. The electric field a vector. According towith the our long investigations, axis it of lies the source in dipole. At a plane together as seen from ourillustrated observations; in it Fig. can be represented graphically as direction of the electric field at the same distance in the figure corresponds tostrength. The the whole magnitude figure of is the just a field, small i.e. the field source free space. electric field of the source dipole. How would the the necessary extension offacts: the image. We start1.ThefieldpatterndrawninFig. by mentioning two Electromagnetic Waves 12 226 Part I 12.6 The Electric Field of a Dipole. The Emission of Free Electromagnetic Waves 227

Figure 12.25 The spatial and temporal alternation of the elec-

tric ﬁeld of an oscillating dipole Part I

S r

2. The field changes periodically at the frequency of the source. The snapshot in Fig. 12.24, if taken a short time later, would be replaced by a similar pattern but with the arrows reversed, that is with the opposite direction of the field. Such patterns alternate continually. With these two facts, we can extrapolate the snapshot image in Fig. 12.24 asshowninFig.12.25. Now we take note of a third fundamental fact: Electric field lines cannot begin or end just anywhere in empty space. In empty space (without electric charges!), there can be only closed-loop electric field lines3. We have to extend the field lines as measured to give closed loops. This is shown in Fig. 12.26. We thus finally arrive at the complete snapshot image as in Fig. 12.27. It shows the electric field of the source dipole at some distance (i.e. excluding the near- field region). This is the radiation field of the dipole, discovered by HEINRICH HERTZ4. As a time-stopped image, it shows the emission of an electric field in the form of a transverse wave which is propagating outwards in free space. Its field strength is indicated by the density of the field lines as drawn. Imagine that the equatorial plane is drawn in and divided up into concentric rings of width =2. Then the surface density of the field lines decreases outwards in these rings as 1=r (r is the radius of the ring), not as 1=r2, like the radially-directed electric field of a charge at rest. This is a fundamental difference between the electric field of an accelerated charge and that of a charge at rest. Figure 12.27, as we mentioned, represents a snapshot, a time-stopped image. Each radial segment of this image can be thought of as mov-

3 The presence of the (relatively sparse) air molecules is quite unimportant for electric processes in space. We emphasize this point once again. 4 Annalen der Physik 34, 551, 610 (1888); ibid. 36, 1 (1889). lting three- r dependence of the field strength r = S ian radiation field of a dipole. If we imagine this to give ERTZ 12.25 The extension of A snapshot (time-stopped) image of the electric field around an ian dipole source to reflect at perpendicular incidence from ERTZ H image to be rotated symmetricallydimensional around distribution its illustrates vertical the axis, the 1 (decreasing resu outwards). the arrows in Fig. complete, closed-loop electric field lines Figure 12.26 of the current in the receiver using an ammeter, and thus the spatial a sheet-metal wall (“mirror”)between and the move mirror the receiver and back the and source. forth We record the relative values ing outwards from the source atthe the vision velocity of of a light. wave This which leads is us propagating to outwardsFor from the the experimental dipole. detectionful of travelling to waves, it convertFig. is them 12.44 always in into use- Vol. 1. standing Correspondingly, waves. we allow the We waves from recall the for example Figure 12.27 oscillating dipole: the H Electromagnetic Waves 12 228 Part I 12.6 The Electric Field of a Dipole. The Emission of Free Electromagnetic Waves 229

Figure 12.28 Measure- 15 ment of the wavelength

of the waves propagat- Part I ing outwards from the dipole shown 10 in Fig. 12.21 (only relative values are given on the ordinate) (Exercise 12.2) 5 Current in the receiver

010203040cm Distance of the receiver from the mirror dependence of the ﬁeld. The result of such a measurement is shown in Fig. 12.28. The nodes of the standing electromagnetic waves can be clearly seen as minima in the recorded ﬁeld strength. The spacing of the nodes is found to be about 0.18 m. Therefore, the wavelength of the standing waves, and thus of the original travelling electromagnetic waves, is about 0.36 m in this example. From Eq. (12.1), the frequency of the dipole oscillations is given by:

3 108 m=s 800 MHz : 0:36 m This experiment has a minor defect: The standing waves are clearly formed only in the neighborhood of the mirror. Further away, the minima become ﬂatter and ﬂatter. The reason for this is the strong damping of the oscillations of the source dipole. The individual wave trains initiated by the spark switch are too short; they resemble the curve shown for example in Fig. 11.18a at the upper right. At larger distances from the mirror, the strong displacements from the beginning of the wave train are superposed onto the weak displacements at the trailing end of the same wave train which are just moving out from the source. This yields only weakly dis- cernable minima (cf. Optics, Sect. 20.11).

The pattern of wave emission from a dipole as sketched in Fig. 12.27 can thus be completely confirmed by experiment. An oscillating electric dipole emits travelling waves with their electric field perpendicular to their direction of propagation (transverse waves) into free space. The field-line pattern of the dipole still requires two complementary additions: In Fig. 12.27, the part of the field nearest to the dipole (the “near- field region”) is not drawn in. In that region, it changes with the momentary charge state of the dipole. We give a brief description in the caption of Fig. 12.29. Furthermore, the magnetic field of the dipole must be discussed. The magnetic field lines are concentric circles (Fig. 12.30). They lie in planes perpendicular to the dipole axis. The density and direction 2 / λ 2 / λ 2 / λ is based upon this close a b c d e 2 / λ S electromagnetic wave The magnetic field lines of an oscillating dipole Five snapshots of the electric field near an oscillating dipole:the a. start Before of the oscillations, boththe halves dipole of are uncharged.no There electric are thus field lines connectingThe them. conduction current b. has begunupwards. to After flow one-fourth of an oscillation cycle, it has charged thedipole upper positive, half and of the the lowerBetween half the negative. halves of thenow dipole field there lines are which loopoutwards. some c. distance During thecycle, second the quarter- magnitudes of theboth charges halves on of the dipoleHere, again they decrease: have already decreasedhalf. by The about outer part offurther the outwards, field and has at moved theodd same constriction of time, the an field linesdipole near has the appeared. d.second At quarter-cycle, the the end two of halves the dipole of are the again uncharged.of The the constriction field lines isquarter-cycle, complete. e. the In conduction the current third isflowing now downwards in the dipole,to leading a negative charge ona its positive upper charge on half its and lowerend half. of At the the third quarter-cycle,resembles the Part pattern b, except that(field the directions) arrows are now reversed. Figure 12.29 Figure 12.30 of the magnetic field linesmoves alternate outwards periodically. together with Thetance magnetic the field from electric field; the atelectric source field. a (“far-field sufficient region”), dis- it isEvery in variation phase in witha the magnetic the electric field. fieldan And (displacement electric every field current) with variation closed-loop creates ofagation field lines, the of by magnetic the induction. field entire The prop- creates interlinking of the electric and the magneticAt fields. a sufficiently great distancecan be from described the as oscillating a dipole, plane the wave wave propagating in the direction of the Electromagnetic Waves 12 230 Part I 12.7 Wave Resistance 231

x Part I By

z y

Figure 12.31 An electromagnetic plane wave propagating in the z direction consists of an electric and a magnetic component, which are polarized in the x and the y directions, respectively, and oscillate in phaseC12.5 (Exercise 12.2) C12.5. For completeness, we mention that the energy transported by an electromagnetic positive z axis at the velocity of light c (Fig. 12.31). Within this wave, wave, i.e. the energy current the electric fields Ex oscillate in the x direction and the magnetic density, is described by the fields (or the magnetic flux density By) oscillate in the y direction. POYNTING vector: Expressed as equations, this corresponds to 1 S D .E B/ Á Á z z 0 E D E ; sin ! t and B D B ; sin ! t : (12.2) x x 0 c y y 0 c (unit: 1 W/m2). With The two waves thus oscillate in phase. The relation between their Eqns. (12.2)and(12.3), we find amplitudes is given by Á 1 z S D E2 sin2 ! t Ex;0 c x;0 c By;0 D : (12.3) 0 c which has the time-averaged Today, we can send free electromagnetic waves all around the globe. value 1 1 They propagate along great circles, experiencing multiple reflections S D E2 x;0 . on the upper layers of the atmosphere. These layers are ionized due 2 0c to radiation from space, and thus have a high electrical conductivity This quantity plays a role in (cf. Optics, Sect. 27.19). The earth’s circumference of 4 104 km is optics. traversed within 0.13 s, i.e. the waves could circle the globe seven times in 1 s. This makes possible a direct measurement of the propagation velocity of electromagnetic waves using their path travelled and their transit time.

12.7 Wave Resistance

For sound waves (Vol. 1, Sect. 12.24), we defined the quotient Pressure amplitude p D % K D Z (12.4) Velocity amplitude .K D module of compression/ as the acoustic wave resistance for parallel beams of sound waves at normal incidence onto a boundary between two media. It determines the reflection coefficient at the boundary interface between the media. We found (Vol. 1, Eq. (12.50)) Â Ã Reflected radiation power Z Z 2 R D D 1 2 (12.5) Incident radiation power Z1 C Z2 is )). E (12.7) (12.9) (12.6) (12.8) 12.5 ECHER (12.10) l and the spac- ; in Eq. ( . A = ld A r E d Cross sectional area Cross sectional 2 0 0 Z ( " )) for parallel beams 333 l r D for the sound waves. I are proportional to each 12.5 = D : : E 377 U H d H : 1 377 D and r in Eq. ( a E H D or D d 1 E Z l A 0 D 0 ( = electromagnetic waves, the value of the " E 16, we find 1 1 el 120 ln 0 Z D r E D D r 0 D A square piece of such a foil is sketched = " D I of U a I U I reflection coefficient p el d made of some material with the specific elec- = conducted Z ic resistance in the direction of the vector d U ), = D C12.6 or . 1 HM c E or the C12.7 D 12.3 D I guided = B , we show an electromagnetic wave which is approach- The calculation of .ItsO U of the two parallel conductors. Their wave resistance is 12.32 reflectivity 12.32 a wave resistance of vacuum ing A numerical example: For wave resistance depends on the shape of the electric field. In a L In the case of system, for example, this shape depends on the diameter 2 it follows that Foils are commercially availablehave which a as resistance squares of of arbitrary size as the in Fig. Such foils preventnetic the plane waves. reflection of normally-incident electromag- other. From Eq. ( of plane waves. For these waves, trical conductivity In Fig. ing a poorly-conducting wall. Wea can thin imagine foil of the thickness wall to consist of given by the resistance Figure 12.32 In a completely analogous fashion, forfine electromagnetic waves the we quotient de- for the a square piece of foil parallel to ’s . 2 ). We ): Á and 2 2 13 n n 0 C " RESNEL 12.5 1 1 1 n n . Together =" 0 ) from the 0 " 2 D 2 is discussed in =" p ) from F 0 Ã 12.11 ) (cf. Chap. 2 2 D 2 Z Z " 1 p 25.8 Z C Electromagnetic Waves 1 1 D and Z Z 12 2 1 have This result is derived later in Sect. ( formulas. following section, we find for the reflectivity Â with Eq. ( Z C12.7. We mention a further application of Eq. ( C12.6. The specific conductivity Comment C1.10. We calculate the reflectivity for an electromagnetic wave which is normally incident on the boundary surface between two dielectric media 1 and 2 (with the dielectric constants " 232 Part I 12.8 The Essential Similarity of Electromagnetic Waves and Light Waves 233 12.8 The Essential Similarity

of Electromagnetic Waves Part I and Light Waves

The HERTZian transmitter (Fig. 12.21, left) has a nearly ideal simplicity and clarity. With such a transmitter, we can readily demonstrate the analogous behavior of electromagnetic and optical waves. We have already observed reﬂection, interference and linear polarization. The vector of the electric ﬁeld always oscillates within a plane which contains the axis of the transmitter dipole. A linear receiver perpendicular to this plane (Fig. 12.21 right) shows no signal.

HERTZ described a very impressive demonstration of this polarization of the dipole radiation, the so-called grid experiment5. We place the transmitter and the receiver parallel to each other. Then we insert a grid made of metal wires with a spacing of ca. 1 cm between the two. First, the wires are oriented perpendicular to the dipole axis and the direction of the electric field; this causes no noticeable weakening of the waves detected by the receiver. Then the grid is rotated by 90ı, so that the wires are parallel to the dipole axis. It now proves to be completely opaque to the waves. The wires parallel to the direction of the electric field produce a “short circuit” and act just like a solid metal wall. A similar experiment can be carried out in optics; however, for this we use invisible infrared waves ( D 100 m). At the shorter wavelengths of visible light, it is not possible to fabricate a sufficiently fine wire grid; instead, one uses “polaroid foils” (Sect. 24.3). These contain oriented long-chain polymeric molecules with alternating double and single bonds, which are electrically conducting along their long axes and thus act as nanoscopic “wires”.

For the demonstration of refraction, a “cylindrical lens” sufﬁces. This is a large bottle which is ﬁlled with a dielectric liquid, e.g. xylol. Its long axis is oriented parallel to the receiver dipole. Using prisms of adequate size, HERTZ was able to determine the refractive index n of several substances in his classical experiments with electromagnetic waves. He found that n is equal to the square root of the dielectric constantp " of the substance of which the prism is made. This relation, n D ", had already been predicted by MAXWELL on the basis of his electromagnetic equations. It plays an important role in the theory of dispersion (Optics, Sect. 27.8). p MAXWELL’s relation n D " follows from Eq. (8.7). In a material with the dielectric constant " and the permeability (this will be treated in Sect. 14.1), the products ""0 and 0 replace "0 and 0 in MAXWELL’s equations. We thus obtain p c ""00 p Index of refraction n D vacuum D p D " : (12.11) cmaterial "00

The permeability of most materials, apart from ferromagnets,p is practically always 1 (Sect. 14.3); we thus obtain n D " (Exercise 12.4).

5 Annalen der Physik 36, 769 (1888). line). along ECHER guided ) using a control ing microphones. . focussed into beams klystron modulated ), modulated electromagnetic oscillations using normal elec- aph key, or by utiliz ARCONI Hz, owing to the finite travel time of the . They were modulated by employing shows schematically an antenna designed 9 10 2.12 3 carriers D Microwaves. Demonstration Experiments for Wave Optics of Electromagnetic Waves In the course of these developments,were shorter and employed. shorter For wavelengths theirthink great of and the often transmission admirableetc.! of achievements images – – of communications engineers thefor had surfaces sources to of of develop Mars, new undamped Jupiterdown technologies to electromagnetic centimeters waves (today, with downagation to wavelengths millimeters) and and for their theirenriched precise prop- physics control. laboratories forsections will research These offer and some technologies examples teaching.to of fundamental have physics. these also The developments as next applied electrons which make up thelimit space charge. for This the means wavelengths that ofsuch the the tubes lower electromagnetic is waves aroundemployed generated 10 cm. to in achieve frequencies However, up thisthe to travel mm 100 time GHz, range. itself i.e. can wavelengths Wehave in be describe proved here to be one practical, of based the on several the methods which waves have beenInitially this increasingly was used achievedin with all as freely directions propagating (Fig. communications waves travelling carriers. to receive such waves).using Later, concave the mirrors waves (as were in a searchlight) or were For generating undamped electrical tron tubes or triodes,its an cathode electron and current the flowsthe in anode space the (plate). charge tube is The between periodically charge varied (number) density ( of 12.10 The Production of Undamped 12.9 The Technical Importance The technology of electronicginnings communications, in from the its first earliestdirect half be- of currents the 19th as century, for many decades used switching devices, e.g. a telegr voltage applied between the gridwith and special the designs, it cathode. is not Incies possible this above to way, 3 generate GHz even oscillation frequen- The transmission lines thenalternating carried current a in chopped thefrequencies, direct audio frequency currents). current range or of an humanBeginning speech (audio in 1896 (G. M double-conductor lines (a further development of the L Electromagnetic Waves 12 234 Part I 12.10 The Production of Undamped Microwaves. Demonstration Experiments for Wave Optics 235

U Part I 2 c β α

Hot cathode b E

1 2 3 by the electrons

Electric field 4 Height traversed in the condenser the in 011 1 3 5 3 a 4 2 4 4 2 Time in multiples of the period T

Figure 12.33 The autocontrol (feedback) of electrical oscillations using a re- ﬂection klystron. At the left, a schematic; at the right, a photograph of a technical device, with rotational symmetry, for frequencies of around 10 GHz, which can be regulated by varying the distance ˇ ˛ (screw at b) (ca. 2/3 actual size; the operating voltages are U1 300 V, U2 150 V; a indicates a coaxial cable connector for outputting the electromagnetic waves).

Figure 12.33 (upper left) shows an oscillator circuit schematically. The plates of its very flat condenser (at ˛ and ˇ) are perforated like sieves. Electrons which are emitted by a hot cathode (below) can pass through the openings. Their flight times t within the gap of the condenser are short compared to the period T of the oscillator circuit. Furthermore, suppose that some quite weak oscillations are already present due to random fluctuations (from thermal motions). Then bunches of electrons leave the condenser at its top with a modulated velocity. Within each period T, when the upper condenser plate is positive, the emerging electrons have higher velocities, up to (u C du) at the time T=4. The electrons which emerge later, at the time T=2, when the upper plate of the condenser is not charged,havethe same velocities u with which they entered through the lower plate. Still later, when the upper condenser plate is negative, the emerging electrons have reduced velocities, down to (u du) at the time 3T=4. During this modulation of their velocities, the number density of the electrons remains constant. In order to modulate it as well, we let the electron bunches be reflected by a negatively-charged plate P,so that they return to the condenser. The fastest electrons (with velocities up to (u C du)) have to traverse the longest paths upwards and downwards; electrons with the original velocity u traverse a path of medium length, and the slowest electrons (with velocities down to (u du)) traverse the shortest paths (Fig. 12.33, lower left). . There, the . Second: They pass 2, when the condenser magnetron = 10 GHz) can be coupled T D is a coaxial connector where

4; the electrons of the second packets with their modulated a = , upper left) are properly chosen, T 4), all the electrons arrive back at = simultaneously . This periodic energy input initially T 12.33 . This occurs periodically, always at times (Fig. , start each time at u 2 U : A cross section through and 4, when the upper condenser plate has its maximum 1 condenser plate is negative. As a result, the electron = 4 and ending at 5 (right side) shows a wave generator based on this U = T : The plate in a plan view (the sec- Above T . Speaking figuratively, it is made as a regular crystal. Its lower 12.33 Below 12.34 then two conditions areeach met. time at First: After each full cycle (beginning Figure 12.34 The fastest electrons ofthe the moment first when group the beginis their positive at charge flight its on upwards maximum, the at i.e.group, upper with each the condenser time velocity plate at a lens for electromagnetic waves.elements The or scattering “atoms” are thumbtackspressed which into are a polystyrene platewaves). (transparent to the tion marked above with an arrow). (See Sect. 27.6) the high-frequency alternating current ( the upper condenser plate nearly downwards through the condensernumber in densities combined when the packets are slowed; they givethe up electric a field of portion the of condenser amplifies their the kinetic weak energy oscillations, and to thenamplitude. maintains them at a constant Figure method, for wavelengths of ca. 3 cm. If the voltages plate is uncharged; and theeach slowest time electrons at of 3 thenegative third charge. group start out of thethis generator, transmitter, e.g. and a for receiverwe feeding antenna can with a a demonstrate transmitter detector allat and antenna. the a ammeter, basic somewhat higher phenomena With cost ofwavelength but sound wave no waves propagation, less (Vol. 1, convenientlyadd than Sects. with to 12.18 short- the throughFig. 12.20). examples given We in Vol. 1 by demonstrating a lens in “atoms” consist of thumbtacksdetails are ordered given on in the a figure square caption. Among lattice. the other devices More used tolations generate at undamped electrical very oscil- high frequenciestimes which of are electrons, based the on most the important finite is transit the Electromagnetic Waves 12 236 Part I 12.11 Waveguides for Short-Wavelength Electromagnetic Waves (Microwaves) 237 energy required to maintain the high-frequency oscillations at constant amplitude is provided by electrons that are moving on circular orbits at their cyclotron frequencies in a magnetic field (see Comment Part I C8.1). With such magnetrons, which played an important role in the early development of radar, electrical oscillations at frequencies between 1 and 100 GHz can be generated, corresponding to wavelengths between 30 cm and 3 mm. Magnetrons with an output power in the range of several kilowatt are used today in microwave ovens (see Sect. 13.11).

12.11 Waveguides for Short-Wavelength Electromagnetic Waves (Microwaves)

In a LECHER line, the distance between the two conductors is small compared to the wavelength of the electromagnetic waves they conduct. The LECHER line thus corresponds to the speaking tubes which were once common in households and in ships. The LECHER system can also be redesigned: One of the conductors can be formed as a tube which surrounds the other concentrically, and the two conductors can be separated by dielectric spacers or a dielectric material which ﬁlls all the space between the outer conductor (tube) and the inner central wire.C12.8 C12.8. However, when the space between the conduc- Starting with such a concentric, coaxial LECHER system, one ar- tors in such a coaxial cable is rives at a waveguide when the axial (center) conductor is left out ﬁlled with a dielectric mate- completely (LORD RAYLEIGH, 1897)6. Then only the outer tube rial (plastic) with an index of remains. In practice, usually a rectangular cross section is employed. refraction n, the waves prop- A waveguide has quite different properties from a speaking tube or agate along the cable only at its electrical analog, a LECHER line: A waveguide transmits only the velocity of light in that those waves whose half-wavelength is smaller than the largest inner medium, that is at c=n. dimension of the waveguide. Here, we must distinguish between two different velocities: First, the velocity u at which a signal carried by the waves, that is a wave train or packet with a beginning and an end, called a wave group, can move on a zig-zag path along the axis of the waveguide. Second, the phase velocity v at which a wave which is modulated transversely to the axis travels through the waveguide. These two velocities are related to the velocity c of electromagnetic

6 Concentric LECHER lines have the disadvantage that they often function in addition as waveguides. This can be prevented only by reducing the cross-section of the outer tube. This reduction, however, leads not only to technical difficulties (centering of the axial conductor, sufficient electrical insulation), but also to an un- acceptably large increase in the damping. The latter increases when the quotient circumference/cross-sectional area increases. ; s). Z / a' b' with 8 mm, D1 direc- : modu- ˇ 250 5 (12.12) z v = configu- 1 . D B (at the right is , ˇ ı 76 produces travelling sin ; that is, the spacing direction, whose am- ˇ to the propagation di- D momentary = z Direction of the modulation λ* T T' ˇ Y D D 2. In standing waves, the l l . Its amplitude, that is the and travels in the = ˇ ˇ ) and the depth of its troughs transverse 0 D : cos cos B B' B' 2 2 = BB c = c l = shows a well-known phenomenon D D , we find for the phase velocity ı D by the equation v v u direction, i.e. the wave is thus 7

90 a b y . (In the example: D 12.35 ˇ axis, and are thus nodes y β wave. Its modulation length is /sin β λ = l /cos ), vary along the 25 mm) 0 between the two wave trains increases, so does the phase velocity direction. Their amplitudes, that is the heights of their crests and standing of the resultant waves propagating along the Please look at this ﬁgure with one eye! The interference between two D z ˇ TT λ* = λ

ˇ modulated wave Wavelength of the Wavelength 2 1 Free space (vacuum) is non-dispersive, and therefore, its signal or group velocity cos tion at the high phase velocity height of its wave crests (e.g. along each other. The twotrain; it wave has trains the superpose wavelength to give a resultant wave from mechanics. At the left is a sketch of the ration, and at thewater right resulting a from photographicTheir image, interference directions of between of surface propagation two waves 1 on linear and wave 2 trains. make an angle of 2 These properties of waveguides, which atstrange, first have may a appear single somewhat origin:wherever The the electric electric field field strength lines must of beWe the must zero wave explain approach this the in metal somewhat walls. more detail: As an introduction, Fig. (e.g. and its phase velocity are identical. 7 waves in free space (vacuum) = c 6 mm, λ λ D Modulation length Modulation β D 2 between neighboring interference minima is interference minima are called l the result is a plane waves of the same frequency which meet at an angle 2 rection; they are modulated with the modulation length waves in the troughs, are directed along the Figure 12.35 a snapshot of waves on a water surface, taken with an exposure time of As the angle v plitudes are transversely modulated.experiment. At This the can limiting be angle readily seen in the demonstration Electromagnetic Waves 12 238 Part I 12.11 Waveguides for Short-Wavelength Electromagnetic Waves (Microwaves) 239

β Z Part I β

Figure 12.36 Construction of a wave which propagates in the z direction and is modulated transversely to its direction of propagation. It utilizes the multiple reflections of a plane wave on two perfectly reflecting walls placed at the interference minima. ˇ has the same meaning here as in Fig. 12.35. Along the zig-zag path, energy is transported at the velocity c, while along the waveguide axis (z direction), it is transported only at the group velocity u D c cos ˇ. The figure also explains the propagation of energy in any interference field: The interference minima act, figuratively speaking, as perfect mirrors. lated transversely8 to the propagation direction: The wave crests are divided by depressions and the troughs by elevations which follow each other at a spacing l D = sin ˇ (called the modulation length). Half of this modulation length is the distance between neighboring interference minima, i.e. the lines along which the amplitude remains zero (for example the lines connecting the points aa0, bb0 etc.). The wave sequence discussed in Fig. 12.35 can be produced experimentally in a simple fashion. This is shown in Fig. 12.36: The region where the waves can propagate freely is bounded above and below by two perfectly reflecting walls (shaded). The spacing of these walls is taken to be an integral multiple N of half the modulation length, i.e. l=2. At the left, a single wave train enters the region between the two mirrors, and it passes along them at the phase velocity c (see footnote 2), following a zig-zag path. As a result, a signal propagates in the z direction only at the low velocity u D c cos ˇ. We thus find, since the phase velocity of the resultant wave with wavelength is v D c= cos ˇ,thatuv D c2. So much for the results with mechanical (surface) waves. Their application to electromagnetic waves in waveguides need only be elucidated with the help of an example. Figure 12.37 shows a first approximation: Instead of a rectangular waveguide, we see a region in the field of a linearly-polarized electromagnetic plane wave which is bounded by two conducting sheets. The wave propagates in the z direction with its electric field in the x direction. The amplitudes of the wave are independent of the y direction, apart from some unimportant edge effects; therefore, we can represent the amplitude all along the y axis with arrows of the same length. This is shown in the figure at two amplitudes. Figure 12.38 shows a different case: Here, the two vertical conducting sheets are joined above and below by horizontal sheets to make a closed profile with a rectangular cross section. Now, a homogeneous distribution of the electric field in the y direction is impossible: The electric field has to be zero above and below where its field lines

8 In communications technology, an amplitude modulation of waves along their propagation direction also plays a significant role. . di- y 12.39 Handbuch der ). In the case of a' b' ı = 13 mm λ 54 λ might be used. D 9 = 22 mm = 16 mm = 13 mm = 54° = 54° λ β ˇ β β /sin /cos is the largest wavelength which X l = λ Z would correspond to Fig. λ* B λ* = λ of the wave amplitude in the 2 E Y D 12.38 waveguide. The waves must be modulated in Z X modulation , the modulation length has been chosen to be direc- can be achieved for them within the waveguide x Y a b N AYLEIGH = l 12.38 B B 2 The corresponding picture after the two vertical conducting Adrawing D l 3 (corresponding in the example to = , Flügge, Vol. 16 (1958). B direction, that is perpendicular to their direction of propagation, so that 2 . y is an integer). Therefore, y D F.E. Borgnis and C.H. Papas, “Electromagnetic Waveguides”, N can be transmitted through thefill waveguide. Longer this waves cannot requirement, ful- at namely the that points the wherewaveguide. electric its field field lines be graze exactly the zero conductingInstead of walls a of tube with the a circularof or rectangular other cross section, forms a number canwire, be a single used wire with as afor waveguides. dielectric demonstration covering, experiments, or For a in rubber the example, hose laboratory a and helical Waves can propagate through this waveguide only when alength modulation of ( the water surface waves, Fig. sheets have been joinedangular above and profile, below a by R horizontal sheets to give a rect- in perspective of a region bounded by two conducting sheets in which an electromagnetic plane wave is propagating with its electric field oscillating in the tion. Its field lines meettwo sheets the at normal incidence and end there. The electricfield amplitude is independent of Figure 12.37 the Physik 9 Figure 12.38 graze the conducting wallsThis (corresponding is to achieved by a a perfect reflection). rection. In Fig. their electric field strengthintersect becomes the conducting zero walls. at the points where the field lines l Electromagnetic Waves 12 240 Part I Exercises 241 Part I

Figure 12.39 The same field distribution as in Fig. 12.38, here as a section from a photograph of water surface waves. The saddle points along the wave crests correspond to regions with field strengths E pointing in the positive x direction, while in the troughs, E points in the negative x direction. At rest, the water surface would correspond to the area bounded by dashed lines in Fig. 12.38, where the vector arrows representing the electric field all begin.

For measurements and experiments which use electromagnetic waves in the centimeter and decimeter wavelength range, waveguides are just as important as wires and coaxial cables for the usual electrical measurement instrumentation. There is a voluminous specialized literature with its own terminology. For example: Reflections on the tube walls are decisive for the formation of tube waves. As is known from optics, reflections are dependent on how the electric field E and the magnetic field H are oriented with respect to the plane of incidence (Optics, Sect. 25.6). If E lies perpendicular to the plane of incidence10, i.e. it is “transverse”, then H has a component parallel to the long axis of the tube. If H lies perpendicular to the plane of incidence (“transverse”), then E has a component parallel to the tube axis. In the former case, waveguide engineering classes the waves as TE or M waves; in the latter case, as TM or E waves. Often, indices are also added to denote the number of modulation lengths parallel to the long or short sides of the rectangular waveguide.

For measurement purposes, a narrow longitudinal slit is cut into the wall of the waveguide. Then a small receiver (cf. Fig. 12.16) can be slipped in, parallel to the waveguide axis. It can be used to determine the wavelength when a reﬂecting wall at the end of the waveguide is used to produce standing waves. Thus, for example, we can measure e.g. the wavelength in Fig. 12.38 which belongs to the high phase velocity v D c= cos ˇ.

Exercises

12.1 In Fig. 12.21, the ammeter indicates a current, although of course it responds only to currents of very low frequency. Explain this. (Sect. 12.6)

12.2 In order to understand the observed standing electrical wave in Fig. 12.28 as the superposition of two travelling waves, consider an electromagnetic wave which is incident along the z axis from z > 0 onto a metal mirror at z D 0. The motion of its electric-ﬁeld component is described by Ex D Ex;0 sin !.tCz=c/. At the mirror, this wave

10 TheplaneofthepageinFig.12.36,theyz plane in Fig. 12.38. https:// 0). b) In D for the mag- z tot B in the element, the direction, opposite to E z and the propagation di- , electromagnetic waves . What is the amplitude x B E , tot E B . It oscillates in phase with the y ) B ) contains supplementary material, which The online version of this chapter ( )), where 12.6 12.2 , a node is present at the mirror ( ). (Sect. (see Comment C12.5.) , a dipole in water is excited to electrical oscilla- S 12.28 19.3 of the magnetic field at its surface, and the resulting 12.12 1 cm. a) At a voltage of 220 V and a current of 4.5 A of the water in this experiment (see Comment C12.7). B vector R ) ) D ). The reflection leads to a standing wave , Sect. . a) Write the equation which describes the reflected wave, b r In Fig. The heating element of a hotplate is 1 m long and has a diam- tot at the surface of the mirror? (The trigonometric formula for 12.6 12.8 E 12.31 tot B OYNTING P of rection are oriented toFig. form a right-handed coordinate system (see electric component (Eq. ( netic component, also. Findwaves expressions which for are the superposed, two magnetic-field and for addition to the electric-field component so that, as in Fig. b) Compare the values obtained withconstant that of the sun’s radiation (solar 12.4 have also a magnetic component the incident wave. The twowave wave trains superpose to give a resultant Electronic supplementary material doi.org/10.1007/978-3-319-50269-4_12 is available to authorized users. is reflected and it propagates in the positive 12.3 eter of 2 (effective values), calculate the electric field the sum of two(Sect. sine functions is helpful for solving this exercise.) flux density tions by electromagnetic waves which aresel radiated into through the its surface glass at ves- normalreflectivity incidence from the outside. Find the (Sect. Electromagnetic Waves 12 242 Part I banfrtedeeti constant: dielectric the for obtain atri nEeti Field Electric an in Matter tafie voltage fixed a At Dielectric the Measuring 13.2 by materials insulating place in decimal fields fourth consider the we in space. when – consider- However, only empty our felt in in digits. is neglected 6 fields be influence electric could their with molecules ations; air only of dealt presence have The we far, Thus Dielectric The Introduction. 13.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © ewe h ltso odne nrae t aaiac.Tu,in Thus, capacitance. its dielectric increases a condenser Sect. Such a of different. plates becomes the between situation the solids, and liquids C est rdslcmn density displacement or density uteto h charge the of quotient t capacitance its h hre n t ufc est,ices hnadeeti sin- is dielectric a when increase density, Figure surface serted. its and charge, The aigcret utemr,tesniiiyo h esrmn is measurement the alter- of utilizing sensitivity by the achieved Furthermore, is this current. employed; series nating periodic is a impulses condenser, current the of of one cycle the only charging of or for instead impulse Usually, current developed constant. dielectric been the of have measurement arrangements experimental Various m " dielectrics slre than larger is D 2.17 Capacitance epooe ento fthe of definition a proposed we , Constant Constant ihadnepcigo tm rmlcls htis that molecules, or atoms of packing dense a with – 13.1 C Capacitance C ( U 0 C hw o ecnmeasure can we how shows C ,sothat h charge the , m D Q facnesrwihi le ihmatter with filled is which condenser a of olsItouto oPhysics to Introduction Pohl’s n h lt area plate the and Q " " " = D U " C sawy ubrgetrta one. than greater number a always is .Fraprle-lt odne,the condenser, parallel-plate a For ). Q 0 Q fteepycondenser empty the of m 0 Q D D nacnesri rprinlto proportional is condenser a on D D D Q m 0 = : A ilcrcconstant dielectric A ie h ufc charge surface the gives (Sect. " basedonthisfact. , https://doi.org/10.1007/978-3-319-50269-4_13 2.13 .Te we Then ). : C13.1 ( (13.1) 2.25 : ) as literature, berofexamples.Inthe num- a with following the in demonstrated is as constant, eoe ipyby simply denoted therefore, (and “permittivity” the called iiyo vacuum”). of tivity 1..I general, In C13.1. product The permittivity”. “relative the called is and vacuum”) " r rmas“eaiet the to “relative means (r " 13 0 " " " 0 r sotndenoted often is sfrequently is ste“permit- the is " " snot is n is and 243

Part I x C C 2 )is R G (13.2) 1.5 2 2.8 4 3.4 4–6 6–8 12.4 method, 1 2 G ). Then 2.3 1 R 1 ). C difference line (Sect. 13.2 or gives the electric field l .Thein- " : ECHER 0 m E E are vari- :AL 2 . D " R 0 m compensation is also known remains constant (Fig. U 1 U and RUDE ). Then we have for the dielectric con- C 1 Q Liquid air Petroleum Amber Bakelite board Polyvinyl chloride (PVC) (50 Hz) Porcelain Glasses ) D R " 2.12 . For a parallel-plate condenser, the quotient U = (Sect. Video 2.1 l ( ic resistors = " ) holds for a fixed voltage. When the voltage source U HM divided by the plate spacing ) A bridge circuit for comparing two Measuring the dielectric constant D U The dielectric 13.1 of various mate- E " Exercise 13.3 The voltage or theelectric electric is field inserted; strength fordrop thus when the decreases the measurement, condenser when we is aproportional completely to determine di- filled the with the voltage material. It is able and known; the capacitance constant rials Figure 13.1 capacitances. The measuring instrumentvanometer (e.g. or a an gal- oscilloscope) is useddetector. as The a O zero crease in the charge ona the current condenser impulse by plates a is ballisticof determined galvanometer. the The as galvanometer deflection is proportional to ( for example some type of “bridge circuit” (Fig. Materials which are poor insulators require aple, special using setup; for the exam- method of P. D Table 13.1 improved by using some sort of Figure 13.2 Equation ( is disconnected, the charge is proportional to 1 of voltage strength, stant: . Matter in an Electric Field 13 Video 2.1: “Matter in an electric field” http://tiny.cc/s9fgoy It is shown how inserting various materials into the electric field of a parallel- plate condenser which has been disconnected from the current source produces a decrease in the condenser voltage. Its capacitance is thus increased. In this experiment, the space between the condenser plates is only partially filled by the material. 244 Part I 13.4 Distinguishing Dielectric, Paraelectric and Ferroelectric Materials 245 immersed in the liquid to be measured and the reduction of the wavelength of radiation along it relative to its wavelength in air (vacuum) is determined at that frequency (see also Sect. 13.11 and Fig. 12.12). Part I Some results of measurements for various dielectric materials can be found in Table 13.1.

13.3 Two Quantities Which Can Be Derived from The Dielectric Constant "

The dielectric constant, which is in general readily measured, yields two other quantities which are important for physics and chemistry. They are: 1. The dielectric polarization:C13.2 C13.2. It is not necessary to D : use vector notation at this P Dm D0 (13.3) point, while we are con- The dielectric polarization is thus the additional contribution to the sidering a parallel-plate condenser; it will be intro- ﬁeld Dm which results from production or orientation of electric dipoles in matter (Fig. 2.56). Its unit is e.g. A s/m2. Another deﬁni- duced later in Sect. 13.5. tion is equivalent: The dielectric polarization of a material is: Electric dipole moment p P D : (13.4) Vo lu m e V

Derivation: We imagine a block of material (base area A, height l)tobe homogeneously polarized. Then on its end surfaces, there is a bound electric chargeC13.3 Q D PA. Furthermore, from Sect. 3.9, its electric dipole C13.3. In contrast to the moment p D QlD PA l D PV, and therefore, P D p=V. “free” charges on the plates of the condenser. 2. The dielectric susceptibility

Dm D0 P e D " 1 D D : (13.5) D0 "0E0

The susceptibility can also be referred to the density % of the material, e=%. This quantity is called the relative susceptibility of the material.

13.4 Distinguishing Dielectric, Paraelectric and Ferroelectric Materials

After explaining the measurement methods, we now wish to give a brief overview of the behavior of insulating materials (“dielectrics”) in an electric field. All these materials can be grouped into three large classes, which can be considered to be ideal limiting cases: e and ma- 30 30 30 30 30 Asm) " by the , inde- and p 10 10 10 10 10 30 " . 1are 1 4 in A s m : : " 1.00006 1.00055 1.05404 1.00095 1.00035 1.464 10 34 13 60 p 6 3 : : p 6 — 5 — 13.3 " 4 : 3 C D ı b e C ı sets out some numerical C ı C ı " 1.0072 — 16 31.2 56.2 81.1 — C ı electrically deformed 13.2 183 molecules; examples are given C C ; they increase with decreasing ı ı C ı and molecular electric dipole moments .Table C C C C liquid, polar ı ı ı ı " 2 20 1atm,0 — 18 18 18 1atm,0 Substance Helium, 1 atm, 20 Air, 1 atm, 18 Air, 100 atm, 0 Carbon dioxide, 1 atm, 0 Bromine vapor, 0.16 atm, 20 O and susceptibilities " permanent electric dipole moments p are quantities which are independent of the by the applied electric field due to influence polarized e ) " , i.e. they are independent of the electric field which )(gas) 3 are set out in the second column of Table induced of some temperature dependent ). The applied electric field tends to align these initially Pa) " " Dielectric constants 5 The dielec- 1!)(1atm ). The non-polar molecules are 10 13.3 > 2.56 013 : of some paraelectric materials (cgs unit: 1 Debye Paraelectric materials with polar molecules Dielectric materials with non-polar molecules 1 p HCl Substance Ammonia (NH KCl (in a molecular beam) Ice Methyl alcohol Glycerine Water Exercises 13.6, 13.7 ( temperature. Interpretation: The molecules ofelectrically paraelectric deformable, materials like arematerials, the not but non-polar only also have molecules of dielectric randomly-oriented microscopic dipoles along its axis; however, the pendently of the applied field ( in Table produced the polarization. At constant density, theydent are of also the indepen- temperature. In an atomic picture, thismaterials means themselves that have the no molecules of electricmoments these dipole dielectric are moments. Their dipole terial constants (Fig. field and thus become values for dielectric materials. 2. For this group of materials, also, the dielectric constants D tric constants strength of the electricexamples field; of they are material constants. Numerical p Table 13.3 Table 13.2 are however 1. Their dielectric constants dielectric materials ( always the susceptibilities Matter in an Electric Field 13 246 Part I 13.4 Distinguishing Dielectric, Paraelectric and Ferroelectric Materials 247

Figure 13.3 The inﬂuence of the voltage I U on the dielectric constant " of a fer-

roelectric crystal. Above: Schematic of U E Part I C C the principle; Below: Observation using an oscilloscope (at the left is a block of U I SEIGNETTE salt crystal with a thickness of d D 1 cm, and two end surfaces of a ca. 3 3cm2. The right-hand condenser C has a capacitance of around 2 106 F, the left-hand condenser about 5 108 F; therefore, nearly all of the voltage drop U occurs across the crystal). Ueff ≈ 300 V molecular thermal motions oppose the alignment and tend to maintain the random orientations of the molecules (see Sect. 13.10). 3. Ferroelectric materials As representatives of this group, we mention potassium sodium tar- trate (C4H4O6KNa C 4H2O), discovered by the pharmacist P. SEIG- NETTE (1660–1719); and barium titanate (BaTiO3), as well as potassium dihydrogen phosphate (KH2PO4, abbreviated KDP). Ferroelectric materials are characterized by the extraordinarily large magnitudes attained by their dielectric constants. Values of several 104 have been observed. These values are however not even approximately constant. They depend not only on the applied ﬁeld strength ( U), but also on the previous history of the sample. For a demonstration, the setup sketched in Fig. 13.3 (top) is in principle suitable. It consists of two condensers in series, connected to an alternating current source. The capacitance of the right-hand condenser is very large compared to that of the left-hand condenser. As a result, for a given voltage U, the current I is practically determined by the capacitance of the left-hand condenser alone; it is proportional to the voltage U of the current source and to the dielectric constant " of the material in the left-hand condenser, i.e. I "U. This current produces the voltage UC between the plates of the right-hand condenser; it is likewise proportional to I,sothat"U UC. We now want to show experimentally how "U depends on U.This can be most simply accomplished by using an oscilloscope (Fig. 13.3, bottom). Figure 13.4 shows an example: It is a complicated curve,

Figure 13.4 A hysteresis loop, registered with the εU setup shown in Fig. 13.3 (the coordinate axes were ~D drawn in after the measurement). The ordinate is proportional to "U or D; the abscissa is proportional U~E to U or E. D is the displacement density and E D U=d is the electric field strength in the crystal. - " EIG is by which " ), which . This can ): The part 0 13.1 E displacement 13.6 D C13.5 . , perpendicular to E D axial channel has the same magni- for the material con- are measured. We find has the same magnitude Within Matter D 13.3 temperature, for the S D temperature; there, one finds ; it can be seen to contain two and and transverse slit URIE E axial channel 13.5 URIE and ; the other is an 13.1 1 as condenser charge/condenser plate transverse slit E 13.1 (Vol. 1, Fig. 8.14). Corresponding pairs of of a dielectric, based on experimental results: found in the , and denote it by the vector C), the hysteresis loop degenerates into a straight ı E . This is easily understood. Imagine that the cavity m interior D . measured in the D D Quantities D in the hysteresis loop C13.4 salt about 25 D within this channel and find that it is the same as in the empty A and = The direction of the field in a parallel-plate condenser is always well defined. In NETTE as that measured in Fig. that they are different infollows: the two cavities. In detail, the1. results The are field as be shown for a channel of suitable dimensions (Fig. of the condenser plate which liesthe above the rest channel of is separated the from Q plate by a slit. We measure the displacement density condenser, independently of the width ofbe the extended channel. in This a result thought can experiment to all axial channels, even if they area, i.e. is parallel to the fielda direction. measuring Both instrument cavities (real servefigure. to or accommodate In virtual), both indicated cavities, by the a fields dot in the only the normal behavior asgroups. seen in the materials of the 1st and 2nd tude as that measured in the empty condenser, i.e. were located directly adjacent tolimit of one vanishing of thickness, the we condenser define plates. this field In as the the density within matter line which has onlyis a small and small constant slope above relative the to C the abscissa: Thus, other cases, we can imaginerial. a sufficiently small The spherical field cavity withinwithin direction the the found matter mate- within the cavity is identified as the field present values on the ordinate and the abscissa show that the quantity 2. The electric field 1 13.5 Definitions of the Electric Field The definitions given in Sects. called a no means constant. Above a certain temperature (the C is initially completely filled withA a section homogeneous of dielectric it material. is sketched in Fig. Let us consider a parallel-plate condenser (e.g. as in Fig. stants of dielectricsonly are to the straightforward average values as and determined bythere. correct, the condenser We setups now but used want they toE refer give general definitions of the field quantities small cavities. One of them is a flat the direction of the applied field . / m . bound 10 is deter- % AXWELL is thus de- )) which elec- 3.10 itten in the E C D 6.21 magnetic free the charges (free .% 0 all 1 " D which were already Matter in an Electric Field E 13 . They correspond to per- div presence of matter as: holds in vacuum can berespondingly cor- rewr These are materials with a “frozen in” polarization P manent magnets made of ferromagnetic substances, in the case of dipoles. For technical applications (e.g. microphones and loudspeakers), electrets are usually applied in the form of thin films ( thick). equation (Eq. ( C13.5. The field C13.4. See for example F. Jona and G. Shirane, “Ferroelectric Crystals”, Pergamon Press, 1962. At this point, we should also again mention the trets described in Sect. and bound). The M mined by termined here by the charges on the condenser plates, the so-called free charges. In contrast, the field 248 Part I 13.6 Depolarization 249

Figure 13.5 The geometry of the cavities: An axial Field direction channel and a transverse slit Part I

Axial channel

Transverse slit

Figure 13.6 The equality of the electric ﬁeld E in matter and in an axial channel

are too narrow to allow direct measurements. We then obtain E in the cavity by dividing this displacement density by theR field constant "0. As a result of the equality of E and E0, the relation Eds D U holds also for E (Eq. (2.3)). For this reason, in the limit of vanishing thickness of the axial channel, we define the quantity E as the electric field within matter and denote it by the vector E. By introducing these vector fields E and D within matter, we arrive with the aid of Eq. (13.3) at the definition of the vector field P,the C13.6. Equation (13.6) electric polarization:C13.6 demonstrates in an especially clear way the dif- P D D "0E : (13.6) ference between the fields E and D in matter which In materials with a constant value of " (and thus no hysteresis), it we already pointed out in follows from Eqns. (13.1)and(13.5) that the relations Comment C2.13. Think for

D D ""0E (13.7) example of the fields within a uniformly polarized disk of and matter (an electret) (see also Sect. 13.6). P D " ." 1/E D " E (13.8) 0 e 0 Note that the direction hold. These three equations describe the relations between the vector of the vector P is defined in fields at each point in space. such a way that it points from the negative (bound) charges to the positive (bound) charges (just like the dipole 13.6 Depolarization moment p of an electric dipole, Sect. 3.9). The vector fields introduced for the interior of dielectric matter will now be investigated in the case that a piece of matter is brought into (13.9) ,which 0 (13.11) (13.10) E , it follows that , we could sup- within the piece E 13.5 13.7 . It then follows, again ). This field is presumed 0 . Then we find - - - E E : 0 are parallel). This is called 13.7 P "" : directly in front of the disk is 0 P : 1 0 " 0 0 D E E E (Fig. " 1 0 D and 2 0 " D 0 0 E D E E E D ( + + + E D . According to Sect. 0 0 0 E E D E ), we can cast the above equation in the form in its interior, 13.8 D ,that ). . Depolarization. The field produced by the influence charges is 13.5 Not to be confused with the field which we have thus far denoted as Exercise 13.5 depolarization In a secondbrought example, into instead the of external field, the parallel disk, to a long dielectric rod is pose that the depolarization would have athese value intermediate two between examples. Indeed, within an ellipsoid of rotation whose This is easy to understand,of since the the rod polarization charges are at fardepolarization the apart effect ends and is the thus surface negligible. area of theWith ends a is sample small. The in the shape indicated in Fig. equal to the field to be homogeneous and produced bydistance charges which away, are so located that some theythe piece are of not matter, influenced i.e.( their by spatial the arrangement polarization remains unchanged of We begin with a simple case,perpendicular an to extended the disk field with its plane oriented Making use of Eq. ( of matter is smaller than Thus, due to the polarization, the electric field as in Sect. was the field ofwhich an would empty be condenser, changedtional produced on charges by inserting which the matter would charges then into flow on the onto its condenser the plates, plates. due and to the addi- 2 Figure 13.7 a constant, fixed external field superposed on the fixed,within the homogeneous matter is external reduced.also field. The modified total field by Thus, in this the superposition. the space outside total the field matter is the displacement density Matter in an Electric Field 13 250 Part I 13.7 The Electric Field Within a Cavity 251

Table 13.4 Depolarization or demagnetization factors N for rotational ellipsoids Length 0 1 0.1 0.2 10 20 50 1 (inﬁnite Diameter (Disk) (Sphere) wire) Part I 1 1 3 0.863 0.77 0.0203 0.0068 0.0014 0

axis of rotational symmetry is parallel to E0, the depolarization effect leads to a homogeneous field E, which is likewise parallel to E0. Then we findC13.7 C13.7. For a derivation, see N for example A. Sommer- E D E P ; 0 " (13.12) field, “Electrodynamics” 0 (Lectures on Theoretical where P is also homogeneous and parallel to E0. N, called the de- Physics, Vol. III), Academic polarizing factor, is a number which is determined by the ratio of Press, 1952, Sect. 13. See the length of the axis of rotation to the diameter of the ellipsoid; see also Richard M. Bozorth, Table 13.4. With Eq. (13.8), we can rewrite Eq. (13.12): “Ferromagnetism”, D. Van Nostrand Co., Ltd., 1953, 1 E D E : (13.13) Chap. 9. 1 C N." 1/ 0 Equations (13.9)and(13.11) represent special cases of this equation, with N D 1 (disk) and N D 0 (rod).C13.8 C13.8. Depolarization also occurs in the experiment of In the special case of a spherical sample, N D 1 and thus in its 3 Fig. 13.1;there,N D 1, al- interior, the field strength is only though the field there was 3 held constant using the volt- E D E : (13.14) ." C 2/ 0 age U.Thisishoweverthe total field. The “polarizing” This likewise homogeneous field can be considered to be the sum of field E0, which is produced the external field E0 and the field produced by a uniformly polarized by the condenser charges (the sphere, Ep, “free” charges), indeed increases when the dielectric is E D E C E : (13.15) 0 p inserted owing to the charges Comparison with Eq. (13.12) yields which then flow onto the condenser plates. This increase 1 in the field is just compen- Ep D P : (13.16) 3"0 sated by the polarization.

The ﬁeld Ep is thus determined only by the polarization. Further- more, it is independent of the manner in which the polarization was produced (if this were not the case, we could construct a sphere with P D 0andEp ¤ 0!). Then, Eq. (13.16) holds e.g. also for a spherical electret.

13.7 The Electric Field Within a Cavity

If we were to exchange the roles of the dielectric material and the empty space in Fig. 13.7, as suggested in Fig. 13.8, then we could expect, due to the polarization charges on the inner surfaces of the ). adi- 3.19 !Fig- (13.18) (13.19) (13.17) (13.20) insulating 1, we find within i ). When the " r D small a " of simple, symmet- : CA 2 5 ) yield : R U ; : R , this (homogeneous) field a R a a 13.4 E @ / E " @ E 1 a 1 " PV 2 a a C 2 )and( " " , we find for the magnitude of the D a 3 ." 3 C R " 0 C i R 3.28 " R 2 " E " @ @ V would be greater than the field outside. 2 p i D D r i E i 6 D E E F ): Eqns. ( D 1 (for example the interior of an empty spheri- t succeeds only for bodies F D 13.19 ) and a large charged sphere (radius i " V A cavity in a polarized dielectric Materials in an Inhomogeneous Electric Field is the field of the material in the outer region at some ). For a shows an example. (volume E 13.14 13.9 Derivation of Eq. ( C13.9 Eq. ( sphere electric material of dielectric constant ure The force thus decreases as the fifth power of the distance where is: In the case of a spherical body with a dielectric constant Figure 13.8 distance between their centers is force 13.8 Paraelectric and Dielectric All paraelectric and dielectric materialsgreater are field pulled strength toward within regionsrecalls an of the inhomogeneous electric oldest field.cloth electrical or observation, paper This to the charged bodies, attraction e.g. a of rubbed amber scraps rod (Fig. of empty space, that the field ric shape, for example for the attraction between a Depolarization makes a quantitativerather complicated. treatment of I this phenomenon cal bubble), it follows that distance from the cavity,present i.e. in the the homogeneous field material that without would the be cavity. For Theoretische ”, B.G. Teubner, 3rd Matter in an Electric Field 13 Physik ed., 1957, Vol. 2, Sect. 18. C13.9. For a derivation, see F. Hund, “ 252 Part I 13.9 The Molecular Electric Polarizability. The CLAUSIUS-MOSSOTTI Equation 253

Figure 13.9 The attractive force on a small insulating sphere in the inhomogeneous electric ﬁeld

of a large sphere, measured using a spiral-spring Part I balance (example: an amber sphere of diameter D 6 mm, V D 1:13 107 m3, " D 2:8, radius of the charged sphere r D 2 102 m, U D 105 V, R D 5 102 m, depolarizing factor N D 1=3 (Table 13.4), F D 2:9 105 N). The reader should look at Exercise 2.16 in this connection. R

At the point of observation, according to Eqns. (2.15)and(2.16), we have

Ur E D (13.21) R R2 and

@E 2Ur R D : (13.22) @R R3 The electric polarization of the small sphere is

P D "0." 1/E ; (13.8)

and the ﬁeld strength in the interior of the sphere is

3 E D E : (13.14) " C 2 R

Combining Eqns. (13.14), (13.20)and(13.22) yields Eq. (13.19).

13.9 The Molecular Electric Polarizability. The CLAUSIUS-MOSSOTTI Equation

The different behavior of dielectric and paraelectric materials has already been explained qualitatively in Sect. 13.4.Itsquantitative explanation is rather important for an understanding of molecular structure and thus for chemistry. To arrive at it, we need the concept of molecular electric polarizability.Wederiveitusingwhatwe have learned about depolarization in the previous sections. In the interior of a body of volume V, let the electric ﬁeld be E,and assume that it produces an electric polarization in the body:

P D "0." 1/E : (13.8) , w E , 0 p (13.26) (13.27) (13.24) (13.28) (13.25) (13.23) (13.29) ). They which are /: 0 ). We then p ) applies. In is interpreted 13.26 p //: /: ) 13.25 13.4 Vm )and( . Note that =. 13.27 : equal is no longer per- As E E 13.24 / 12 0 1 p 2 1 / and 10 V V : 1 C : N w N w w ." " V " p ), obtaining: E 86 E 0 : E V " N D 8 1 V ˛ ." N 0 c polarizability that was suggested 0 0 V D ˛ p " D " D D 0 N 3 13.24 is constant, and thus the average con- V 0 N D : particle) density of the molecules P ;" P . p " D which occurs in Eq. ( . For this reason, we set P V D V w 1 D = ˛ quantities, acting at the level of atoms or N E 2 E ˛ P , for gases, vapors and dilute solutions, we )and( this equation reverts to Eq. ( ): We start from Eqns. ( w D ; number 0 E 1 OSSOTTI p 13.8 D in A s m 13.28 V " ˛ = molecular electric polarizability N for and M . microscopic e.g. D . individual molecules, that is: is proportional to the magnitude of the field acting on the the V N 0 N p ˛ . are all LAUSIUS ˛ Derivation of Eq. ( give parallel to the direction of the field. Then Eq. ( due to the and call and tribution Experimentally, we find that molecules, which we call missable. In these “condensed phases”, thepacked, molecules are and too densely therefore, theirtaken into interactions account with in polarized one liquidsequation and another for solids. must the This is molecular be done electri by in C the molecules. As the effective field can simply take the field In an atomic picture, the total electric dipole moment In liquids and in solid bodies, setting We combine Eqns. ( With this polarization, the bodyp acquires an electric dipole moment vector form, it states that obtain as the vector sum of the average molecular contributions Matter in an Electric Field 13 254 Part I 13.9 The Molecular Electric Polarizability. The CLAUSIUS-MOSSOTTI Equation 255

To calculate the effective ﬁeld Ew, we consider a single molecule a.The remaining molecules are divided into two groups of unequal size. In the

first, smaller group, we include all those molecules in the immediate neigh- Part I borhood of molecule a. The boundary of this nearby region is arbitrarily chosen to be a spherical surface with a at its center. The second, larger group includes all the molecules outside this sphere. In amorphous substances and regular crystals, the neighboring molecules within the virtual boundary surface around molecule a are arranged with spherical symmetry. Therefore, their influence cancels mutually, and only the effect of the second group remains. The molecule a floats, figuratively speaking, within a spherical “cavity” inside a homogeneously polarized body. To determine Ew, we cannot use Eq. (13.18), since it was derived under the assumption that " D 1 within the cavity. Then, however, the electric field in the neighborhood of the cavity would not be constant, in contradiction to the case we are dealing with here. We therefore assume that at the position of molecule a,thefieldE can be expressed as the sum of the field of a uniformly-polarized sphere (Sect. 13.6) and the field we are seeking, Ew:

E D Ew C Esphere ; (13.30)

where E is the ﬁeld in a uniformly-polarized body (without a cavity) and Esphere is the ﬁeld in a sphere with the same polarization, P (see Eq. (13.16)); then

1 Ew D E C P : (13.31) 3"0

Then, using Eq. (13.8), we find:C13.10 C13.10. Thus, the internal field E that we defined in " C 2 E D E ; (13.32) Sect. 13.5 does not act on the w 3 single molecule within the and from this, using Eqns. (13.25)and(13.26), we finally obtain Eq. (13.28). dielectric material, but instead the field Ew acts there. The two equations (13.27)and(13.28) permit a relatively simple It – depending on the value determination of the molecular electric polarizability ˛: We need of the dielectric constant " only measure the dielectric constant " and insert it, together with (Table 13.1) – may be consid- the relevant number (particle) density NV.Table13.5 contains some erably larger! numerical values for the polarizability ˛ of non-polar molecules in liquids (Exercise 13.6).

ı Table 13.5 Electric polarizabilities of some non-polar molecules in liquids (at 20 C) (NA is the AVOGADRO constant, 6:022 1023 mol1) % Substance Molar mass Density Number density of the Dielectric Electric M % D NA constant " polarizability ˛ Mn molecules NV D n Mn kg kg Asm2 in in in m3 in kmol m3 V 27 39 Carbon disulﬁde, CS2 76 1250 9:9 10 2.61 0:94 10 27 39 Biphenyl, C6H5–C6H5 154 1120 4:37 10 2.57 2:1 10 27 39 Hexane, C6H14 86 662 4:63 10 1.88 1:3 10 .The (13.34) (13.33) 2.56 is due to “in- /: a K = of the permanent x Ws 23 10 /K Pa); measurement frequency –3 5 38 , p. 400 (1924)). :

: w

b 24 are randomly distributed owing p a E D p kT 2 p dependent part is caused by an 013 p HCl : p xp in 1 V 3 1 D D A s m 0 p constant ), and was explained in Fig. x –39 = reciprocal temperature=

024·10

2 1 1 T α ·10 polarizability 2.17

. Each molecule then has a net component

Physical Review p Molecular electric Molecular p , averaged spatially and over time, is equal to zero. OLTZMANN ,thatis p B p p p is given by D The polarizability of a dipolar molecule as defined by Moments of Polar Molecules k x even in the absence of an external electric field. . as the contribution of a single molecule. The magnitude of p shows a typical example for a gas, HCl. We can recognize 0 p ), as a function of temperature. The constant part ) (at a pressure of 1 atm ( p 13.26 1 MHz) (C.T. Hahn, 13.35 13.10 is only a fraction (usually a very small fraction) D is due to alignment of the thermally disordered polar molecules with their 0 along the field axis, averaged overmoment time, and this results in an average to the thermal motionsdipole moments of the molecules. The sum of the electric

An electric field however providesthe a dipole preferred axis moments of orientation for p The fraction fluence” or “molecular deformation”,b while the temperature-dependentpermanent part dipole moments.Eq. ( Only this second part should be inserted into Eq. ( temperature-dependent part adds to it,molecules and of is due paraelectric tomoments materials the fact have that permanent the electric dipole Figure 13.10 13.10 The Permanent Electric Dipole In paraelectric materials, themolecular measurements electric show polarizability aure with decrease increasing in temperature. the Fig- Without a field, the orientations of dipole moment electrical deformation of the molecules,tric as materials is (see well Sect. known for dielec- two different contributions, one which is independentture of (the the thin tempera- horizontal line below),the and temperature the (above the other constant which part). depends on Interpretation: The temperature- Matter in an Electric Field 13 256 Part I 13.11 The Frequency Dependence of the Dielectric Constant " 257

The fraction x is thus essentially equal to the ratio of two energies: (i) the work pp Ew would be required in order to align the dipole perpendicular to the ﬁeld axis; (ii) kT is the thermal energy which is on Part I the average transferred by molecular collisions (and tends to disorient the dipoles). A precise calculation would have to take not only the perpendicular alignment into account, but also all other possible orientations by computing the appropriate average (LANGEVIN-DEBYE formula). This leads in the ﬁrst approximation to the numerical factor of 1=3. We combine Eqns. (13.33)and(13.34) with Eq. (13.26), denoting the temperature-dependent part of ˛ in Eq. (13.26)as˛T, and obtain for the permanent electric dipole moment of the dipolar molecules: p pp ˛T3kT : (13.35)

For the example of the HCl molecule, the measurements in Fig. 13.10 give the molecular electric polarizability at 273 K:

Asm2 ˛ D 1:05 1039 : T V Inserting this value into Eq. (13.35), we obtain for the permanent electric dipole moment of a single HCl molecule, pp 3:4 1030 A s m (cf. Table 13.3)(Exercise 13.7).

One could therefore imagine the molecule from an electrical point of view as consisting of two electrical elementary charges, each of 1:6021019 As, at a spacing of around 0:2 1010 m. (For comparison: The order of magnitude of the molecular diameter is 1010 m.)

Making use of this value of pp, we can calculate the fraction x in Eq. (13.34). The field strength may be large, namely E D 106 V/m, and the temperature T D 300 K. Then we find x D 3 104,thatis x 1. Thus, the average contribution p0 of the permanent dipole moment pp to the electric polarization P is still proportional to the field strength, and the susceptibility, P="0 E D e D ." 1/, is constant (linear region). Only at very low temperatures can x approach a value of 1 with increasing field strength, causing the electric polarization to saturate at its maximum possible value.

13.11 The Frequency Dependence of the Dielectric Constant "

In order to determine the capacitance C D Q=U, we measure the charge Q which is stored in a condenser at the voltage U. In doing this, we make a tacit assumption: The magnitude of the charge Q stored at a given voltage U is supposed to be time-independent. This can however not be generally true. The processes which occur within , s k s, k 8 4 2 0 11 Hz to 10 ω 9 10 of water cm 10 (13.36) with an 10 ! 3 is reduced, : " ω j 5 ε " ! 10.6 D " (right) shows 9k Hz D r Dj ! 2 ). 11 13.12 / r 10 2.16 Circular frequency .! 1 C 1 , Sect. , the absorption coefficient p ), is also plotted. = 0.5 1 5 10 30 is reduced by the factor λ RC 12 )). Second, the AC current is no D ! D 10 R 25.3 75 50 25 13.11 r ω 100 , we would obtain a different value R . However, if it becomes comparable ε 10.30 r ! C13.11 . ! " . In this frequency range, electrical waves r )and( ). In Fig. = 1 on the other, is quite general. It is unimportant 10.29 r 37 % is still missing from the equilibrium values. D R ! e , at the left, we show the model system: We imagine the = shows as an example the dielectric constant Current amplitude with The frequency Current amplitude without of water. The Exercise 13.8 s. For pure water, the relaxation time is k 13.12 D 11 Hz, that is for fields with oscillation periods between 10 13.11 R ; C 11 C I I , then the measured (apparent) dielectric constant r C) The resistor in the equivalentof circuit an has AC two current effects: of First, circular the frequency amplitude an equivalent circuit, the well-known series circuit from Sect. (This quantity is called thefollows relative from dielectric Eqns. constant. ( Equation (13.36) dielectric to consist offectly layers. insulating, The while shaded theelectrical layers resistances. dotted are For layers supposed this are layered to poor dielectric, be Fig. conductors, per- with large In Fig. AC current source (relaxation time 10 ı , a fraction 1 r 2 determined by the rateapplied of alignment field. of thewhich This dipolar molecules consumes is in energy hindered the (dissipation).cular by frequency It a is resistance greatest around similar the to cir- friction, and 10 at 18 °C in the range of circular frequencies from Normally, the time during whichlonger the than the field relaxation time remains constant is much to are strongly absorbed byovens; water see (this is the principle of microwave dependence of the dielectric constant and the absorption coefficient precisely which physical processes are responsibleWe for explain the this relaxation. by referring to a simple model. Figure 13.11 Figure the dielectric after thethey field require a is certain time applied on the are average. not After instantaneous; one “relaxation rather, time” often used in optics (Optics, Sect. corresponding vacuum wavelengths are shown on the abscissa above (temperature 18 and at each circular frequency of the dielectric “constant”, The relationship between thesipation dielectric constant (absorption and accompanied the bythe power heating) relaxation dis- on time the one hand, and . , under the topic 27 27.7 Matter in an Electric Field 13 C13.11. Another cause of a frequency dependence of the measured dielectric constant is discussed in Chap. of optics; see in particular Sect. 258 Part I 13.11 The Frequency Dependence of the Dielectric Constant " 259

Figure 13.12 At left:Alay- abaa ered dielectric. At right:Its equivalent circuit C Part I U U R

b b

Wavelength λ 104 105 106 m 107 1.0 1.0 R R R εω with C without without I C C I 0.5 0.5 I Amplitude of Amplitude of

ω Amplitude of 0 R 0 106 105 104 103 Hz Amplitude of the active current Circular frequency ω

Figure 13.13 The frequency dependence of the relative dielectric constant (Eq. (13.36)) and the loss factor (Eq. (13.39)) in the equivalent circuit of 4 Fig. 13.12 for a relaxation time of r D RC D 10 s( is the wavelength of electromagnetic radiation at the circular frequency !)

longer phase-shifted by 90ı relative to the voltage, but rather only by a smaller phase angle '.FromEq.(10.31), we ﬁnd

1 tan ' D (13.37) !r .the current leads the voltageI cf. Eq. (10.24)/:

In addition to the reactive current (or “idle” current), there is also an active current of amplitudeC13.12 C13.12. In deriving Eq. (13.38), note that: !r Iact D U!C : (13.38) C !2 2 D ; ': 1 r Iact IC R cos

The ratio of the active current amplitude to the total current amplitude in (see Sect. 10.8) a loss-free condenser is called the loss factor, and it is found to be

!r "!; D : (13.39) act C !2 2 1 r

The results of Eqns. (13.36)and(13.39) are shown graphically in Fig. 13.13: At a frequency of !R D 1=r, the loss factor "!;act exhibits amaximum. )to . ) of the U " , similar 6 ) is placed 0 : . Neglect 13.28 13.6 0 " ) U 13.6 D to 19.3 0 13.9 U U equation cannot be ) equation ( , 6th ed. (Springer, , Sect. and is slid into a charged . (Sect. 5 GHz), a liter of wa- 2 0 l used to determine the : " E 2 5 : is slid between the con- , so that it ﬁlls the entire 13.10 0 l D Q 13.2 in the rod? (Sect.

D < OSSOTTI OSSOTTI D ) d 37 kW/m 00612. Explain the difference d : : -M -M 1 1 . (Sect. which is absorbed by the water. c) D 13.11 D e 13.3 P of a dielectric sphere with a dielectric " W Landolt-Börnstein E P of liquid water using the data given in of the microwave radiation in water. b) (it however was determined for the gas ) LAUSIUS LAUSIUS p W p 13.2 13.3 with its long axis parallel to the field direction. , see Comment C12.5) with the solar constant for E S ) at a temperature of 22.5 °C and a pressure of 1 atm 3 Pa) a value of ? (Sect. 5 x in a homogeneous electric field ). This causes its voltage to drop from l C vector " 10 , and compare the result with the correct value, which ) How is the bridge circuit in Fig. Make use of the C In the reference work A long dielectric rod with a dielectric constant of Find the polarization A sheet of glass of thickness A dielectric slab of dielectric constant In a microwave oven (frequency 13.3 13.2 013 : 1 OYNTING D P 13.3 to what is shown in Video 2.1. Find the dielectric constant glass, and compute it for the case that capacitance Table find the dipole moment reflection and heat losses. (Sect. 13.6 applied to water in the liquid phase. Why not? (Sect. 13.7 Berlin 1959), Vol. II, Part 6, p.of 874, ammonia we (NH find for the dielectric constant Determine the average power in an electric field How large is the displacement density 13.4 constant of 13.5 The water is contained in a cube-shapedon plastic beaker a which side, is 10 and cm thepare radiation the impinges on radiation it power from per all surface directions. area Com- (i.e. the magnitude of the ( is also given in Table phase). Apparently, the C denser plates of athe charged plates is parallel-plate condenser (the spacing of 13.2 volume between the condenserin plates. the How condenser does change, the(Sect. and energy where stored does the energy difference go? parallel-plate condenser with a charge of Exercises 13.1 onto the surface of the Earth, compared to the value in Table 13.8 ter is heated by 10 °Ca) in Find one the minute. wavelength Matter in an Electric Field 13 260 Part I ag atr(n ofil ie r on usd h oli hscase; this in coil the outside Fig. found are cf. lines field no (and factor large a ligmtra ob h ratio the be to material filling ˚= 41Itouto.TePermeability The Introduction. 14.1 Field Magnetic a in Matter iue14.1 Figure wound is coil ring a When (Fig. effect. core drastic iron rather an a around has field, iron, magnetic the of in materials e.g. other some of presence the contrast, In decimal the after place 6th results. the our units). only on 4 changes by effect space air negligible the empty a of in had influence only The molecules air fields of magnetic presence considered (the have we far, Thus .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © bu 0 SeSect. (See 10. about ae-ae aeilwt permeability a iron-containing with an material paper-based “Ferrocart”, with filled can be coil also ring the experiments, density. demonstration flux For magnetic the of measurements by iiigtemgei flux magnetic the Dividing ooeeu antcfil,oeotisits obtains one field, magnetic homogeneous uaiglyro hla.Teefcso hs poswsas negligible also was experiments. spools demonstration these our of for needed effects in- precision an forms The the with coil within covered shellac. on tubes of wound wooden layer or were sulating cardboard particu- rather thin-walled in but e.g. considered, self-supporting, spools, or we not that were coils, conductors lar current-carrying the of Some A hnfrtedfiiino the of definition the for Then . 14.2 enn antcmtra constants material magnetic Defining .Bsdo hsfc,w en the define we fact, this on Based ). D antcflux Magnetic antcflux Magnetic 5.4 ) olsItouto oPhysics to Introduction Pohl’s 14.1 ˚ C14.1 ,ismgei flux magnetic its ), ˚ ˚ ytecosscinlarea cross-sectional the by D 0 m fteepyrn coil ring empty the of B B ftefildrn coil ring filled the of permeability m 0 : of u est B density flux ,wefind permeability , ˚ https://doi.org/10.1007/978-3-319-50269-4_14 sicesdby increased is : G J ,i.e. A fthe of (14.1) fthe of B D h ieaueby literature the permeability The C14.1. h product the Frequently, permeability”. “relative name the given is it vacuum”); to “relative for preblt fvacuum”. of “permeability permeability”. eoe by denoted sotnas eoe in denoted also often is 14 n ald“the called and , 0 r 0 ssimply is r stu the thus is rstands (r 261

Part I D D (14.2) (14.3) (14.4) m ˚ of the . The unit % ,whichis A . C14.2 14.1 : : 0 : 0 B / m M H 1 . Then the magnetic flux l D V ), its magnetic moment is . M 0 0 as an additional contribution to B magnetic polarization D 8.20 M 0 B D Vo lu m e 1 , or to the amount-of-substance density, of the block produces is given by m =% M m B Magnetic moment is defined by the equation D D D . ; and from Eq. ( : M m M 0 ) is also called the MV MA 0 Some examples can be found in Table 4.8 D M 0 D C14.3 MA l A . The outside space / from the Permeability / 0 D V B = 0 n magnetic susceptibility magnetization M = =. which the magnetization is for example 1 A/m. A different definition is also equivalent: m l m B ˚ . homogeneously magnetized along its length ˚ Sometimes the susceptibilitycorresponding is substance, also referred to the density Derivation: We imagine a block of matter with a base area M We thus define the magnetization the magnetic flux density inapplied matter, field. which is in turn produced by the Figure 14.2 of It refers to the magnetic moment within a material: around a ring coil (toroidalwith coil) an iron core is field-freeholds (this also for a ringiron coil core; without see an Fig. 2. The 14.2 Two Quantities Derived Two other, often-used physical quantitiesto are the defined permeability with reference 1. The M , n V ). and cgs ,the ; m m m M filling fac- , a vec- B D is given. Re- ; i.e. instead 4 of a sample / % O V , which can be m D =% = V m . n SI ; is the amount of m =. n m 14.5 Matter in a Magnetic Field , i.e. whether the material – and therefore the sus- and the magnetization are collinear, the vector 14 ity is often referred to the mass density of simply quoting influenced by the tor is porous, gaseous etc. For this reason, the susceptibil- computed from them – are A note on expressing itunits: in cgs (In the cgs unit system, this quantity is often (and somewhat incorrectly) called the “molar susceptibility”. where substance (unit: mol) is also convenient. quantity of volume ceptibility C14.3. The magnetic moment m M this point. However, compare Sect. occurring case that B notation is not necessary at C14.2. The magnetization is, like the field tor. Since we assume here the simple but frequently- ferring the susceptibility to the amount-of-substance density, 262 Part I 14.4 Distinguishing Diamagnetic, Paramagnetic and Ferromagnetic Materials 263 14.3 Measuring the Permeability Part I There are many setups suitable for measuring the permeability. If the sample is available in the form of a ring (torus), then the scheme illustrated in Fig. 14.1 may be used. In carrying out the measurements, one can if necessary increase the sensitivity by several orders of magnitude by applying a difference method. The induction coils of the empty and the filled ring coil are connected in opposition, and the difference between the two magnetic fluxes is measured directly as a voltage impulse. An example is illustrated in Fig. 14.6.

For many substances, 1. In that case, one does not try to measure directly, but determines instead the magnetization M. can then be calculated using the deﬁning equations as given in Sect. 14.2. Instead of large, ring-shaped samples, one can use small samples of volume V with arbitrary shapes. They are inserted into a magnetic ﬁeld, and the magnetic moment which is produced by magnetization is measured:

m D MV :

To this end, the magnetic ﬁeld is made inhomogeneous and then the force acting on the sample along the direction of the ﬁeld gradient is measured by some method (e.g. Fig. 14.3):

@B @B F D m D MV (8.18) @x @x (e.g. F in N, m in A m2, @B=@x in V s/m3, V in m3). The field gradient @B=@x is measured as described in the text following Eq. (8.18) in Chap. 8. For the determination of , one uses the magnetic flux density B0 which is present without the sample. It is measured using a small induction coil as the average value at the location of the sample. The application of this field in Eq. (14.2) is only approximately correct. It is however permissible if the permeability is 1. We will see the reason for this later from Eq. (14.17)(Exercise 14.1).

14.4 Distinguishing Diamagnetic, Paramagnetic and Ferromagnetic Materials

After explaining the measurement procedure, we now want to give an overview of the magnetic properties of materials. All substances can be collected in three large groups: 1. Diamagnetic materials.

Their susceptibilities m D .1/ are, like , material constants and are independent of the strength of the applied magnetizing field. They are also independent of the temperature (for a constant density %). The permeability is somewhat smaller than 1. Table 14.1 contains some examples. mol mol kg kg = = ’s law (14.5) = = 3 3 3 3 .Some- m m ILHELM m m 6 9 10 6 9 10 /: URIE (W 14.1 is somewhat 10 10 10 10 10 10 14.8 8 2 : : 3 293 K 165 16 35 S 2 Bi D Dy T 17 200 2839 11 960 is not even approximately 9 : 5 78 : : derivation in Sect. T 13 6 3 I T 2K : C NaCl urement. They are always forced 90 Atoms or molecules of diamagnetic D (liquid) Pa) D 2 m 1000. With increasing temperature, 5 constant decreases with increasing temperature 06 06 63 : : : O T 3470 3020 967 O 2 9 9 1 10 m . H . . They can be identified in a magnetic field URIE C) 013 (gas) C) ı : 2 ı 1 6 08 69 O (1 atm) 1.88 1300 416 20 : : : . D 9 1 0 . Explanation: can exceed Cu 2 Pt 264 12.3 24 14.3 , 84th Edition, 2003/4 (CRC Press, NY). 293 K is called the C , Parts B and C). In simple limiting cases, C (1 atm 293 K (20 temperature), ferromagnetism vanishes and the material a T 002 5 : : C 2 0 25 0 . b, opposing the current in the field coil, without losses as T 14.5 Al 20.7 7.67 2.07 T , 1852). The induced currents flow in circles, as shown in H (1 atm) URIE 1), 1), 8.16 > Diamagnetic materials Ferromagnetic materials Paramagnetic materials < EBER W holds: a material constant. Itnetizing depends not field, only but on also theon strength on its of structure, the the e.g. mag- magneticmagnitude whether history of it of is the bulk sample material and or powdered. The the permeability decreases.(the C Above a certain critical temperature In these materials, the permeability larger than 1. Examples are likewise collected in Table Fig. (cf. Fig. exhibits only paramagnetic behavior. So much for the general classification; now some1. details: even without a quantitative meas out of the regionexample, where a the piece magnetic of bismuth field willshowninFig. strength be is pushed upwards strongest; in for the apparatus materials have no permanent magnetic moments.ment They only acquire in a an mo- applied magnetic field, through induction times, their susceptibility Their susceptibilities are also practicallyof independent the of applied the strength magnetizing field. The permeability 3. 2. always always / / 1 1 . . Amount of substance) D D Amount of substance) m m D D n Density) n Density) ( ( Dia- and paramagnetic substances D D / / % % CRC Handbook of Chemistry and Physics ( V V ( = = n n Matter in a Magnetic Field =% =. =% =. m m m m 14 See the Susceptibility Susceptibility Paramagnetic substances ( Diamagnetic substances ( Table 14.1 a 264 Part I 14.4 Distinguishing Diamagnetic, Paramagnetic and Ferromagnetic Materials 265

Figure 14.3 A sample in an inhomogeneous magnetic ﬁeld is

hanging from a spiral-spring bal- Part I ance (Exercise 14.1)

80 mm 5 mm

Figure 14.4 Small diamagnetic objects (V 1mm3) made of bismuth or well-tempered arclight carbon are levitated on the fringes of a magnetic field with a flux density of B 2Vs/m2 (an electromagnet as in Fig. 14.3). The concave region milled into the top of the magnet pole piece (radius of curva- C14.4. Note here the differ- ture 6 cm) stabilizes them in the horizontal direction. ence between magnetic and electric moments (Chap. 13): Both dielectric and para- long as the field is present. These induced currents must occur in all electric materials are always atoms, so that all substances are originally diamagnetic. Their dia- pulled towards regions of magnetic behavior can however be concealed by other, stronger phe- stronger fields; that is, e is nomena; this is the case for para- and ferromagnetic materials.C14.4 always positive. Diamagnetic samples can be made to levitate in an inhomogeneous C14.5. Superconducting ma- magnetic field of a suitable shape (Fig. 14.4), and they will oscillate terials show strong diamag- C14.5 if given an impulse along the direction of a field gradient. netic behavior due to their 2. Paramagnetic materials. In contrast to diamagnetic materials, they expulsion of magnetic fields are pulled into the region of greatest field strength. Some demonstra- (MEISSNER-OCHSENFELD tions are shown in Figs. 14.5A–C. effect). They can be used for impressive demonstrations Explanation: The molecules of paramagnetic materials not only ac- of magnetic levitation. See quire a magnetic moment by induction in a magnetic field, but also Comment C10.3. they have permanent magnetic moments mp, independently of the C14.6 field. C14.6. Exception: PAULI paramagnetism, for exam- The dipole axes of these permanent moments are however distributed ple in the metals Al and Pt. randomly over all directions as a result of thermal motions. There- More details can be found fore, the paramagnetic sample exhibits no overall magnetic moment in C. Kittel, Introduction without an applied field. In an applied magnetic field, however, the to Solid State Physics,7th atomic moments acquire a preferred direction. Nevertheless, under ed., Chap. 14 (John Wiley, usual conditions, the orientation of all the permanent moments paral- NY 1996), especially Figs. 1 lel to the field is far from complete. The time-averaged contribution 0 and 11. mp of individual atomic or molecular moments to the total moment m, and thus to the magnetization M, is only a fraction (usually a small ). m P , ,the ) will A is thus 2 up to CCC m .Coldair B m ; ). B and C: 1 ), where it dis- T P 14.3 0Kand : .Otherwise,the 4 p . Division by m H 0 < ˚ T of paramagnetic materials ,at 1 m M ˚ ). Both field coils are carry- In contrast, rising warm air (from of the magnetic flux in the two are proportional to . 14.1 % ). We use two ring coils as shown 8.4 BC are easily recognized even by non- , 1898). Physically, the ferromagnetic , 559 (1952). of the field; or, expressed differently, difference 88 (Sect. B (Video 14.1) PP ) is forced away from the magnet into a region of which can be attained. It, in turn, depends in H of the magnetization EUSLER . and the density in sulfate crystals (alums) M NNN SSS A =% in strong applied fields would no longer be propor- ). As a result, a cloud of cold air is pulled into the region m 14.8 2 CCC M T saturation A: Liquid oxygen is strongly paramagnetic and is therefore ; they have the same dimensions and number of turns. Physical Review . 2 ,andGd 14.6 would no longer be constant. An example for a gas (O CCC Ferromagnetic materials 5Vs/m can be observed onlytemperatures. at It has extremely been high demonstrated for field a strengths number and/or of ions, very e.g. low Cr In general, Fe creeping galvanometer W.E. Henry, be given in Sect. has a larger susceptibilityceptibility. than (Both room air, while warm air has a smaller sus- magnetization tional to the flux density materials are characterizedtheir magnetization by the extremely large magnitude of a complex way on thetreatment strength of of the the magnetizing sample. field and the prior We want toa demonstrate this dependence experimentally using The left-hand coil contains anwound iron on core, a while wooden the core right-hand (cf. coil Sect. is coils, with and without an iron core; i.e. ing the same current, andloops, both but are these surrounded are by connected identicalgalvanometer in induction indicates opposition. the Therefore, the creeping in Fig. proportional to (The electromagnet used is similar to the one shown in Fig. The temperature dependence of the paramagnetic susceptibility pulled into regions of greater field strength (here, from a cardboard tray 1 and 3. Figure 14.5 fraction) of the permanent molecular moment physicists.Examples: They will Thecopper be pure alloys (F.R. attracted metals H by Fe, any Co, permanent Ni; magnet. manganese-containing smaller field strength by the more strongly attracted room air (Part C). a flame or a heating coil of greater magnetic field strength (from the cooled paper tray places the room air (Part B) cross-sectional area of the coils, yields the additional magnetization Matter in a Magnetic Field 14 Video 14.1: “Paramagnetic materials” http://tiny.cc/kdggoy The video shows the experiments from Parts A andthe B figure. of 266 Part I 14.4 Distinguishing Diamagnetic, Paramagnetic and Ferromagnetic Materials 267

G Part I J1 J2 Iron Wooden core core

R A

Figure 14.6 Measuring the hysteresis loop of iron using a creeping galvanometer G. Two ring coils as shown schematically in Fig. 14.1.The induction loops J1 and J2 are wound oppositely; the galvanometer thus indicates the difference between the two magnetic fluxes ˚, with and without iron. If the slider L is pushed past the midpoint of the resistor, the direction of the current in the two field coils is reversed (why?). which is due to the iron: 1 M D .Bm B0/: (14.2) 0 We carry out a series of measurements, increasing and then decreasing the field current for both directions, and thus obtain the curve showninFig.14.7, the “hysteresis loop” for wrought iron. From this curve, we read off the following facts:

1. For each value of B0, the ﬂux density of the empty coil (wooden core), there are two corresponding values of 0M. The right-hand

2 μ0M 3 V s/m

B0 32211 3·10–3 V s /m2 Remanence 1 Coercive field 2

Figure 14.7 A hysteresis loop for wrought iron, measured as in Fig. 14.6. B0 is the flux density of the empty (i.e. iron-free) field coil. The saturation value of the magnetization (multiplied by 0 in the figure) is found at 2.1 V s/m2. At this value, each iron atom contributes a moment of m0 D 23 2 M=NV D 2:0 10 Am (compare Sect. 14.8,Table14.2). The dashed line indicates schematically a new curve or initial curve from a sample which had been previously tempered and thus had no remanent magnetization. axis) (14.6) 0 temper- B EUSLER remanent or URIE be performed. C14.8 W C14.7 . In magnetically . 2 2 /: Vs/m remanence 6 1Vs/m 0 10 B d C, nickel loses its ferromagnetism, M ı remains when the field of the coil Z M V D W approaches a “saturation value” at large val- is the volume of the sample , with a remanence of M V 2 . C. A piece of one of these alloys will stick to ı . The iron core has become a permanent magnet. . A wheel made of nickel and heated on one side will rotate in . 0 B Materials with a small coercive fieldvery are pure termed iron “magnetically which soft”. has With beenreduce tempered the in coercive a field hydrogen to atmosphere,hard one as alloys can low made as of ca. Fe, Ni, 3 belonging Co and to Al, the e.g. AlNiCo Oerstite family), (athe hard the order magnetic coercive alloy of field 0.1 V can s/m be increased up to corresponds to increasing magnetization, and thedecreasing left-hand magnetization; branch to in the(and lower the half, magnetization is their oppositely roles directed). are reversed 2. The magnetization ues of ature”. This critical temperature lies for example in the H the field of a permanent magnet. At 356 is placed in frontrotation. of or behind this plane, depending on the desired sense of and then the magnet pulls inThe a midplane cooler, still of ferromagnetic the region of magnet the passes wheel. through the axle of the wheel; the flame is reduced to zero.magnetization It is called the magnetic 4. In orderreversed to and remove its the fluxcoercive remanence, density field the increased to field a in certain the value, coil called must the be 3. A portion of the magnetization alloys below 100 Figure 14.8 5. The cyclic magnetization process,teresis i.e. loop, a requires full that circle a aroundIt certain the amount is hys- of equal work to the area inside the hysteresis loop, that is 6. The old problemin of a how magnetic to field levitate wasductivity a solved (see ferromagnetic Comment only object C14.5.). after freely the discovery of supercon- An important property ofdependence. ferromagnetism is Ferromagnetism its vanishes strong above temperature the “C branch of the curve in the upper half of the graph (above the and 5 .These Classical (John Wiley ; OHL B d H ). B attain much higher 14 Z Matter in a Magnetic Field 8.28 Fe D 2 14 V W from small electric motors to the “undulators” employed in free-electron lasers. modern materials are used in many applications today, C14.8. This equation follows from the expression for the energy density in a magnetic field terials such as SmCo C14.7. Modern “hard” ma- mentioned by P stored in the magnetization (cf. Comment C14.8.), is up to a factor ofthan 10 in larger the AlNiCo alloys magnetic remanence. In addition, their “energy product”, a measure of the energy Nd values of the coercive field and similar values of the netic field energy given in Eq. ( Electrodynamics & Sons, NY, 1962), Section 6.2). In vacuum, it leadsthe to expression for the mag- in textbooks on theoretical physics or electrodynamics (e.g. J.D. Jackson, whose derivation is given 268 Part I 14.5 Definition of the Magnetic Field Quantities H and B Within Matter 269 a horseshoe magnet at room temperature, but when it is dipped into boiling water, it falls off. Another demonstration experiment for the critical temperature of ferromagnetism can be seen in Fig. 14.8. Part I

14.5 Deﬁnition of the Magnetic Field Quantities H and B Within Matter. The MAXWELL Equations

The definitions of the magnetic material constants given in Sects. 14.1 and 14.2 are straightforward and correct, but they refer only to the average values of B0, Bm and H0 measured in the ring coils used there. We now want to define the vector field quantities H and B in the interior of matter, based on experimental results: Imagine a ring coil which is initially completely filled with a homogeneous material of permeability . We suppose it to contain two small cavities, similar to those already shown in Fig. 13.5 (cf. also the footnote in Sect. 13.5, where you should replace “parallel-plate condenser” by “ring coil” in the present context). One of the cavities is a flat transverse slit whose plane is perpendicular to the direction of the field, while the other is a very narrow axial channel, parallel to the field direction. Both cavities are again used to accept a measurement probe (real or virtual), indicated by dots in the figure. In both cavities, the fields H and B are measured. In the limit of vanishingly small cavities, these measured fields are defined as the fields within the matter. We find that the fields in the two cavities are different! In detail, the results are as follows: 1. The flux density B measured in the transverse slit as a (voltage impulse)/(field area) has the same magnitude and the same direction as the field measured by an induction loop surrounding the coil on its outside, i.e. B D Bm, independently of where the cavity is located within the material. In the limit, we can conclude that the flux density is the same at every point within the material filling the coil. We define it as the magnetic flux density within matter and denote it by the vector B. 2. On repeating this experiment in the axial channel, we measure a different flux density, one which is not the same as measured with an induction loop outside the coil. However, we also find that the measurement in the axial channel gives a very simple relation for the field H: It has the same magnitude and direction as the magnetic field H0 measured in the empty ring coil. This is determined as shown in Fig. 14.9. The material filling the coil contains a channel along its long axis which has an arbitrary but constant cross-sectional area2.

2 Note for the experimentalist: The ring channel can be cut into the surface of the iron core of the ring coil as an open groove. In practice, this means that one need only make the iron core somewhat smaller in diameter than the inside of the coil. Then the gap between the core and the coil forms the “channel”. . M H ,the is the equa- (14.9) (14.8) (14.7) M j (14.12) (14.10) (14.11) (14.13) V ). In empty ); and 4.4 AXWELL 2.17 : ,theM (and thus no hysteresis), within matter, we arrive H ). 13 m : H ; P ) the following relations: D ; H P H D and B : 0 , these equations are simplified . The following section demon- C in an j %; 0 14.4 H B H B / 0 6.5 H 0 , and we denote it by the vector 1 D D D D 1 defined in the presence of matter, as D Exercise 6.3 E B D H B D D B . given in Chap. div div B is placed in this axial channel. It is used curl curl M ) at a definition of the vector field and ) through ( D D J H and M 14.2 14.1 and E 0 in matter (see Comment C5.5.). Think for example E " B C14.9 D field within matter D and The equality of the field H is the charge density of the free charges (as opposed to the % as in an empty coil. We can extend this result in a thought experi- well as the fields These three equations describe thefields at relationships every between point the in vector space. With the vector fields and of the fields within theis material homogeneous of and a constant permanent ( magnet, for which This equation demonstrates especially clearlythe the fields difference between to measure, independently of the widthH of the channel, the samement field to an infinitely thindefined channel. as the The field measured in this way is By introducing these vector fields, tions take on their general form: Figure 14.9 axial channel to the field in the empty ring coil we find from Eqns. ( For materials with a constant permeability to those which we set out in Sect. strates some of their applications. with the aid of Eq. ( The slim induction coil Here, bound charges within polarized matter, see Sect. magnetization: space, with “free” current density (as opposedthe to molecular the currents “bound” in current magnetized density matter, of see Sect. , / M 0 = B . is simply an after defining H equations in M . The magnetiza- B Matter in a Magnetic Field AXWELL 14 M abbreviation for tion is defined in terms ofmagnetic the moments in the material. Then a simpler form. which can be used among other things to write the the field C14.9. In textbooks on theoretical physics, it is usual to introduce the magnetization vector 270 Part I 14.6 Demagnetization 271 14.6 Demagnetization Part I The vector fields introduced for the interior of magnetic materials will now be investigated in the case that the field coil is not completely filled by the material, but instead, only a smaller piece of the material of some arbitrary geometry is placed in the field. In analogy to the geometric properties of the electric polarization, we find that in a magnetic field, also, the observed magnetization depends on the geometry. An ellipsoid of rotation is brought into a homogeneous magnetic field H0 .D B0=0/ in such a way that its long axis coincides with the direction of the field vector H0. Then the magnetic field in its interior is homogeneous and we find (in analogy to Eq. (13.13)):

1 H D H : (14.14) 1 C N. 1/ 0

Values for N, here called the demagnetizing factor, are collected in Table 13.4. N is a number whose values range from 0 to 1 (0 Ä N Ä 1). Except for the case N D 0, the magnetic field H within magnetic materials is thus reduced in comparison to the externally-applied field H0. This is called demagnetization. Some examples may help to elucidate this reduction of the magnetic field H depending on the geometry (as we can see from Eq. (14.14), it is important when 1, that is for ferromagnetic substances): 1. For a long, thin rod parallel to the direction of the applied magnetic C14.10 field H0, N D 0 and therefore the field within the material is C14.10. This result can also be obtained from H D H0 : (14.15) MAXWELL’s equation (14.12). j and DP are The magnetization both zero. Then, curl H D 0, and in integral form, D . / H Mrod 1 H0 (14.16) H ds D 0. We choose the closed path to be a long, is thus the same as when the material completely fills the field coil, narrow rectangle, whose long i.e. no demagnetization is observed. B in the rod is equal to 0H0. sides are parallel to the long 2. For a sphere, N D 1=3 and thus the magnetic field within the axis of the rod, one of them material is given by within the rod and the other outside it. From this, we 3 obtain Eq. (14.15). H D H ; (14.17) C 2 0 and the magnetization is

3. 1/ 3 M D H D M : (14.18) sphere C 2 0 C 2 rod

Therefore, for 1 (ferromagnetic material), the magnetization of a sphere is considerably smaller than that of the rod (Exercise 14.2). is . r d is the B d (14.19) (14.20) (14.21) B magnetic . The air gap in I , I 0 C14.11 H ; : within the gap and in the iron 1 0 Fe H H B 1 C14.11 ). The field inside the ball vanishes D and D D d , which is sufficiently small that stray ). B B d H 14.10 disk M 1), we find D N 1, it is another factor of three smaller than in the turns of wire which carry a current ), it follows that the magnetic flux density ). The fields Exercise 14.3 ( N

Schematic drawing of an electro- Magnetic shield- 13.5 ). In the figure, an iron ring of circumference 2 14.13 C14.12 14.11 From Eq. ( same as in the filled part of the coil (core) and thus for the magnetization: then, for magnet (ring coil wound on a toroidal core) ing inside a hollow magnetic shell Figure 14.10 neous magnetic field (Fig. sphere. 4. As aferromagnetic body, practical e.g. a example hollow of iron ball, demagnetization, in a we previously place homoge- a hollow due to the cancellation ofexcept the magnetization for from a the opposite smallshielding sides, remnant field. This is the principle of 5. Demagnetization(Fig. effects alsowrapped with play athe role iron core for hasfields electromagnets a are width negligible. of Itpoles forms (Fig. a transverse slitcore are between denoted the by two the indices iron d and Fe. We want to compute 3. For a flat disk ( Figure 14.11 ) T) b 0 D ,and ). B ( 0 / within 0. We 2 (Ni 79, B 3 AXWELL b B = ), in integral D D 14.19 3 a Vs/m A B 9 3 d ,coercivefield 14.13 1): 5 1 T). A practical , outside radius 10 (W. de Gruyter, B 10 7

. a supermalloy 2 H 2 Matter in a Magnetic Field D Classical Electrody- 10 D D 14 is from this, Eq. ( “SQUID” systems (extremely sensitive superconducting magnetometers). used for shielding the earth’s magnetic field (and other ambient fields) to allow medical investigations using application is found in the large field-free chambers p. 231). A suitable material would be for example the alloy (see for example J.D. Jack- son, namics Berlin). 2nd ed., 1983, (for a hollow sphere made of a magnetic material (inside radius C14.12. As an additional example, consider the homogeneous field equation ( form C14.11. This can also be found from the M 2 Fe 15.7, Mo 5, Mnweight 0.3 %; its permeability at B B and one outside it. Itfollows then that to be in the shapepillbox of whose a lid flat and bottom surfaces are parallel to the disk, one within the material choose the closed surface 272 Part I 14.7 The Molecular Magnetizability 273

The ﬁeld H, however, is discontinuous at the boundaries between the gap and the iron core. We ﬁnd Part I 1 H D H Fe d (14.22)

(demagnetization). Both fields, H as well as B, are reduced in comparison to a coil which is completely filled with an iron core (no air gap).C14.13 We obtain C14.13. In Video 8.4 http://tiny.cc/xbggoy 0NI (see Fig. 8.24), this field re- Bd D 0 Hd D : (14.23) 2r C d duction can be very clearly seen by making a gap of Derivation of Eq. (14.23): The relation between the current I and the field 0.4 mm width using a few H follows from the MAXWELL equation (14.12), where the second term sheets of paper. The force, vanishes for a stationary state. In integral form, it is then given by which is proportional to I B2 (Eq. (8.26)), decreases D ; H ds NI (14.24) from more than 530 N to 15 N, i.e. the field is reduced where the path integral passes around the current NI. In the present case, by more than a factor of 6 the integration path follows a circle 2 r through the iron core and the air (Exercise 14.4) gap. We obtain

HFe.2r d/ C Hdd D NI ; (14.25)

and, together with Eq. (14.22) and the assumption that d 2r, we arrive at H d 2r C H d D NI : (14.26) d

This leads directly to Eq. (14.23).C14.14 C14.14. The depolarization in a filled parallel-plate con- As the gap width d increases, the field thus decreases. Assuming denser (Fig. 13.1) can be a constant susceptibility of e.g. 1000 for iron, one can find determined in perfect anal- from this equation that even a relatively narrow gap will have a strong ogy using the method shown effect on the fields, both within the gap and in the iron core itself. here. We begin with a narrow air gap. At the transition Qualitatively, we have already observed similar behavior on changing from the air gap to the di- the iron magnetic circuit in a coil with an iron core (Fig. 8.12.) electric, the field D remains unchanged, i.e. it is continuous (Eq. (14.10), see also 14.7 The Molecular Magnetizability Comment C14.11), while the field E is reduced due to depolarization. At a con- The different behavior of paramagnetic and diamagnetic materials stant condenser voltage, the was already indicated qualitatively in Sect. 14.4. Its quantitative ex- amount of free charges, and planation is important among other things for an understanding of thus also D, must increase. molecular structure and thus for chemistry. For this, we will need the concept of molecular magnetizability. ) ) 1) A 14.3 14.7 N ( ( (14.27) (14.30) (14.28) (14.29) is the V N is constant, is interpreted is the macro- m ; m parallel to the field n individual atoms or A V m : N m N the molar volume, 0 B (Vol. 1, Sect. 13.1), and : n / : 0 = 1 1 V V B m . We then find D N / V 0 of the 1 mol : V N 0 B , an external magnetic flux den- . 23 N m magnetic susceptibility, )). We demonstrate this in the m V ˇ m V D 0 10 D 0 D . D D 0 m m 0 1 are proportional to the magnetic flux M 14.30 D V N 022 m M 0 : V / 1 6 ), obtaining D m N 1 V D D N M /(V s), (Eq. ( D 0 M 4 0 14.27 ˇ . m Vs/(Am)). D 6 in A m constant of Paramagnetic Molecules ˇ ˇ )and( is a microscopic quantity, while 10 molecular magnetizability p 0 m m 14.7 the 257 . For this reason, we set : ˇ 1 VOGADRO B , assuming a negligible demagnetization effect (since D 0 B (for example, the A number (particle) density of the molecules, as the time-averaged contributions In an atomic picture, the overall magnetic moment Note that molecules, that is density and thus the contributions Experimentally, we find for diamagnetic materials that and call direction, with Then the body will acquire a magnetic moment scopic magnetic momentequations ( of the whole body. We combine the sity 14.8 The Permanent Magnetic Moments These can be calculated fromular the magnetizability experimentally-determined molec- In the interior of a body of volume present section. will produce the homogeneous magnetization Matter in a Magnetic Field 14 274 Part I 14.8 The Permanent Magnetic Moments mp of Paramagnetic Molecules 275

Without an applied field, the directions of the moments mp are randomly distributed in space as a result of thermal motions. The sum of the magnetic moments mp averaged over time and space is equal to Part I zero. However, in the presence of a magnetic field, the moments have a preferred direction. As a result, each individual molecule makes a contribution m0 averaged over time to the total magnetic moment m of the body. This contribution is usually only a small fraction x C14.15 of the paramagnetic moment mp which every molecule carries. C14.15. Note, however, the We thus have remarks in small print in Sect. 14.4, Point 2, paramag- 0 D : m x mp (14.31) netic materials. This fraction x can be calculated, as long as the interactions between the molecules can be neglected, as is the case in gases and dilute solutions. We then find 1 m B x p (14.32) 3 kT

(k D BOLTZMANN constant D 1:38 10 23 W s/K, cf. Vol. 1, Sect. 13.10, as well as Sects. 14.6. and 16.6).

The fraction x is thus essentially equal to the ratio of two energies: The work mpB is necessary to rotate the carrier of the magnetic moment mp to an orientation perpendicular to the field direction. kT is the thermal energy which can be transferred to it by a molecular collision. A precise calculation would have to consider not only a perpendicular orientation, but rather all possible orientations of the molecular moments relative to the field direction (LANGEVIN- DEBYE formula). This leads in the first approximation to the numerical factor 1/3. Combining the equations (14.29), (14.31), (14.30), and (14.32) yields

2 1 m 0NV const D p D ; (14.5) m 3 kT T that is CURIE’s law (Sect. 14.4,forNV D const). In paramagnetic materials, the molecular magnetizability therefore decreases with increasing temperature. Figure 14.12 shows an example (see also Fig. 14.5). Furthermore, combining Eqns. (14.5)and(14.30) gives an expression for the permanent magnetic moment of a molecule p mp D ˇ 3 kT: (14.33)

A numerical example for the O2 molecule: From Table 14.1 (p. 264), we read off the value of the magnetic susceptibility of oxygen referred to the amount of substance, m=.n=V/, and using Eq. (14.30), we calculate its molecular magnetizability to be

Am4 ˇ D 5:5 1026 : Vs –1 D I with K in .We CCC r –3 x (14.35) (14.34) 2 Cr 3.54 O 14.2 /: 293 K, into CC mass 4 Ni 3.00 D D : (from Sect. 6.6 in V s m A m ; T CCC = reciprocal temperature eur 1 T –25 2 1 Fe 4.92 . Further examples are 0 3 6 9·10

2 1 2

mru ·10

β Molecular magnetizability magnetizability Molecular mentum D D A s is the elementary charge, Am 2 r u r 19 Mn 5.40 23 2 angular velocity r mr 10 10 D ), it gives rise to a ring current of .TheGyromagnetic eu . 2 D 2 4.4 ;! O 2.58 58 602 : : 13.10 D 2 ! 1 D IA D D L molecule NO 1.70 e D p ( 2 . p m Koninklijke r , m Magneton 2 2 14.2 = moment of inertia , then it has an angular mo et al. The effect of the tem- u ). Compare the analogous computation for electric dipole Magnetism and Mat- eu Am D or Ratio and the Electronic Spin Permanent magnetic moments of paramagnetic molecules )gives , 494 (1931).) 23 ), and this current is associated with a magnetic moment of . 34 14.32 , D 3.6 l 14.33 = in 10 Eq. ( moments at the end of Sect. With this value for the moment, we can estimate the fraction (Methuen, London 1934), p. 343. p eu Molecule or Ion m Eq. ( perature on the magnetizability of the paramagnetic O giveninTable Figure 14.12 Sect. Akademie von Wetenschappen te Am- sterdam (E.C. Stoner, ter For measurements below 200 K,E.C. see Wiersma Vol. 1) of In addition, according to Eq. ( 14.9 The Elementary Magnetic Moment In this section, wemagnetic want moments to discussed above investigate and the collected origins in of Table the permanent Table 14.2 Inserting this value and room temperature, the velocity start with the momentwithin atoms. which If is an produced electron traverses by a the circular orbit orbiting of electrons radius Matter in a Magnetic Field 14 276 Part I 14.9 Magneton. The Gyromagnetic Ratio and the Electronic Spin 277

The quotient

m Magnetic moment of the particle Part I p D (14.36) L Angular momentum of the particle is called the gyromagnetic ratio. For an electron in a circular orbit, combining Eqns. (14.35)and(14.34) gives for the gyromagnetic ratio

m 1 e As p D D8:8 1010 (14.37) L 2 m kg .e=m; the speciﬁc charge of the electron; is equal to 1:76 1011 As=kg/:

In BOHR’s model for the hydrogen atom (e.g. as described by POHL in the 13th edition of “Optik und Atomphysik”, Chap. 14), an electron on the innermost circular orbit in the hydrogen atom has the elementary angular momentum

h L D (14.38) 2 .h D PLANCK’s constant; 6:626 10 34 Ws2 D 4:36 10 15 eV s/:

Substituting this expression into Eq. (14.37) yields the elementary magnetic moment or BOHR magneton

e h m D D 9:27 1024 Am2 : (14.39) Bohr m 4 The measured values in Table 14.2 are of the order of magnitude of the BOHR magneton. We consider this agreement to be an indication that the orbital angular momentum can play an important role in forming the magnetic moments of atoms and molecules. An additional contribution is made by the electron’s spin, which we will discuss in the following. The measurement of a gyromagnetic ratio was first carried out for a ferromagnetic substance, iron. With iron, we observed in Sect. 4.4 the first of the phenomena which are now termed “gyromagnetic”: A ferromagnetic object during the process of magnetization acquires not only a magnetic moment, but simultaneously a mechanical angular momentum. They are the sums of all the magnetic moments mp and all the angular momenta L of the N participating electrons in the object.C14.16 C14.16. See the EINSTEIN- de HAAS effect (Sect. 4.4). For the quantitative evaluation, we proceed as follows: The rod in Sect. 4.4 The inverse effect has also has the moment of inertia . At the remanent magnetization, the rod ac- been observed: A rotating D ! quires an angular momentum of NL 0. Here, NL is the angular iron rod becomes magnetized momentum of all N participating electrons. The rod leaves its rest position (BARNETT effect). with the maximum value !0 of its angular velocity and reaches its impulse deflection of ˛0. The quantity !0 is found from the relation !0 D !˛0, where ! is the circular frequency of the rod which acts as a torsional pendulum. After measuring the angular momentum NL, we take the rod out ) of di- is and , p 14.40 (14.41) (14.43) (14.40) (14.42) Nm ). D B m ERLACH relative values : even when they are permanent magnetic e m : : p of the electrons, as in spin of an electron m and defines its “g-factor” . (In practice, it is expedi- D : : e ). The m 8.15 (often denoted as kg As circular orbit h ) yields the 2 Á 11 h p Bohr 2 1 4 L 10 m m orbit. This gyromagnetic ratio there- orbital motion D 14.40 )), circular orbit S D . 75 : L )). p p 1 L OHR m m g-factor) by the equation illations (Vol. 1, Sect. 11.10). Adjustment of the 14.39 found experimentally by W. G É 2 for the electrons in iron using Eqns. ( D D evunix.uevora.pt/~stadler/FAN-06-07/History_ g p D ) into Eq. ( L AND m g (Eq.( , like that of a spinning top. For that reason, the an- 14.42 OHR 0023.) : (see e.g. Its magnitude was determined by making use of the 2 ). (Precision measurements on free electrons later gave the D resonance enhancement and an associated magnetic moment ’s model. Instead, the electrons must have an angular momen- g TERN 14.37 C14.17 L . of the field coil and measure its remanent magnetic moment frequency of the currentsional in pendulum the gives coil a to largerod the increase ( resonance in sensitivity frequency to of the the rotations tor- of the ent to use an alternating currenton in the the magnetic coil, causing moments aresponds in periodic with force the forced to rod. osc act The rod (torsional pendulum) thus e.g. by using the procedure described Fig. OHR proper rotation B O. S We find the value It thus has the samenamed magnitude for as B the elementary magnetic moment fore cannot originate from the Today, it has proven to be expedient to quote only (often called the L tum gular momentum of a non-orbitingspin electron has been given the name not moving on circularat orbits, i.e. rest. when their Botha centers quantities of are gravity most are simply explained as the result of Inserting Eq. ( and ( value of_the_Stern-Gerlach_experiment.html In this way, weelectron obtain in experimentally iron: the gyromagnetic ratio of an rectional quantization gyromagnetic ratios: A measurednetic value ratio is of referred an to electron in the a gyromag- moment of the electron associated with its spin The gyromagnetic ratio ofpractically an twice electron as in large ferromagnetic asrent iron would of is be the expected thus electron from on the a orbital B cur- have been spin proper angular or orbital angular mo- and Matter in a Magnetic Field 14 C14.17. To distinguish between these two angular momenta (of the orbital motion and the proper rotation), the terms mentum momentum introduced. 278 Part I 14.10 The Atomic Interpretation of Diamagnetic Polarization. LARMOR Precession 279 14.10 The Atomic Interpretation of

Diamagnetic Polarization. Part I LARMOR Precession

Now, how does diamagnetism come about? In diamagnetic atoms, the electrons in the electronic shells have their permanent spin moments pairwise oppositely directed (antiparallel); furthermore, their orbital moments cancel pairwise. Only in this way is it possible that the electronic shells have no overall permanent magnetic moments. A moment is formed by induction when the atom is brought into a magnetic field. In Fig. 14.13, the equatorial plane of a planar atomic model is shown as a shaded disk. The symmetry axis A of the atom is tilted by an arbitrary angle # relative to the direction of the field B. At the moment represented by the drawing, an electron is at a distance rn from the vector B. For simplicity, we assume that the magnetic field increases linearly with time after being switched on; its maximum strength B is attained after a time t. During the period when the field is increasing, Eq. (6.2) applies. There is thus an electric field along the circumference of the circle 2rn with the magnitude

BPr2 r E D n D n BP : (14.44) 2rn 2 It causes the electron to accelerate: Ee 1 e a D D r BP : (14.45) m 2 m n This acceleration increases its orbital velocity along the circular orbit 2rn within the time t by an amount 1 e u D r B ; (14.46) 2 m n corresponding to the angular velocity ! D u=rn,thatis 1 e ! D B : (14.47) Larmor 2 m This angular velocity or circular frequency, named for its discoverer, is independent of the radius rn; it holds for all the elementary charges

Figure 14.13 ARMOR B The origin of L precession. The A double arrow indicates that the electrons in a diamagnetic ϑ rn molecule are circling as pairs in opposite directions.

ϑ , ) 0 2 2 D m B V ,the N 0 14.9 +++ m ). Only 0 Gd A). attain the A Ni 0 Here m .Atasmall , Point 3.2). Its 3 14.15 14.15 O 14.4 2 1, as observed for 0 0.05 0.1 Vs/m 00.51Vs/m nickel (Fig. Bohr 8H 0.1 0.6 0.3

Magnetic flux density

m 0.05 3

< m' ) moment magnetic Average , top) and inversely pro- 4 Bohr )), increases in an applied m ’s law, Fig. , then it moves around that (SO 2 14.14 B 14.27 : ). According to equations ( URIE (Fig. Top 8.3 ) exhibit a behavior which is sur- )). (C B H (Eq. ( hydrate (see also Sect. magneton, 0 T is negative and MV V 14.4 m N D 300 K). = D OHR 0 iontothe M B D C m ( , in units of the 3 , in the alum Gd and the particle number density of the Gd ions is T V D ’s law, Sect. 3 C ; N 0 3 : Ferromagnetic, mi- = 14.7 m Bohr ENZ M m O (measured as described 2 D 01 g/cm . : Antiferromagnetism, and Ferrimagnetism bottom The average contribution 3 0 3 on a precession cone. The sense of the rotation ensures 8H ); m D B ), this means that cm 3 % ). ( / 4 14.3 21 magneton 10 14.7 SO 14.9 . defined as in Fig. of a Ni atom and a Gd 2 OHR 0 86 0 Crystalline gadolinium sulfate octa : B which is given by average contribution of an ion to the overall magnetic moment at low temperatures and in high magnetic fields does order of magnitude of a B Gd direction that the magnetic moment that it generatesmagnetic opposes field its (L own origin, the crocrystalline nickel (magnetic flux density B Ferromagnetic solids (Sect. prisingly different. As an example, we choose and ( diamagnetic materials. We can thussimple explain models. both paramagnetism and diamagnetism with m (Eq. overall magnetic moment magnetic field proportionally to portionally to the temperature density is 4 Figure 14.14 3 When paramagnetic atoms are bound together incontinue a solid, to they as be a paramagnetic. rule ionized As an Gd example, ion, we Gd mention the triply- 14.11 Ferromagnetism, within the atom. Asgether, a that is result, the all atom thosethe as elementary a magnetic whole charges field. rotates rotate around An to- atom the observer would axis see supposed defined no to by changes be inelectrons within at the an electronic the orbits. atom center can The ofaxis have orbits a of the is common the not axis of parallel rotation. to If the this direction of in Sect. Matter in a Magnetic Field 14 280 Part I 14.11 Ferromagnetism, Antiferromagnetism, and Ferrimagnetism 281

∙10–26 A∙m2 ∙105 kg/m3 12 7.0 m per Gd+++ ion Bohr Gd (SO ) ∙8H O 6000 2 4 3 2 Part I AA' Gd2(SO4)3∙8H2O 8 4000 Paramagnetic, 4 2000 2 B0 = 2.6 V s/m 0 0 05101520K 0 100 200 300 K

600 0.61 mBohr 20 BB' Ni 15 400 Ni

m' Ferromagnetic, 10 200 B = 2 V s/m2 0 5 Curie temperature TC 0 0 200 400 600K 800 600 650 700 750 K 20 C MnO 10 C' Antiferromagnetic, MnO 10 2

Average magnetic moment Average B0 = 2.4 V s/m – reciprocal reduced susceptibility

m 5 χ /

Néel ϱ

temperature TN 0 100 200K 300 0 100 200K 300 4000 3.9 mBohr D' 3000 D 20 2000 Fe O Fe3O4 3 4 10 1000 Ferrimagnetic

0 0 200 400 600 800 K 900 1000 K Temperature Temperature

Figure 14.15 Influence of the temperature on solids with different magnetic behaviors: The average contribution m0 of one molecule or one atom or ion to the overall magnetic moment m, and the reciprocal magnetic susceptibility (%=m) relative to the density of the material (the flux density B0 is defined as in Fig. 14.7). (Literature to Parts A and A0:seeLandolt/Börnstein,6th edition, Vol. II/9, “Magnetische Eigenschaften” (Springer-Verlag 1962), in the article by I. Grohmann and S. Hüfner, pp. 3-200 ff. For the other parts of the figure, see E. Kneller, “Ferromagnetismus” (Springer-Verlag 1962), Chap. 4–6). fraction of the magnetic flux density which we required for Gd3C, the average atomic contribution m0 in Ni has already reached its saturation value (Fig. 14.14, bottom). It increases below the CURIE temperature (TC D 631 K) with decreasing temperature, and already at room temperature, it has a value of 0:58 mBohr (Fig. 14.15, B). A similar behavior is observed for iron: In Fig. 14.7, already at . , 4 D 0K 2 each Bohr m power D Am 3 T O B). Above 23 2 . The smaller ,sothatall 10 14.15 0 : 2 (Fig. D 0 which can be observed C m T M superparamagnetism Macroscopically, the directions from the individual atoms to the t the alignment of the elementary Two points have emerged in more 0 ey bring about an orientation of the m temperature temperature, since not the thermal motions, reach the order of magnitude of C14.18 ;thisiscalled p m URIE in spite of the thermal motions URIE m domains or magnetic domains) are sponta- of all the domains are distributed statistically , 1891). and room temperature, M 2 EISS W WING C14.18 Vs/m In solids with ferromagnetic behavior, new forces ap- ). 0 3 temperature, the crystal behaves simply as a paramagnet B is observed. 10 URIE 3 14.15 Bohr m D 1 0 : This holds of course only for objects which contain many spontaneously- the domains are oriented inThe the saturation same value of direction, the parallel magnetization to the field. in an applied field decreases from its maximum value at monotonically up to the C onto the surface. The powder achieves the same result for micro- this C Thermal motions limit themation spontaneous of domains magnetization, in i.e. which allmoments the the for- are microscopic elementary oriented magnetic parallelthere is to not each some other,An external fraction and field can of thus now them align fortization, the which directions which example of is spontaneous is magne- antiparallel present to the rest. then a large fraction ofparallel the atomic to moments each must already other.temperatures be can oriented achieve Only such very a parallel largematerials orientation; which magnetic that behave is fields shown paramagnetically. at by explanation: very There low remains onlypear one within the crystal lattice.atomic Th magnetic moments within microscopicThese regions of regions the ( crystal. 2 recent times: 1. The recognitionmagnetic tha moments within thenot spontaneously be saturated explained regions by can- means magnetostatics alone; to and observe 2. thesespontaneous The saturated magnetization experimental microscopically. regions and Thesebe the discussed two in direction points the will of following. their In order to visualizecrystal microscopically the regions spontaneously-magnetized (domains), onenetized polishes iron the crystal, surfacebrings preferably of a using a suspension electropolishing. non-mag- of extremely Then fine one ferromagnetic Fe overall magnetic moment over all angles; therefore,not magnetized a before ferromagnetic it is crystal brought is into “as an external a magnetic whole” field neously magnetized up to saturation. of the magnetization but instead the surface tension limits their spontaneous magnetization. (Fig. When the average contributions the particles, the lower their C magnetized domains. If the body is in the form of fine powder in which powder particle containsdomain only particles”), then one the spontaneously-magnetized particlesvery act large domain like magnetic moments giant (“single- paramagnetic molecules with 4 The assumption ofcrystal microscopically regions in small, materials whichnot magnetically exhibit new (I.A. ferromagnetic saturated E behavior is B D m temper- of the fer- constant, URIE law: m ; / URIE C EISS T is the C -W C . T Matter in a Magnetic Field T is the C =. URIE 14 T T C C and ature A recent review of magnetic microstructures, as well as an overview of ferromagnetism, can be found in the“Magnetic book Domains” by Alex Hubert and Rudolf Schaefer, Springer-Verlag (1998). C C14.18. In this temperature range, the magnetic susceptibility romagnet is described by the 282 Part I 14.11 Ferromagnetism, Antiferromagnetism, and Ferrimagnetism 283 Part I

Figure 14.16 Left: Visualization of the domain boundaries between spontaneously-magnetized regions, and the directions of the magnetization M within them. Right: An explanatory sketch. In polycrystalline material, the domain boundaries often assume rather complex shapes.

ab NS

Figure 14.17 Furrows, scratches etc. produce poles N S on the surface of a magnetized object (so long as they are not parallel to the direction of the magnetization, which is perpendicular to the plane of the page in Part b) scopic dimensions as iron filings on a macroscopic scale (Chap. 4). Figure 14.16 shows an image of this type on the left, and on the right, a sketch which explains the image on the left. At the domain boundaries, poles occur. The dark powder collects within their magnetic fields. These images not only allow us to recognize the boundaries of the domains, but also the directions of the magnetization M within them. For that purpose, a brush with glass bristles was used to scratch fine furrows on the surface and thereby to localize magnetic poles along them. Such poles however are formed only when the furrows are nearly perpendicular to the direction of the local magnetization M,as can be seen from Fig 14.17a. Therefore, the darker lines filled with the powder can be seen in Fig. 14.16 only where the furrows cross the magnetization directions (arrows in the sketch). In order to investigate the magnetization process, let us first heat a ferromagnetic object (made e.g. of iron) temporarily to above its CURIE temperature; after cooling, it will be in an overall non-magnetized state. In which manner can an applied magnetic field then convert this macroscopically non-magnetized object into a magnet, so that it acquires an overall magnetic moment m? The microscopically observed, spontaneously-magnetized domains in Fig. 14.16 (left) are not bounded by surfaces in the mathematical sense, but rather by separation layers of finite thickness (ca. . , is M S - as the which 14.18 N B which ex- rotate wall motion . These also M 14.16 comes to an end . The coil is connected J ). walls. The transition of the ; this occurs through a shift and the local magnetization ˇ which can make its effects felt B m LOCH . As a result, the regions containing or B ˇ to the remaining domain wall 1. There- Exercise 14.5 2 domain rotation process symmetric domain walls . The short arrows in Part III indicate the sense of rotation. is smaller than =0 M , since the surface was cut parallel to one of the cubic The magnetization process. The lines denoted by 1 and 2 indi- β B ˛ , surrounded by an induction coil , α have become zero; this schematic can be seen in Fig. M l 14.16 β I II B III B IV B m), called 14.19 111 2 grow at the cost of those containing α vectors is parallel to a facescopic diagonal magnetization of of the the cube. cubeduces This a an field preferred overall gives direction magnetic the and moment macro- thus pro- direction of magnetization ofdomain a occurs domain within into these thatwalls domain of with walls. its the surface neighboring The ofin intersections the Fig. sample of show the relatively simple patterns in the space around the cube. How doesOf this occur? the two angles between the field cate domain walls. The arrows showmagnetization the direction and relative strength of the Figure 14.18 0.1 field increases further, andthe both field domains (Part approach III). the This direction of of the domain wallonly 2 two to domains the are right still (in present. the figure, Their Part magnetization II). vectors In Part III, initially have the same orientationsPart as II; those they on are the left and above in fore, this wall will not be shifted further; instead of a a new process occurs: The directions of magnetization ˛ Now suppose that the iron cube is placed in a magnetic field crystallographic faces. The sketch at the right in Fig. plains the observed pattern can belengths reduced to a schematic in which the can be readily observed. A polycrystalline material,Fig. a thin, soft iron wire Fe, is shown in The wall motion during the process ofand magnetization recorded can be microscopically. observed An impressiveof effect domain is walls the on “sticking” ferromagnetic disturbances inclusions. in the When crystalthere the lattice, are walls such sudden, break as jerky loose non- jumps irreversibly, in the magnetization when the whole crystal hasmagnetization a is unified saturated ( magnetization (Part IV), i.e. its a small bar magnet; it can be moved back and forth in the direc- to an amplifier andflection thence proportional to to an the oscilloscope time, with and its also horizontal to de- a loudspeaker. Matter in a Magnetic Field 14 284 Part I 14.11 Ferromagnetism, Antiferromagnetism, and Ferrimagnetism 285

Figure 14.19 The demonstration N SJ of statistically distributed jumps in Fe the magnetization during a uniform Amplifier Part I variation of the magnetizing ﬁeld (BARKHAUSEN effect)

Oscilloscope Loud- speaker

Figure 14.20 Resolution of the BARKHAUSEN M effect (steps in the magnetization) in a hysteresis loop, schematic

tion towards or away from the iron wire, along a ruled scale. As it approaches the iron wire, the magnetization of the wire increases. When the magnet is removed, the magnetization again decreases. In this way, the hysteresis loop of the iron wire can be repeatedly traversed. Earlier, in Fig. 14.7, the loop appeared to ba a smooth curve. Now, we find that it is composed of many small steps, as can be seen in Fig. 14.20; each step represents an irreversible jump due to a jerky wall motion. These motions can be seen on the oscilloscope as individual steps along the curve. In the loudspeaker, they produce a crackling noise which is named for H. BARKHAUSEN (the Barkhausen effect). Most of these jumps occur in a purely statistical manner. A few of the strongest, however, occur again and again at the same positions of the bar magnet. Domain-wall motions occur predominantly in weak applied magnetic fields, while domain rotations occur in stronger fields. There, the situation is complex. Briefly, one can summarize the whole process as a magnetic recrystallization, which usually begins at locally-formed “nucleation centers” (they play a role not only in crystallization, but also in evaporation and condensation processes). Based on these experimental findings, we can readily understand the sketch in Fig. 14.21A: Iron crystallizes in the body-centered cubic (bcc) system, i. e. its crystal lattice consists of two parallel, inter- meshing cubic sublattices. The vertices of the cubes of the second lattice lie at the intersections of the body diagonals of the first lattice. The cubes sketched in the figure belong to a magnetic domain which is spontaneously magnetized to saturation. Instead of the atoms, only the magnetic moments of the atoms are drawn at the vertices of the cubes, differently colored and readily distinguishable for the two sublattices. In Fig. 14.21, Parts B and C complement Part A: They contain the magnetic structures which are discussed in the following. In Fig. 14.21B, equally strong atomic moments in the two sublattices in- M 200 URIE (named ttice; C: again m 150 MnO = 100 of the atoms in- 50 p 0 m 100 200 .However,theinflu- B behavior. The antifer- temperature, 1 Temperature Néel temperature C14.19 Ni ÉEL , i.e. its magnetization ,the N and on the reduced susceptibility ). Explanation: In order to destroy T V W s/(mol K)W s/(mol K) W s/(mol N , an antiferromagnet behaves at a con- 14.22 = 0 400 500 600 700 K K

20 40 , antiferromagnetic materials behave as

M p T c N heat Specific antiferromagnetic T paramagnet ). The reason: The orientation of the atomic mag- 0 C give examples for antiferromagnetic MnO. 0 in an applied magnetic field becomes less effective when p 14.15 ). Above m temperature (Fig. Ferromagnetic (A) and antiferromagnetic (B) ordering of Variation ÉEL ÉEL CandC temperature, temperature and is more complex than for a simple paramagnetic material. Fig- ÉEL 14.15 creases relative to the thermal energy. The specific heat capacity ofmagnetic the behavior solids is with anomalously ferromagnetic large and in antiferro- the neighborhood of the C the antiferromagnetic coupling between the moments netic moments increases (Fig. or the N With decreasing temperature below the N the ordering of thesample. atomic magnetic moments, heat must be added to the abc =% URIE m C romagnetic order is reducedferromagnetic by order, as random the thermalorder temperature motions, vanishes increases. just at like The a long-range temperature for Louis N the atomic magneticSchematic moments representation of in a ferrimagnetic a material body-centered cubic crystal la paramagnets in an applied magnetic field. of the specific heats of ferromagnetic and antiferromagnetic materials near their magnetic transition temperatures (the respectively) ence of the temperature on creases linearly with the applied flux density stant temperature like a the N Also at temperatures below Figure 14.21 are oriented antiparallel tois each spontaneously other. magneticallymagnetic Each saturated, moment of of but the the theof two domain a sublattices resultant is body zero. overall which This exhibits is the simplest case ures Figure 14.22 ,

,where tempera- / EISS C T -W is described by =. T N C T URIE Matter in a Magnetic Field D in the temperature range 14 m m ture, is positive. C14.19. In antiferromagnets, the magnetic susceptibility above the C 286 Part I 14.11 Ferromagnetism, Antiferromagnetism, and Ferrimagnetism 287

There are solids in which the two sublattices are not equivalent, but instead have differing overall resultant magnetic moments (Fig.

14.21C). Then the difference between the sublattice magnetizations Part I gives rise to a spontaneous magnetization. Such materials are termed ferrimagnetic (ferrites). A well-known example is magnetite, known for over 2000 years as “lodestone”, Fe3O4.

This mineral has the spinel structure. Its chemical formula is FeOFe2O3. The negative oxygen ions form a face-centered cubic (fcc) sublattice, into which one divalent and two trivalent iron ions per formula unit are included. The divalent iron ions can be partially or completely replaced by other metal ions; this gives rise to a large variety of cubic ferrites, as these compounds are called. The ferromagnetism of the ferrites comes about as follows: One half of the trivalent iron ions forms one sublattice, while the other half forms another sublattice together with the divalent metal ions. The divalent oxygen ions are diamagnetic, and thus have no permanent magnetic moments mp. The magnetic moments on the metal ions are oriented antiparallel to each other in the two sublattices. The moments of the trivalent iron ions thus mutually cancel. The resulting spontaneous magnetization is formed by the magnetic moments of the divalent metal ions.

In ferromagnetic materials, the magnetization M can be increased only up to a maximum or saturation value by applying an external magnetic field of increasing flux density B. The same is true of ferrimagnetic materials. The temperature dependence of the saturation values of M isshowninthecurveinFig.14.15D; it is similar to that for ferromagnetic materials. However, the influence of the temper- 0 ature on the reduced susceptibility m=% (Fig. 14.15D ) is different from the case of ferromagnets (Fig. 14.15B0). As a result of the inequivalence of the two antiferromagnetically coupled sublattices, not only are the saturation values of their magnetic moments m different, but also so is the temperature dependence of the spontaneous magnetization of the sublattices. For this reason, it can happen that the resulting spontaneous magnetization goes to zero with increasing temperature before the CURIE temperature is reached. This can be shown using a lithium-chromium ferrite in a surprising demonstration experiment.

In Fig. 14.23, a rod made of this material is hung from a thread. The rod has a remanent moment m. In ﬁeld-free space, it will be oriented perpendicular to the plane of the page. Between two magnetic poles, it orients itself parallel to the plane of the page, for example with the arrow pointing to the right. Then the rod is warmed by hot air and thermal radiation from a glowing heating coil beneath it. At T D 38 °C, it turns until it is again perpendicular to the plane of the page; it has thus become non-magnetic. If the temperature is increased still further, the rod again turns into the plane of the page, but this time with the arrow pointing to the left. Therefore, the overall magnetic moment m of the rod has reversed its direction by 180°.

The ferrites are ceramic-oxide materials (first described in 1779). As semiconductors, they have resistivities which are many orders of magnitude greater than those of metals. Therefore, perturbations caused by eddy currents play no role in their properties. , . B g of S 13.6 m in the in the 14.24 p m 5mg E H within the ). m B N( 5 14.17 Heating coil 10 )to( 0 T, and for the field : 2 1 : N D 14.14 B direction is oriented upwards, echnology. Complex ferrites, ) z 6 mm is measured in an inhomo- ) ). For the magnetic flux density 14.6 (the , 3 D , can be powdered and bound together 3 ). An example is shown in Fig. )), derive the magnetic field 14.3 d O 14.3 2 . Is it necessary to take the demagnetizing 14.6 m 13.16 0Vs/m 6Fe : Exercise 14.6 20 38 °C, the ferrite ) ). BaO ), beginning with Eqns. ( > 2 16 D 3 O M 14.6 z O T ) through ( d 2 A “compass needle” The magnetic moment of C = 3 6 B Fe 4Fe quency and communications t 13.12 a) In analogy to the derivation of the electric field In order to determine the magnetic susceptibility C 2%).Calculate 2 6 Cr C 2 interior of a uniformly(Eqns. polarized ( sphere, as described in Sect. interior of a uniformly-magnetized sphere (amagnetization permanent magnet with sphere? (Sect. material with a diametergeneous of magnetic field (Fig. at the position of the sphere, we find graphite (arc-lamp carbon), the weight of a sphere made of this opposite to the force of gravity).on, When the the magnetic weight field is of switched the sphere decreases by 5 whose long axis points fromwest, east because to its north andare south located poles on its sides ( a ferrite is formed as thetween difference two be- oppositely-directed magnetic moments with differing temperature dependencies. At rod rotates by 180° (here,Li the ferrite is factor into account? (Sects. 14.2 for making strong permanent magnets. For example, one can fabricate short,ferrites with thick a permanent high magnets from coercive field,magnetizing although factor this (Sect. shape has a large de- with a glue, and they haveapplied very for high example in coercive fields. magnetic data They storage are or widely as low-cost materials Figure 14.23 Exercises 14.1 Figure 14.24 This makes thehigh-fre ferrites exceptionally important as materials for gradient, d b) What do you find in this case for the flux density i.e. e.g. PbO Matter in a Magnetic Field 14 288 Part I Exercises 289

14.3 As a simple example of magnetic shielding against an external magnetic field Ha,thefieldHi within a spherical cavity in a magnetiz- able plate of permeability a is to be computed. The plate is oriented Part I perpendicular to Ha (analogously to Fig. 13.8). Ha is parallel to H0, as in the electrical case; Eq. (13.18) holds also for magnetic fields (see e.g. M.H. Naifeh and M.C. Brussel, Electricity and Magnetism (John Wiley, New York 1985), p. 308). Find the relationship between Hi and Ha. (Sect. 14.6)

14.4 With the pot magnet in Fig. 8.24 (Video 8.4), we want to estimate the forces F0 for a gap width of zero and Fd for a gap width d D 0:4 mm. The permeability of the iron is D 727, the number of turns in the magnet coil is N D 175, and the current is I D 0:75 A. The end surface of the inner pole has an area of A D 7:55 cm2 and is the same as that or the outer pole (the ring), so that Eq. (14.23)may be applied. The length of the path integral (Eq. (14.26)) is l D 12 cm (note that in the calculation of the path integral, the quantity d occurs twice). (Sect. 14.6)

14.5 In Video 10.1, “The inertia of the magnetic ﬁeld”, the slow buildup and decay of a magnetic ﬁeld are shown (Sect. 10.2,Com- ment C10.5). a) Initially, switch 1 (Fig. 10.6) is closed and the increase of the current up to its maximum value of 15 mA is followed (the OHMic resistance of the coil is R D 130 ). Evaluate this rise by plotting it on semilogarithmic graph paper and test it for an exponential dependence. b) In a second experiment, the decay of the current is determined by closing switch 2 and shortly thereafter opening switch 1. Again, look for an exponential dependence. c) Finally, the leads to the coil are exchanged, switch 2 is opened and switch 1 closed, and again the rise in the current is registered. What time dependence is found now? How can you explain qualitatively the observed time dependence in all three experiments? (Sect. 14.11)

14.6 Explain why the magnetic ﬂux density B in front of the ceramic disk in Fig. 14.24 is smaller than that in front of a long bar magnet with the same homogeneous magnetization M. As a simpliﬁ- cation, approximate the shapes of the disk and the bar by ellipsoids of rotation. For the disk, the ratio of thickness to diameter is l=d D 0:1. (Sect. 14.11)

Electronic supplementary material The online version of this chapter (https:// doi.org/10.1007/978-3-319-50269-4_14) contains supplementary material, which is available to authorized users. Optics II 291

Part II lcrmgei ae fvr hr aeegh.W fe althis word call often We of consists wavelengths. light radiation short this very that of waves learn electromagnetic children reach school Today, stars radi- The the space. and there. of sun way It its the on eyes. of medium our ation transporting at of arrives kind “something” no sources, requires light or objects diating h pia ain Power Radiant Optical the Measuring Introduction. h aito hc xie u esr ecpinof perception sensory our excites which radiation the a accomplish can eyes Our Detector. Radiation a as Eye The 15.2 will you Then eye. your of un- corner head the your on stick press see bedroom, and darkened covers the your der in night black Some Introduction 15.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © h od rne eei tlc r sdb u aua language natural our by used are italics describe in to here printed words The aeu uhfrhrta u asd nteaaoospolm of problems analogous the in do ears our than further much us take t esrmn pooer,seChap. see (photometry, measurement its utailt nrrd.Ti obemeaning, double This infrared). (ultraviolet, of n brilliance and perceptions, visual our of excitation usual fruitless The many avoid can outset diversions. fun- the and this from discussions Taking account into physiology. fact and psychology damental of jurisdiction the are ovnini cutc,a el(o.1 et.1.41.0.There, 12.24–12.30). of perception Sects. the 1, (Vol. also, well language as the acoustics, to in corresponds convention radiation, physical a as light and ception, ae” cutcrdain rsml on.I hscs,as,the also, case, this In sound. simply word or “sound radiation, called usually acoustic is waves”, sound of perception the causes which diation on,infrasound). sound, colors a pouigrdain“pia aito” rsml ih.The light. simply or radiation”, “optical radiation -producing light sound rgtlight bright and M ntesneo aito sue vnfrivsberays invisible for even used is radiation of sense the in esr perceptions sensory sue ihu eiainfriadbesud (ultra- sounds inaudible for hesitation without used is brilliance ACH scue yafr of form a by caused is ntefr fa of form the in , sStripes ’s sound olsItouto oPhysics to Introduction Pohl’s ontbln nteramo hsc;they physics; of realm the in belong not do ra eli h hsclivsiainof investigation physical the in deal great sectdb yeo aito.Tera- The radiation. of type a by excited is vr novmn with involvement Every . su hog h atemptiness vast the through us es radiation ooe,ylo,shining yellow, colored, 29 ,adeeyinvestigation every and ), light ih,bihns,color brightness, light, rgntn ihra- with Originating . , https://doi.org/10.1007/978-3-319-50269-4_15 sasnoyper- sensory a as light hycan They . light ring. and 15 293

Part II 1 2 ,astar bc 15.1

a abc Reflected optical radiation optical Reflected .InFig. s stripes ’ two. This is shown as a drawing ACH ’s . ACH the dark ring bounded on its inner side by the inner, bright circle surrounded by a still 15.1 see see ’s stripes at the – quite a different light distribution from what is – both on the rotating disk and in its photographic ACH see 15.2 M The origin of M as a photograph is seen. 15.2 Optical Radiant Power brighter border. We stripes. When the disk istated, rapidly the ro- image which is shownFig. in eyes, the most light would seem to come from the bright border, and cut from white paper has been pasteddisk onto is a dark illuminated cardboard disk. atwith The a a window motor. or The eye byinnermost perceives a zone three has lamp concentric the and circular greatestoutermost zones: is brightness zone The rotated per is rapidly surface the area, leasttinuous and bright. transition the between The the middle other zonein provides the a lower part con- of Fig. However, we image, Fig. in fact present. We a still darker border. According to the strong impression given by our and between grey and black boundaries between white and grey, Figure 15.1 A drastic example is provided by M acoustic radiation. But, likeeyes every fail organ when of it sensorygive perception, comes us our an to exact quantitative numerical measurements. description of They “less” fail or “more”. to Figure 15.2 Introduction. Measuring the 15 294 Part II 15.2 The Eye as a Radiation Detector. MACH’s Stripes 295

Figure 15.3 On “inverted seeing” (image of a surface which is partially wetted). Look at the image alternately as printed and then rotated by 180ı. Part II the least light from the dark border. Every objective observer would be led inevitably to the wrong conclusion that the ambient light is most strongly reflected by the bright ring, and least reflected by the dark ring. The light distribution sketched in the lower part of Fig. 15.1 can be observed in many arrangements and experiments. For this reason, these “MACH’s stripes” have caused many an error in diverse physical observations. Nevertheless, we should not hastily dismiss them as an “optical illu- sion”. The phenomenon of MACH’s stripes is of great importance for our visual sensory perceptions. Think for example of reading dark printed letters on white paper. The lenses of our eyes by no means produce a perfect image. The outlines of the printed letters are not sharply imaged on the sensitive layer in our eyes, the retina. The transition from the dark letters to the bright background of the paper is washed out, as in a poorly-focussed photo. But our visual perceptions can compensate for this error with the help of MACH’s stripes. The eye sees, figuratively speaking, a bright border at the boundary of the bright paper, and a dark border at the boundary of the dark letters in its image of the printed page. In spite of the fuzzy image on the retina, this gives us the impression of sharp outlines. Another useful hint shows how our vision is decisively influenced by processes within our brains: Such “central processes” depend in a very complex way on the processes within the retina of the eyes. An example is “inverted seeing”, as explained in Fig. 15.3 (an exchange of deep and high levels in an image). Keep in mind also what is said in Sect. 18.15 about the “depth of focus” in images. So much for these important phenomena which are generally typical of the operation of our visual sensory perceptive apparatus. radiant (a detailed 0 A d A = P W d .There,ametal , and for the receiver ). D V ˝ e d Thermometer 15.4 ). Heating coil = E 15.5 P 19 W ; thus we define the 0 d is radiated into the solid angle A D P W I A' area d area watt. This electrical power is equal to Irradiated receiver D , that is, they receive energy. In the sun’s Ω (or irradiation intensity) as ampere warmed Calibration of an optical radiometer. The voltage of the current C15.1 irradiance Solid angle d . Direct Measurements of Radiant Power of the lamp as emitter or source as ,the 0 A and is then absorbed in the receiver area d Optical Radiant Power ˝ treatment of these quantities will follow in Chap. intensity d source can be adjusted. The radiant power d area d the previously-absorbed radiant power: Thus, weradiometer have calibrated the plate is being irradiated byblackened an with incandescent soot lamp. so The platediation. that has it A been absorbs thermometer practically andplate. all an the electric incident heater ra- are also builtWe wait into until a the constant temperature has beenlibrium established. Then has equi- been reached:energy is brought During to each the plate timeconduction by etc. interval, the Then radiation just we as block it as offcurrent loses the so much through that radiation heat the and same adjust temperature thetain is heater maintained; electrical this power, that requires a is cer- ameasured certain in product volt of current and voltage, Figure 15.4 By comparing this calibrateda but more not very sensitive radiometer, sensitivemopile, e.g. instrument we with a can then radiation calibrate thermocouple the or latter (Fig. ther- 15.3 Physical Radiation Detectors. Our eyes are by nois means the emitted only indicators by for glowing theradiation radiation objects: will which be All bodies which are struck by the radiation or the radiation fromeffect an directly arc with lamp, the we temperature can sensorsthe feel in hand this our is warming skin. especially sensitive. The palm of This heating effect of theradiant radiation power, offers that a is methodradiation. for the measuring quotient its The of energy/time principle transmitted is by explained the in Fig. m), 9 , with with the 10 . D 29 Introduction. Measuring the radiant intensity luminous intensity 1 nanometer 15 the unit candela (one ofseven the SI base units). These quantities will be discussed in detail in Chap. several additional, specialized quantities have been defined which take into account the perception of brightness by the human eye. For example, analogous to the unit watt/steradian, we have the C15.1. For a quantitative description of the optical radiation in the visible wavelength range of light from around 400 to 750 nm (1D nm 296 Part II 15.4 Indirect Measurements of Radiant Power 297

Figure 15.5 Schematic of a thermo- 1 cm couple (e.g. cubic SnTe-Constantan), into which a soot-covered Ag foil has Tellurium rod been inserted for measuring the radiant 0.4 mm diameter C15.2 power Sootcoated Ag foil C15.2. The thermocouple 0.05 mm thick in Fig. 15.5 is a simple ≈ 10 Ω) i radiometer. The active Constantan wire (R 0.03 mm diameter the voltmeter To (e.g. cubic SnTe-Constantan) junction is underneath the Ag foil, which serves as receiver for the radiation. The open 15.4 Indirect Measurements of Radiant circles indicate two other Part II junctions where the Cu leads Power to the voltmeter are attached; they must be kept at the In the case of radiometers which are based on the production of heat same constant temperature by the radiation (thermal radiometers), the incident radiant power (“reference temperature”) to is distributed over all the components of the absorbing object. The avoid unwanted additional observed increase in its temperature corresponds only to the average thermovoltages. A general energy increase of all the molecules in the detector. This limits the summary of the use and sensitivity of such radiometers. Much more sensitive radiometers design of thermocouples are based on allowing all the radiant energy to be absorbed by only can be seen for example a small component, namely only by some fraction of the electrons at www.msm.cam.ac.uk/ which are a component of the radiometer. The electrons which re- utc/thermocouple/pages/ ceive the energy can be conveniently measured in the form of an ThermocouplesOperatingPrinciples. electric current. This is the case for example with vacuum photo- html. Today, radiometers cells (Fig. 15.6, left) (The photoeffect or photoelectric effect was often use a thermopile, described in the 13th edition of POHL’s “Optik und Atomphysik”, a series of thermocouples Chap. 14,), with photodiodes (Fig. 15.6, right) (cf. 21st edition of connected together to POHL’s “Elektrizitätslehre”, Chap. 27), with ionization chambers increase sensitivity. An (Fig. 15.7), and with GEIGER-MÜLLER counters in their different example of a commercial forms (ibid., Chap. 20). In all of these devices, the measured electric instrument is shown at www. currents are proportional to the absorbed radiant power. They thus kippzonen.com/Product/ provide an indirect measurement of the radiant power. Unfortunately, 36/CA2-Laboratory- Thermopile#.V6xVelfuKrW.

Selenium layer Transparent Opaque metal electrode metal electrode

A A Light

Light Ammeter Ammeter

Alkali metal

Figure 15.6 A vacuum photocell (left) and a photodiode (right). Both are convenient to use as radiometers in demonstration experiments, but they are unfortunately very selective. That means that their indicated values are indeed proportional to the radiant power, but they must be separately calibrated for every type of radiation. photo- lting from intermittent V accelerated by a first-stage volt- accelerated in turn by a second d on the type of radiation being is an aluminum foil entrance window F R ). B 9 10 is an insulator. U R B F V, illumination is filtered out by the AC amplifier. 3 10 tubes, which are technically advanced vacuum photocells with A gas-filled ionization chamber for detecting X-rays, in the form U ( ). This has the added advantage that measurements can be carried out V points: 1. A very high sensitivitymultiplier and broad applicability are exhibiteda by built-in amplifier (electron multiplier): Theejected primary from the electrons which photocathode are byage light onto are a metal plateemitted (e.g. there, whose AgMg). number They is causeelectrons. a secondary large multiple electrons These of to secondary the be voltage number electrons of stage are primary onto anotherthrough a plate, number ejecting of stages tertiaryall (called electrons, amplification ‘dynodes’), factor. providing and a so very2. high forth In over- order to makeuses use a of “chopper” the to convenient irradiate aspectslight the of AC instrument amplification, with one lightwithout pulses requiring ( a darkenedthe background room: The constant current resu Technical details have no place in this book. Nevertheless, we mention two Optical Radiant Power of a cylindrical condenser, usedmeter in connection with a DC amplifier and a volt- for the radiation, and Figure 15.7 the proportionality constants depen measured. Their application thereforeknowledge requires much than more in physical measurement the instruments case (radiometers) appear of inthis a the book, thermoelement. illustrations we of cana Where generally more take radiation sensitive them measurement to iswill required, be be the provided thermocouples. necessary in information the When description of the experimental setup. Introduction. Measuring the 15 298 Part II Observations Optical Simplest The iue16.1 Figure us- experiment first our to out used carry (Fig. be we air can Now, room smoke ing sus- tobacco many haziness. necessary, contains its usually If increase it hazy; particles. less dust or pended more always is air Room casein suspen- and a fat milk, of very of particles contains drops microscopic few which water of a ink, clear sion with India or make powder, of by carbon example amount divided clouded small finely for are a can liquids adding We manner, by similar murky large a particles. as a In suspended to contains small referred dust. air usually or Hazy smog particles, liquid. smoke, suspended murky microscopic a of and air, number hazy clear and a clear between between difference and the knows being human Every from treat- experiences simplest our the begin with to fields, life. reasonable other daily in is as It optics, of laws. ment physical points ar- and starting other concepts the in provide for as to optics, have experiments In and science. observations empirical eas, an remains and Rays is Physics Light and Beams Light 16.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © a itdi o.1a n ftebscpoete fwv propagation wave of properties basic the of (Sect. one as 1 Vol. borders in straight-line listed with was beams in Travelling diaphragm. aperture an imtro h prue(Fig. aperture the of diameter he-ea osn slgtsuc.Tefotedo h osn has housing the opening of end circular front a The source. light as housing sheet-metal pnn angle opening ntefiue.I scle the thus called is light lines It The dashed straight, figure). by the room. bounded in is the (which into cone far a light within of opening propagates cone this shimmering white, from a extends see which can we side, the from Looking later) in drawn (the were air rays hazy dashed in beam light a of 12.6 ,poie httewvlnt ssalcmae othe to compared small is wavelength the that provided ), h iil trace visible The ! ti eemndb h opening the by determined is it ; 16.1 B hc evsa h xtaetr o h light. the for aperture exit the as serves which :W mlyacro-r api h usual the in lamp carbon-arc a employ We ): olsItouto oPhysics to Introduction Pohl’s ih beam light 16.2 ). hslgtba a large a has beam light This . Vdo16.1) (Video B , B https://doi.org/10.1007/978-3-319-50269-4_16 hc evsas serves which ω ω . Fig. (see cuvette water-filled a in visible beam light the make to used is (styrofan) persion dis- plastic a video, this In http://tiny.cc/5dggoy light” “Polarized ie 16.1: Video 24.4 16 ). 299

Part II of . ! ). There, the C16.1 become. In the visible trace 16.2 exist only on the ! parallel-bounded beam . It defines the direction appear practically parallel . In drawings, we represent Light rays illustrates the ”, Chap. 18), the more narrow a small fraction of it in all di- . beam axis light beams experimentally in a corre- 16.1 scatter B observed curved (arc discharges were discussed in the 21st collimated beam and will represent them using curved lines or rays. rays (e.g. chalk marks or pencil strokes) at the Elektrizitätslehre 16.1 can be boundaries of the beam C16.2 ’s “ OHL The propagation of mechanical waves as a beam with straight- beams boundary , or, for short, a sponding manner Later, we will demonstrate limiting case, the as viewed from the side. We then speak of a of light a light beam in one of two different1. ways: By two the light beam and the smaller its opening angle For demonstrations toorder a to large make the audience, tracecan of we avoid the this need problem; light instead beam very ofa sufficiently hazy hazy bright. trough, air, or we But air use we even in a moreWe cloudy conveniently, can a liquid obtain in flat the table latter withcovering by it a with painting matte a a finish. sheet flat of board white paper. with matte white or The experiment shown in Fig. Figure 16.2 rections, and some portionIsotropic scattering of from this small particles scatteredics is light familiar of from waves. reaches the We our mechan- a recall eyes. pond: a stick When water standing wavescircular up strike “secondary” in wave it, trains the the with propagate smooth stick inthe becomes surface all water the directions of surface over source (cf. of Vol. 1, Fig. 12.17). The further we movecarbon the arc) opening in away Fig. from the light source (the a light beam in aor hazy illuminated medium. by The dust the particles light which are struck edition of P line boundaries.through The a sketch wide opening shows (schematic, water after Fig. waves 12.12 before in Vol. 1). and after passing blackboard or on paper.mathematical representation. They are merely an aid to graphical and rays in the drawing clearly represent lines normal toOnly the light wavefronts. 2. By a single ray representing the sides of the beam. They define twice the opening angle, 2 of the light beam relative to some fixedWe direction. thus use the samefor techniques the to cones or represent beams light of beams mechanical waves as (cf. we Fig. did and . ). 27.11 The Simplest Optical Observations 16 Video 27.1: “Curved light beams” http://tiny.cc/wfggoy C16.2. See Sect. C16.1. This is only a schematic sketch. In the case of mechanical waves, e.g. water waves, the boundaries of the beam canonly be roughly discerned due to diffraction (see Vol. 1, Video 12.2: “Experiments with water waves” http://tiny.cc/tfgvjy 300 Part II 16.2 Light Sources of Small Diameters 301

L C F B S

Figure 16.3 The visible trace of a collimated light beam along a white Part II painted board S (B is a round aperture, F a red ﬁlter). To avoid having too great a distance from the light source with the associated problems, a lens C (called a condenser lens; cf. Sect. 18.3) with a focal length of around 7 cm (Sect. 16.7) is placed in front of the exit aperture of the light source.

The “dust” in commercial white paints consists of a very fine powder made from a transparent material. Clear rock salt when powdered appears white. Clear ice gives white snow in the form of small particles. If “light” or “dark” beer is spread thinly in the form of fine bubbles, it appears as a white foam “head”. White paper has a similar structure to white pigments; instead of a suspension of very finely powdered crystals, it consists of fine fibers which are tangled together and held by a layer of glue (cf. Sect. 26.10). C16.3. See Comment We can thus allow the light to shine at a glancing angle along a white C1.1. in Vol. 1. painted board. Then we see its trace with a nearly blinding brightness. For the demonstration of collimated light beams, we make use C16.4. Light sources which of the convenient trick shown in Fig. 16.3. With this arrangement, fulfill these conditions in an we can also readily demonstrate a “colored”1 light beam, for exam- excellent manner are avail- ple a beam of red light; we need only place a red filter in front of the able today in the form of lasers.(Theword“laser” light source, for example a darkroom filter. We will continue to work is an acronym and stands in the following with red-filtered light. for “Light Amplification by For the light which we commonly encounter in daily life, that is the Stimulated Emission of Radi- light of the sun, light from the sky, light from electric incandescent ation”. Described in the 13th lamps, candles, gas lamps,C16.3 or carbon-arc lights, we use the com- edition of POHL’s “Optik pact collective term natural light. The usual word “white” light is too und Atomphysik”, Chap. 14; see H.J. Eichler/J. Eichler, vaguely defined. “Lasers” (Springer Ver- lag, Berlin 2003)). Some of the experiments described 16.2 Light Sources of Small Diameters in this book can therefore also be demonstrated using a laser as light source (see To demonstrate many optical phenomena, we require light sources e.g. Video 27.1: “Curved with a small diameter and a strong luminous exitance (source bright- light beams”). In many ness). The choice is limited.C16.4 cases, the use of a laser however provides no particular Among them are the carbon tips (the “crater”) of small arc lamps (di- advantage, so that an aperture ameter 3 mm), or the small arcs in high-pressure mercury lamps or slit illuminated by an arc lamp and condenser lens has 1 “Colored light” or “red light” is in terms of language at the same level as a “high by no means lost its useful- note”. Both expressions are justifiable only because of their convenient brevity. ness as a light source. B falls I (16.2) (16.1) two partial index of re- . All the rays with the “axis 0 into ˛ into the material split and refracted air ˇ : : , C16.5 , is reflected towards the B ˛ n II , the “mirror image” of the . A thin red light beam 0 D L ; , these angles are shown for the ’s law) 0 16.4 ˛ , enters the glass block, changing const 16.4 D III NELL ˛ D (S law of reflection ˛ ˇ . Some numerical values are collected in . At the surface, it is B B sin sin .InFig. lens. One of many examples can be seen in . In general, the boundaries of the source in 2 N . One of them, beam 3 mm) : III 0 , we show a boundary surface between air and glass. condenser law of refraction of Reflection and Refraction . In comparing two materials, the one with the higher in- . and ). In acoustics, in contrast, we can easily make the apertures of radia- . The details of a proper illumination will be given later in of the material 16.4 16.1 16.9 16.3 18.12 , often written without the subscript, is called the B Even this diameter is still very large compared to the wavelength of visible light n of normal incidence” center axes of theThe beams; set the of angles boundary obeys rays the are left off for clarity. and for the transition of the light from the such lamps are notas sufficiently light sharp. source, we often Therefore,behind, employ instead a or circular of a aperture a slit illuminatedhind, lamp from with we place straight a edges. lensaperture: of For short a focal the length illumination between from the lamp be- and the its direction of propagation at the surface; it is object point. The other beam, drawn in the figure(the lie plane in of the thecase; same page). they plane, Three make the of the “plane three these of adjacent rays incidence” angles belong together in each (glass), the (Sect. tion sources (e.g. of pipeswaves. and whistles) smaller than the wavelength of the sound 2 16.3 The Fundamental Facts Making use of the experimentalby aids described recalling above, we school now physics begin Vol. 1 (Sect. and 12.7): the the law laws ofemploy reflection the treated and setup the illustrated in law in of some Fig. refraction. We detail in (diameter at a slant from the uppersurface left of through a the glass air onto block the planar, polished Table dex of refraction is referred to as theIn more “optically Fig. dense” material. Instead of this, we could employ a boundary surface between two fraction Fig. upper right. After the reflection,from the the rays as “virtual” drawn intersection seem point to originate beams II Fig. NELL S ,the matter c for = n 12.8 vacuum c D ILLEBRORD ). n 25 The Simplest Optical Observations 16 (A detailed description for light follows later in Chap. C16.5. W (1580–1626), a Dutch mathematician. In Sect. index of refraction electromagnetic waves was introduced; it thus holds for light, as 302 Part II 16.3 The Fundamental Facts of Reflection and Refraction 303

Figure 16.4 Demonstration of re- L N flection and refraction of a light beam Air at the planar surface of a glass block B* α α' II (flint glass). The block is standing in I front of a matte white screen, and its A back face is also ground to a matte B finish. (Red filter light, L; the light source has a small diameter, and B is β its aperture diaphragm) III L' Glass N Part II Figure 16.5 Reflection and refraction at N the planar boundary surface between two materials, water (A) and flint glass (B), IIIα with different indices of refraction, n A Water A and nB (red filter light; only the center axes of the light beams are drawn) Flint glass B β III

arbitrary transparent materials A and B (with the indices of refraction nA and nB), for example, as in Fig. 16.5, between water and flint glass. The law of reflection is unchanged, while in the case of refraction, we find for the transition from material A into material B:

sin ˛ nB D nA!B D ; (16.3) sin ˇ nA

1:60 ! D D : e.g. nwater ﬂint glass 1:33 1 20; cf. Table 16.1.

A comparison of Eqns. (16.2)and(16.3) yields nA D nair D 1. We have thus deﬁned the index of refraction of a material (as in general and expedient usage) in terms of the transition of the light from room air into that material. For the transition from vacuum ! material, the

Table 16.1 The indices of refraction of some materials For the transition of red filter light Index of refrac- ( 650 nm at 20 °C), from air into tion n D Fluorspar 1.43 Quartz glass 1.46 Light crown glass (lead-free silicate glass) 1.51 Rock salt crystal 1.54 Light flint glass (silicate glass with 25 wt.-% PbO) 1.60 Heavy flint glass (silicate glass with 40 wt.-% PbO) 1.74 Diamond 2.40 (!) Water 1.33 Carbon disulfide 1.62 Methyl iodide (iodomethane) 1.74 , α A B A B ·sin III (16.5) (16.4) , is con- = b A s wavefronts 2 β b ,howeverhere 0003. : 1 α 1 16.5 D : β B air A ·sin ! ! A n : B I A = b optical path length B B n n s Water Flint glass B* D n vacuum n D B B s ˛ ˇ C16.6 D sin sin or A n D B A . The rays drawn continue as normals to A B A n n s s s D 16.6 ’s principle. B A waves, we observed reflection and refraction in the ERMAT describes the same experiment as Fig. The definition of the op- Reflection and refrac- 16.7 mechanical form sketched in Fig. the wavefronts after reflection. Quantitatively, we find: for the special case oflight a rays collimated at light the edges beam. of Inthe the addition intersection beam, to two lines cross-sections the 1 are and two drawn 2. in as In a wave picture, these are e.g. they represent crests of the waves. From this sketch, we can read off the following results: For stant. This is F In words: Between twoof cross-sections path and of index a of refraction, light called beam, the the product We will have occasion to apply this equationFigure later to light, as well. Figure 16.6 tical path of a collimated lightThe beam. reflected light beam isto not keep shown, the drawing simple and clear. tion of mechanical waves (e.g.waves) water at the boundary between two materials with different wave velocities (above higher than below; thus shorter wavelength below) (schematic) Figure 16.7 or corresponding indices of refraction wouldhigher. be Thus, taking about this 0.3 transition thousandths asair the would basis have of the the index definition, of room refraction , 0 , - P s .It ’s t ER F B c or . matter is thus A B D c ERMAT s s s A B B s is a constant of n n n , so that the / / IERRE DE or lity). This quan- vac vac vac t c , a light beam to another point c c t = 1970)), Sect. 15. = = optical path length P ; 1 1 A . . Principles of Op- c s The Simplest Optical Observations , 1601–1665, French d (Pergamon Press, 4th minimum D D D n 0 t t 16 P A MAT P follows from this, according to Comment C16.5, that it is found that thepath optical length, in general the path integral tity, the proportiona will repeatedly play a rolethe in following sections. We mention an important application here: For the path which is chosen by awave light in order to go froma point The quantity principle (P is a light covers the distance most rapidly along this path (this holds also in inhomogeneous media). This is F (available online – cf. Com- ment C25.11). tics edition ( mathematician). See for example Max Born and Emil Wolf, Z (where 1 C16.6. In equal time intervals passes through a distance s with the velocity a measure of the timequired re- by the light inpass order through to the distance 304 Part II 16.4 The Law of Reflection as a Limiting Case. Scattered Light 305

C L

Figure 16.8 The reflection cone formed by reflection of light from the sur- Part II face of a cylindrical glass rod (C is a condenser lens; its mount carries an iris diaphragm about 8 mm in diameter. L is a lens of focal length f D 20 cm) (Video 16.2) Video 16.2: “Reflection cones” http://tiny.cc/qdggoy To illustrate the law of reflection (16.1), we mention a practically- In the video, a reflecting important but little-known special case: In Fig. 16.8, a thin light beam stainless steel tube is held strikes the smooth surface of a cylindrical rod at a grazing angle. The by hand in a light beam. De- light reflected from the surface forms a cone. The axis of the cone pending on the angle between coincides with the axis of the rod; thus, a screen perpendicular to the tube’s axis and the light the rod shows a circular line at the intersection with the cone of the beam, reflection cones are reflected light. The direction of the incident light beam lies within formed, whose circular sec- the surface of the cone. The more steeply the incident beam strikes tions can be seen on the wall the rod, the larger is the opening angle of the cone. of the lecture room. They change depending on the po- Knowledge of this type of reflection is important for example in the inves- sition of the tube. At an angle tigation of rod-shaped formations using dark-field illumination, e.g. with of 90ı, the section becomes an optical microscope (Sect. 18.12) or an electron microscope. It is also a linear band of light. important for diffraction of X-rays by crystal lattices (Sect. 21.14)andfor the explanation of atmospheric halo phenomena, in which a ring touches the image of a star or planet on the outside (for references, see Com- ment C21.1). A beautiful collection of haloes can also be seen at https:// en.m.wikipedia.org/wiki/Halo_(optical_phenomenon) .

16.4 The Law of Reﬂection as a Limiting Case. Scattered Light

According to the illustration shown in Fig. 16.4, the reflected light should be restricted to beam II, that is to a three-dimensional cone with its apex at L0. This description however holds only for an ideal limiting case: In reality, we can see the point at which the light beam I intersects the surface from any arbitrary direction. Thus, some portion of the incident light must be “scattered” in a diffuse manner in all directions and thus reaches our eyes. This scattered light is a cause of annoyance to physicists and engineers, as a disturbing source of errors; but it is a blessing for fathers: Without the scattered light, their children would continually run into plate glass doors and windows. . ). oc- )to B graz- 16.6 (16.7) (16.6) 16.10 and For light, we 16.9 : C16.9 Total reflection B cannot become larger ! 1 ˇ A B n . Experimentally, we find . The scattered light van- m. Light scattering thus I ! C16.8 1 A D n terial. The diameter of dust B A n n D , there is no longer a refracted for total reflection. The critical by innumerable small, randomly- ˛ B A do not emit light become visible to T D propagates at a larger angle to the n n ˛ A C16.7 burner over it. ; only the optical path is reversed in the III D ! B T n ˛ diffuse reflection 16.9 reflection ), this time, exceptionally, from right to left. D A UNSEN sin ˛ ˇ and than the incident beam sin sin . Quantitatively, the angle N 16.4 16.10 corresponds in the optically less dense medium to a . All of the incident light is reflected: , that is, its sine cannot become larger than 1 in Eq. ( T ı III ˛ beam, i.e. to a beam which propagates parallel to the interface The dust particles which fallwaving the onto flame a of mercury a surfaceMica B can sheets be must be burned cleaved off both by on their upper and their lower sides. interface normal the less dense medium ( The corresponding angles are again drawn inof only for the the central light axes beams.figures: We can reach two1. conclusions The based refracted on light these beam determines the “critical angle” The light beam passes from the more optically dense material ( demonstrate it with the arrangement sketched in Figs. angle The axes of the incident andpatterns the in refracted Figs. light beams show the same Thus, the formula us only through scattered light. ing (compare Vol. 1, Fig. 12.25). Total reflection is also treated in detail in Vol. 1. particles is seldom less than around 10 16.5 Total Reflection For all objects which themselves Scattered light issmooth produced surface, in forishing example the and due main inhomogeneities to in by dust the imperfections particles, ma in imperfect the pol- occurs primarily through oriented mirror surfaces.expediently Therefore, referred to this as type of light scattering is ishes almost completely when the reflectingproduced surface is without nearly mechanical perfect, treatment;prepared an surface example of is pure mercury, themica or freshly- crystals. the recently-cleaved surfaces of two figures. 2. At large angles of incidence than 90 curs in Fig. beam . Physics , July 2004, p. 82. The Simplest Optical Observations Video 12.2, “Experi- 16 “. . . all objects which them- C16.9. See Vol. 1, Sect.and 12.9 ments with water waves” http://tiny.cc/tfgvjy (at 5:30 minutes). C16.8. Mercury surfaces are used in particular toduce pro- concave parabolic mirrors. The shape is obtained through rotation (Vol. 1, Fig. 9.3). See e.g. theby note R.F. Wuerker in Today C16.7. this important statement should be particularly emphasized here. The decisive role of scattered light for “seeing” objects is often not noticed by the novice. selves do not emit light become visible to us only through scattered light” 306 Part II 16.5 Total Reflection 307

Figure 16.9 Reﬂection and refrac- L' tion of a light beam in passing from an optically more dense to a less N dense medium (red-ﬁlter light). The III angle of incidence is again denoted β Air by ˛. A

B I II Glass α α' Part II

Figure 16.10 The continuation of L' Fig. 16.9. When the angle of inci- N dence ˛ is increased, the refracted Air light beam is no longer present: Total reﬂection occurs. A

B II I C16.10. Light guides made α' α of glass fibers (“fiber optics”) work on the same principle; Glass L they are used extensively today for optical-digital data transport, for example in telecommunications, and Total reflection is a favorite object for demonstration experiments; for endoscopic examina- there are many setups for showing it. The best-known is a gimmick, tions (technical details can using a jet of water as a light guide (a “luminous fountain”).C16.10 be found e.g. in H. Kogelnik, In nature, total reflection can frequently be observed with air bubbles “Optical Communications”, under water; think of the bright, silvery shining bubbles on the sides in the Encyclopedia of Ap- of water beetles. plied Physics, Vol. 12, p. 119 The critical angle for total reflection can be rather precisely deter- (VCH Publishers, 1995). mined by a variety of methods. This fact is used in the construction of refractometers.C16.11 These are apparatus for the rapid and con- C16.11. Refractometers are venient measurement of the index of refraction (usually of a liquid commercially available in sample); they are popular with chemists and medical researchers. An many forms. Along with the example is described by Fig. 16.11. measurement of indices of refraction, they can be used Total reflection can occur at the interface between two media with for example to indicate the a very small difference in their indices of refraction. The light beam alcohol or sugar content of must be incident at a grazing angle, i.e. ˛ must be nearly 90ı.In solutions directly as a digital this way, we have seen how sound waves can be reflected from the output. interface between warm and cool air (Vol. 1, Fig. 12.56). The corresponding effect is also seen with light beams (Fig. 16.12): A collimated light beam enters at a grazing angle from below into a box which is electrically heated. The inside surfaces of the box are blackened. When the heater is turned on, the box fills with hot air; some of it expands over the rim of the box, while the rest forms a rather flat interface (diffusion boundary as a substitute for a surface; cf. Vol 1, ). in F 16.7 ca. 8 cm with a red filter is the crater of an arc K is then computed from K ca. 5 m A n ). The totally-reflecting boundary is filled. At left, parallel to the flat A n mirage or 1 m Heater wires KF fata morgana A refractometer which is suitable for demonstration exper- carries a glued-on rectangular glass cuvette into which the liq- ). B n ), or else the angular scale is calibrated directly in units of the index can thus be rather precisely read off, and 22.6 T 16.7 Total reflection of a collimated light beam at the interface between hot ˛ layer usually appears as if it were the surface of a pool of water. This total reflection byThe a hot warm ground layerthe of in air air a just is desert above oftenat region it. observed a or grazing in angle, a nature. A sometimesnear hot traveller the containing sees horizon asphalt the ( a mirror road image mirror surface of image some warms of object the bright sky ca. 0.5 m This is indicated in the figure by two dashed convergent rays. of refraction. The round glass plate acts as a cylindrical lens (Sect. front of it. The lightat which passes a through grazing the angle liquidright and appears vertex enters on the as glass the seen plate protractor bygle the scale observer as is a quite thin sharply red defined. beam The whose critical an- Eq. ( uid of unknown index of refraction face of the glass plate and about 30 cm away, is a lamp iments. Arefraction thick, semicircular glass plate with a known, large index of In physics, totalcant role reflection in at spectral(Sect. apparatus grazing (“monochromators”) for incidence X-ray plays light a signifi- Sect. 9.9). This interfaceplanar between mirror. hot Strong andavoided. cool drafts air perturb acts the like experiment a and fairly should be Figure 16.11 Slits, about 1.5 mm wide C K lamp (light source point)) and cool air (the beam is about 2 cm thick at its right-hand end; Figure 16.12 The Simplest Optical Observations 16 308 Part II 16.6 Prisms 309 16.6 Prisms

Prisms represent important applications of the law of refraction in optics and metrology. In Fig. 16.13, we see the two planar faces of a prism which make an angle with each other, the “angle of refraction”. Perpendicular to the two faces, the plane of the page forms the principal plane. Within the principal plane, a collimated beam of light passes through the prism; only its central ray is drawn in the figure. Refraction at the two faces of the prism changes the direction of the beam by the angle of deflection ı. Quantitatively, we Part II can apply the equation sin ˛ D n sin ˇ (16.2) after some rearrangements (for the relation between ı, ˛, ˇ,and , see Exercise 16.1) Â Ã ı C Á tan ˛ 2 tan ˇ D tan Â Ã : (16.8) 2 2 ı C tan 2 The minimum deflection can be determined experimentally when the collimated light beam passes symmetrically through the prism, as in Fig. 16.13, right. Then we find 1 1 ˇ D and ˛ D .ı C / 2 2

(here, ˛ is the angle of incidence to the normal, ˇ is the angle of the refracted beam, is the angle of refraction (apex angle) of the prism, and ı is the angle of deﬂection of the light beam).

Then we obtain from Eq. (16.2)

sin 1 .ı C / n D 2 (16.9) sin. =2/

A A

γγ δ NNN δ α β α γ β= 2

Figure 16.13 The deﬂection of a monochromatic beam (the ray drawn indicates the central axis of the light beam) by a prism with a non-symmetric optical path (left) and with a symmetric path (right). The dashed lines marked ‘N’ are surface normals to the prism faces. The line perpendicular to the principal plane (the plane of the paper) at the apex point A is called the refracting edge of the prism. S

or (16.10) (16.11) and can S )bytheir Video 12.2: ). One thus which deﬁnes 16.9 L' , we can replace 16.14

in the object with ) two-dimensional )and( ) by placing a lens in , the beam axis and its aperture of the lens. The center . Analogously, in optics, point 16.8 0 S L : 16.15 16.15 / ” 2 / ; ˛ 1 . , then we must not employ three- . = sin n http://tiny.cc/tfgvjy sin . point D D ”). In Fig. . . In this case, the axis of the light beam ı image point n ˛ is proportional to the angle of refraction S ı or principal ray ı optical axis object point A lens converts a divergent beam of mechanical waves into The Focal Length (the “ L L . We measure either we can allow a divergent light beam to fall on an aperture These two equations can be usedn to determine the index of refraction a convergent beam (schematic, as in Fig. 12.20 in Vol. 1; see also “Experiments with water waves” convert it into a convergent beam (Fig. can be made convergent by means of a lens (Fig. Figure 16.14 A divergent beam of water waves which is defined by an opening 16.7 Lenses and Concave Mirrors. and the aperture. In this way,source we can form an “image” of a pointlike light obtains a narrow “waist” indenoted the succinctly beam of as waves an within “ a short region, In the limiting case of small angles of refraction of the prism. i.e. the deflection angle arguments (angles). Thenfor we the find symmetric for optical paths both a the deflection non-symmetric angle and of the sine and tangent functions in Eqns. ( has a special name, the point of the aperture thus liesof here on the the (dot-dashed) lens, symmetry the axis two limiting boundary rays are drawn. The The quantitative treatment ofIf lenses we begins with wish cylindrical to lenses. form an image of each planar light beams. This means that we must use an aperture which dimensional light beams, but rather practically a cylindrical lens as an image the beam is at the same time the mount The Simplest Optical Observations 16 310 Part II 16.7 Lenses and Concave Mirrors. The Focal Length 311

S L'

Figure 16.15 A lens converts a divergent beam of light which is deﬁned by the aperture (lens mount) S into a convergent beam (L0 is a real image point, schematic)C16.12 C16.12. To distinguish it from virtual image points, B which will be introduced later, we refer here to a real

L' image point. Part II

Figure 16.16 Imaging of a distant object point by a cylindrical lens as an image streak L0. One must limit the width B of the incident light beam using a slit in order to convert the “image streak” into an “image point”. takes the form of a narrow slit which is perpendicular to the cylinder axis of the lens. Continuing the experiment, we open the aperture until it reaches the width B marked in Fig. 16.16 with a double arrow. Then a cylindrical lens produces for each object point L an image streak L0 rather than an image point. Only with two crossed cylindrical lenses with the same radius of curvature can we obtain the effect of a spherical lens; i.e. for each object point L they produce an image point L0 (Fig. 16.17a) and thus yield good (sharply focussed, well resolved) images. Two crossed cylindrical lenses with different radii of curvature produce two perpendicular image streaks L0 and L00 at different spacings, instead of an image point for each object point (Fig. 16.17b; this is called astigmatism, cf. Sect. 18.5). Starting from the cylindrical lens, we can relate the action of a lens to the action of prisms. We restrict ourselves to a nearly ﬂat cylindrical lens, i.e. with a very limited curvature (“thin lens ”, Fig. 16.18)and a light beam which is very narrow in both its dimensions and close to

b B C L' L''

Figure 16.17 Imaging of a distant object point by two crossed cylindrical lenses (a) with the same radius of curvature: One obtains a single image point L0; (b) with different radii of curvature: One obtains two separate image streaks L0 and L00 (cf. Sect. 18.5, Astigmatism). With a slit, one can reduce either the width B or the width C of the incident light beam. In the first case (width B small), we convert the image streak L0, and in the second case (width C small), we convert the image streak L00 into an “image point”. . above (16.12) (16.13) h 16.20 (16.14) of the light 0 focal length ! L' and /: 2 which are valid only ! ˇ ; aces, in order to make 0 1 f 2 ,wehave .˛ ı D ω' : ) is illustrated by Fig. (left) and the object-side focal C 0 Ã 0 / 1 f ) is called the “lens-maker’s f 1 2 ). (Unfortunately, in the draw- 1 r ˇ 16.13 D lens formulas C 1 b 1 16.12 1 F 1 .˛ r , and only the central axis ray of each 1 C )and( 2 δ Â D δ 1 a / 2 1 is termed the image-side ' ) is the “imaging formula”. 1 16.12 thin light beams close to the optical axis 16.18 h paraxial rays 0 C f 1 n C16.13 ' . . 2 16.13 r D ı (called on the object. They are produced by subdividing a wide and ab ω L’ F’ 1 L The relation between the action of a lens and the action of r , left). Equation ( )as L 16.19 16.12 f’ f There, the lens isspace thick for and the has largetriangle strongly number with the curved of outer f necessary (supplementary) symbols. angle For the small shaded The derivation of Eqns. ( In these equations, beam for clarity!) This lightbeams beam as is shown then in divided Fig. up into small sub- sub-beam is considered. At thea same time, series we of divide prisms the which lens up are into one above the other at altitudes the optical axis. We thus arrive at the well-known (Fig. prisms. The radiiEq. ( of curvature of the two faces of the lens are denoted in ings, we have to exaggerate the opening angles the optical axis Figure 16.18 in the limiting(“paraxial case rays”): of equation”, and Eq. ( The definition of the image-side focal length path (right). The former is demonstrated using a series of collimated light beams. They all f ’s principle . 0 L optical to collimated light beam using a lattice aperture. length originate at the same distant point Figure 16.19 ERMAT L ,the ı The Simplest Optical Observations 16 lengths are the same. Thus, the light can use all ofpaths the sketched in the figurepass to from (Comment C16.6): Although the geometric path length depends on the angle offlection de- C16.13. This is a goodample ex- of F 312 Part II 16.7 Lenses and Concave Mirrors. The Focal Length 313

δ = φ1 + φ2 α δ α1 2 β1 β2 r2 r1 φ1 φ2 χ χ Object point2 1 Image point

Figure 16.20 The derivation of Eqns. (16.12)and(16.13)

Then, according to the law of refraction, Part II sin ˛ sin ˛ 1 D 2 D n (16.2) sin ˇ1 sin ˇ2 or, for small angles, approximately

˛1 D nˇ1 and ˛2 D nˇ2 : (16.15)

We then obtain from Eq. (16.14)

ı D '1 C '2 D .n 1/.ˇ1 C ˇ2/: (16.16)

Furthermore, the large triangle with angles 1 and 2 and the small triangle with angles ˇ1 and ˇ2 have the same apex angle; therefore, ˇ1 C ˇ2 D 1 C 2 and Eq. (16.16) takes on the form:

ı D '1 C '2 D .n 1/.1 C 2/: (16.17)

Note that the right side of Eq (16.17) is simply an application of the prism formula, Eq. (16.11). Now, introducing the altitude h of the common apex in Fig. 16.20, we ﬁnd tan '1 D h=a and tan '2 D h=b, and also sin 1 D h=r1 and sin 2 D h=r2. With the small-angle approximation (tan ' ', sin ), we then rewrite Eq. (16.17): Â Ã h h h h ı D C D .n 1/ C I (16.18) a b r1 r2 or, dropping the leftmost equation and cancelling the common factor h, Â Ã 1 1 1 1 .n 1/ C D C : (16.19) r1 r2 a b Now we consider the case that the object is very distant, i.e. a ! 1; 1=a ! 0. Then the image will be formed in the focal plane at a distance b D f 0 from the lens plane (cf. Fig. 16.19,left),sothat1=b D 1=f 0. Substituting these expressions into Eq. (16.19) leads immediately to Eq. (16.12), and comparing again to (16.19)gives(16.13).

The distances a and b and the focal length f 0 are measured tenta- tively from the central plane of the lens (more details will be given in Sect. 18.2). The set of all the image points of all the distant object points make up the image-side focal plane. Its intersection with the optical axis deﬁnes the image-side focal point F0. In a corresponding manner, we deﬁne the object-side focal plane, the object-side focal point F, and the focal length f ;cf.Fig.16.19, right. The light rays which originate from the point L on the object-side focal plane (and are divergent on that side of the lens) form a collimated y' 2 0 Image ! ! (16.20) (16.21) (16.22) of the ). From tan tan C16.15 18.25 principal rays a b thus has a focal 1 '; optical power 3m

2tan D 1 diopter (analogously to ω' b D ' 1 (e.g. as in Fig. B 2 , this is the lens mount) lies on the Image distance Object distance 3 = /tan 3 diopter the image-side opening angle). tan 2 ω' b 0 2 D D ! 16.21 b 0 f y C16.14 y = D 0 y for the uppermost and lowest points of an 2 Image distance is the name of the axis of the light beam in the case . The unit is 1 m f D = 0 1 y φ' 16.21 B 2 Object size 2 Image size 2 D P 33 m. When several lenses are placed one behind the other, D : 0 principal ray are called the object-side and the image-side opening angles. 0 D ! f 1 Hz). A lens of power 1 , we can read off the frequently-used relations Imagesize2 D and Lens diameter Lens diameter

1 is the object-side opening angle, is the angle of incidence between the principal ray and the optical axis). ! φ ω 16.21 2 ! ' Image scale lens, i.e. power 1s ( ( Practitioners call the reciprocal of the focal length the For angular units, see the footnote at the end of Chap. 17. their powers add (approximately). length of We repeat: The B 2 or, for small angles, Fig. that the midpoint of the beam (in Fig. Furthermore, we have: 3 Formation of the imageimaging of of an each extended of object itsis can individual illustrated be points in traced by Fig. back a to single light beam. This beam after passing through theical lens. waves, For several the wave comparison crests tolenses mechan- are in air drawn (or in in asobject-side general and transverse with the lines. image-side the focal same For lengths material are on the both same. sides), the symmetry axis of the lens (Fig. 16.15). object. For manywhich purposes, are it shown suffices here to as sketch heavy the lines The imaging of an extended object by light beams originating from its indi- ω = /tan 2 a are the inclination angles of the principal rays, which are equal here (see also Com- 0 ' http:// and is given and y d 2 vidual object points. ' Figure 16.21 Object ment C18.1). 18.11 . It can also lenses ( , thick 18.10 for optical instruments. The Simplest Optical Observations ). http://hyperphysics.phy- 16 18.13 C16.15. The image scale or magnification as defined here should not be confused with the increase in the viewing angle introduced in Sects. C16.14. The exact expression for the power of twolenses (thin) at a spacing be used to calculate the power of by the ‘Gullstrand formula’; see astr.gsu.edu/hbase/geoopt/ Gullstrand.html hyperphysics.phy-astr.gsu. edu/hbase/geoopt/gullcal. html 314 Part II 16.7 Lenses and Concave Mirrors. The Focal Length 315

Figure 16.22 The object point lies inside the object- To the virtual side focal plane. The lens image point L1 only reduces the divergence of the beam. f

Figure 16.23 The action of a concave lens (F0 is the virtual image-side focal point)

F’ Part II

An example of Eq. (16.22): The disk of the sun has an angular diameter of 2' D 320 (arc minute), or 0.53°. Its image is at a distance of b D f behind the lens, that is 2y0 D tan 320 f D 9:3 103 f . A lens with a focal length of 1 m produces an image of the sun with a diameter of 2y0 D 9:3 mm.

A light beam from an object point Linsidethe object-side focal plane (Fig. 16.22) is not made convergent by the lens (i.e. it is not ‘focussed’), but rather its divergence is only reduced. The dashed backwards extrapolations of the two boundary rays drawn lead to the virtual image point Ll. For the comparison with mechanical waves, a number of wave crests are also indicated in Fig. 16.22. Concave lenses exhibit no essentially new properties.C16.16 They in- C16.16. Concave lenses, or crease the divergence of the light beam. Figure 16.23 illustrates this more generally, lenses which for the case of a collimated beam of light which is incident from the are thinner at their centers left. It also provides the deﬁnition of the image-side (virtual) focal than at their outer rims, are 0 point F . Equations (16.12)and(16.13) remain valid with a proper also termed diverging lenses. choice of the signs. In contrast, convex lenses, or more generally, lenses which Concave mirrors are mainly important for physics and astronomical are thicker at their centers applications in a particular mode of employment: The object or im- than at their outer rims, are age points are in the focal plane, near the symmetry axis (optical axis) called converging lenses. of the mirror, and the opening angle of the light beam is only moderately large. The action of the concave mirror can then be found from simple geometrical considerations employing the law of reﬂection. The focal length of the concave mirror is equal to half its radius of curvature R (Fig. 16.24).

Figure 16.24 The action of a concave mirror (Exercise 16.2) F Z R . G 16.25 .There,it G the apparatus , nor should they α G α α in front of ) light beams. In experiments G G . A collimated light beam is incident parallel 16.25 ,wesetalens ). It converts every collimated light beam into 16.27 16.26 C16.17 Incomplete The unwanted receives nearly collimated light beams. Rays by Image Formation G G separation of two parallel light beams after passing through some sort of apparatus overlap is eliminated after concentrating the two beams into one image point each The light is incidentplane is on shifted far the to lens thebeams right, as which usually converge several a to meters. divergent the Thenapparatus the beam. image light points are The very image slender, and the a convergent beam. The observation iswhere carried the out “waist” in of the the image plane, beams isFor narrowest. demonstration experiments, ancient. approximation As is in always Fig. suffi- Figure 16.25 Figure 16.26 The incomplete separationing of a the lens beams (Fig. can be eliminated by us- 16.8 The Separation of Parallel Light Many optical phenomena take their simplestserved form using when they collimated are (i.e. ob- of this type, onelight is beam often into dealing two with orthe more a scheme beams. splitting shown In of in the a Fig. simplest single case, parallel we observe be broadened. However,with it real is light impossible beams. toin All fulfill fact so-called these somewhat collimated divergent. conditions or Among parallelmention the beams here various are reasons only for one,light this, sources. namely we the finite diameter of all available further to the right. Both of these suggestionsbeam. presuppose a The strictly beams parallel musttheir form not cross-sections of become the nor unfocussed, at neither a on large limiting distance from is separated into two parallelcomplete, the light beams beams. still strongly But overlap. the separationHow is can not we obtain aFrom sufficiently the effective geometrical separation appearance, of wewe would two be beams? inclined must to say: makesmall; First, the and cross-sectional second, we areas shift of the the plane of collimated observation beams in Fig. from the left and passes through some sort of apparatus . 19.6 oducing The Simplest Optical Observations 16 truly parallel, collimated light beams, see also Sect. C16.17. For a discussion of the feasibility of pr 316 Part II 16.9 Description of Light Propagation by Travelling Waves. Diffraction 317

Figure 16.27 A simpliﬁcation of the arrangement sketched in Fig. 16.26, which is adequate for demonstration experiments. For comparison with beams of mechanical waves, in Figs. 16.25–16.27 a number of wave crests are indicated by transverse lines. Part II 16.9 Description of Light Propagation by Travelling Waves. Diffraction

The propagation of waves can be transversely bounded by an obstacle, e.g. the blades of a slit. This transverse bounding of a wave can be represented by straight lines or rays, but only to a more-or-less good approximation. For this approximation to be valid, two conditions must be fulfilled: The geometric dimensions of the obstacle, e.g. the width B of the slit in Fig. 16.2, must be large compared to the wavelength; and secondly, the point of observation should not be too far from the obstacle. In reality, the geometrically-constructed boundaries of the beam are always exceeded; the waves “run over” their boundaries. This behavior of waves is absurdly described in the passive form; we say: The waves are diffracted. Diffraction is in- separably intertwined with every attempt to limit the boundaries of a beam.C16.18 Figure 16.28 shows a model experiment and reminds C16.18. Diffraction was de- us briefly of the phenomenon of diffraction by a narrow slit. Accord- scribedindetailinVol.1 ing to Vol. 1 (Sect. 12.13), the angular spacing of the first diffraction (Chap. 12). Here, we are con- minimum from the optical axis is given by the equation cerned with showing that this phenomenon also occurs with light waves. A more exhaus- sin ˛1 D ; (16.23) B tive treatment will follow in Chap. 21. and for the angular spacing of the first sub-maximum, we found

3 sin ˛0 D : (16.24) 1 2 B

By measuring these angles and the width B of the slit, we can obtain a rather precise determination of the wavelength . These facts as we already know them from Vol. 1 are observed as well for the propagation of light waves. Thus, light waves cannot be “cut out” into an arbitrarily narrow beam by the two blades of a slit. Light also exceeds the geometrically-constructed boundaries of a beam; it is “diffracted”. In the region of diffraction, we can observe a periodic distribution of the radiation intensity, with maxima and minima. For a demonstration, we use the setup sketched in Fig. 16.29, top. Compare it with the model experiment illustrated by Fig. 16.28.Note is intended C16.19 II B 0 . Slit amplitude of a light (center). 16.29 ; it provides a quantitative (i.e. the quotient of radiant ntity which is proportional 11 2cm). : 16.29 16.29 1 22 irradiance ) and the dimensions given in the figure cap- 16.23 A model (see Comment C19.4). We place a narrow slit as aperture the light beam to a narrow width, and it should, according is in the first approximation a qua 16.29 limit experiment on diffraction by a narrow slit (compare Vol. 1, Sect. 12.12) Figure 16.28 also the scale shown in the legend of Fig. to to the geometrical construction, give aa bright width spot of on the around screenthat 2 with mm. is shown as Instead, a we photograph find in Fig. the pattern on the screen power/irradiated area) in our first experiment onFig. diffraction, shown in With light, sharply-bounded beams and their representation byof means straight-line rays or lines onproximation. the However, this blackboard are approximation also is merely oftenin an particularly optics ap- good because of the small wavelengths ofIn the order light to waves. obtainmeasure a the quantitative distribution of description the of our observations, we in front of our radiometer,wide so of that its we detector usetion area. only of a the strip Then screen about in wetransverse 0.5 the to mm put light the the beam optical radiometer and axis.radiometer is at slowly noted At and slide the then each it plotted posi- position, as inthe a the curve a graph. shown deflection direction In at this of the way, we bottom the obtain of Fig. complement to the photograph of thepart diffraction of pattern the in figure. the center Making use of Eq. ( tion, we determine the wavelength650 of nm. our It red filter isthe light around sound to 20 waves be and 000 water about timesfor surface waves demonstration smaller that experiments than ( we the used in wavelengths of mechanics to the square rootsatisfy of our the need deflection fora of concreteness quantitative a treatment and of radiometer. clarity, numerous optical but This phenomena. it may is not sufficient for The irradiance plotted ontransported the by ordinate the is light proportionalall waves, types to and of the this, radiation to in power theSect. turn, square 12.24). is of the Therefore, proportional wave we for amplitudewave can (see Vol. state 1, that: The The Simplest Optical Observations 16 C16.19. See also Com- ment C12.5. 318 Part II 16.9 Description of Light Propagation by Travelling Waves. Diffraction 319

C I Illuminated slit Width 2y as a line-shaped light source a

II Diffraction Width B

slit Part II

b α1'

α1 Screen

210 1 2

210 1 2 0 Irradiance

2211 20' 10' 010' 20' Angular spacing from the midline of the slit Video 16.3: “Diffraction and coher- Figure 16.29 Bounding of a beam of light (red ﬁlter light) by a slit. Top ence” part: The experimental setup. The angles between the dashed rays are greatly http://tiny.cc/xdggoy exaggerated. Center: A short section of the diffraction pattern as observed on Beginning at 9:30 min., the the screen ( a photographic negative shown actual size for B D 0:3 mm, b D diffraction pattern of a sin- : D D : 3 8m,a 1m,2y 0 2 mm). Bottom: The distribution of the irradiance in gle slit is demonstrated. Both the diffraction pattern of a slit as measured with a photodiode (i.e. measured red and blue ﬁlter light beams as the quotient of radiant power/area, e.g. in W/m2). This is the diffraction are used. The experimen- pattern of a slit (B D 0:31 mm, b D 1m, a D 0:75 m, 2y D 0:26 mm; the width of the radiometer aperture used here (Fig. 15.6, right) was 0.55 mm) tal setup is explained at the (Video 16.3). beginning of the video. S of can b after g Screen , plane- e , d onwards to , continuous Spectrum shows only a c ca. 25 cm wide , the spectrum is is a narrow slit b 16.30

4 m .From C 16.31

ca. is compensated and the 2 l 2 l 20 cm c to be incident on the

to project the exit position ~15 cm B f )oflens b 2 l 18.8 Flint glassFlint blue red through a condenser lens K , however with two modiﬁcations. First, we , the dashes do not refer as usual to rays, but rather = 1.62 2 g l β red ad N n 16.4 = 69° α e B

ca. . There, the spectrum is about 8 mm wide owing to the longer g Wavelengths. Dispersion Production of a spectrum by refraction in a plane-parallel glass 20 cm

1 mm) from the aperture Slits . We can make this spectrum visible to a large audience; to glass block. Second, we observe the refracted beam ca. 0.3 < mm wide , it can be projected onto the screen where it has a width of around 2 C l K the right and out to lens around 2.5 mm wide. However, light reflected along the paths band of the spectrum has a practically uniform width. At path length of thelens light within the glass (about a factor of three), and, with be observed at to a diverging lightthat beam. the chromatic The aberration screen (Sect. must be set up at a grazing angle, so block. The line-shaped light source here and in Fig. illuminated by an arc lamp the light beam with sufficient magnification onto a screen. 75 cm. Figure 16.30 16.10 Radiation of Different We now repeat in Fig.tion 16.30 shown the in fundamental Fig. experiment on refrac- a red and a blue sub-beam.the In glass reality, we block see with a a continuousspectrum band whole below series of brightthis colors, end, i.e. a we need only use a lens use ordinary natural lightof (from red an filter incandescent light.(width light We bulb) allow instead a narrow, nearly collimated light beam parallel it has emerged fromactly parallel the to lower its surfaceThe upper beam of surface. of light the We from find block, theglass an incandescent block lamp which important into spreads is new out a result: within series ex- emerge the from of the sub-beams. lower surface Parallel, of the colored block. light Figure beams The Simplest Optical Observations 16 320 Part II 16.10 Radiation of Different Wavelengths. Dispersion 321

K ca. 50 cm B C S

ca. 3° Part II

ca. 5 m violet red

Screen Spectrum ca. 30 cm wide

Figure 16.31 Production of a spectrum with a prism, as a demonstration experiment. The light beam which is incident on the prism is only approximately parallel. The lens images the slit (line-shaped light source) onto a wall screen at a distance of several meters. Here, out of the “colored light beam”, only the red and violet sub-beams are drawn. The grazing angle of the screen is again necessary for the reason given in the caption of Fig. 16.30. A typical setup for accurate measurements using a precisely collimated light beam is shown later in Fig. 22.2.

Refraction in a plane-parallel glass block thus generates aseriesof colored beams from a beam of colorless natural light. These colored beams are fanned out within the glass block, but after emerging from the block, they are parallel to each other. As previously, we will accept the somewhat vague expression “colored beams” and will first try to continue the fanning-out of the sub-beams outside the block. This can be readily accomplished: We simply have to abandon the parallel form of the upper and lower glass surfaces and instead make the block in the shape of a prism. With the stronger fanning-out, we can use a much wider light beam than in the case of the plane-parallel glass block. At first, however, we are bothered by the overlap of the individual colored sub-beams; therefore, we make use of the trick explained in Sect. 16.8. We place a lens within the wide aperture B and make all the emerging light beams convergent, i.e. we form an image of the illuminated slit S on a screen (Fig. 16.31). There, we can clearly see the brightly-colored band of a continuous spectrum. Now we carry out a quantitative evaluation of these observations. First of all, we need to eliminate the unphysical terms “red”, “blue” etc. light beams and instead characterize the various radiations of the , 4 16.29 between D 405 nm 1.5236 1.6507 1.8060 2.4621 D on the wavelength is 436 nm 1.5200 1.6421 1.7913 2.4499 n wavelength range a variety of radiations from D s of a measurable quantity. This 578 nm 1.5101 1.6200 1.7552 2.4175 : We separate a narrow light beam produces along the band of the spectrum so as m. The technical details of the measurements D 9 16.31 1.7473 2.4099 Index of refraction at the wavelength 656 nm 1.5076 1.6150 wavelength 10 * . D C16.20 . 16.2 of a particular material. In principle, the setup shown in Indices of refraction at different wavelengths n is adequate for this purpose. Then, for various materials 1 nanometer dispersion * * D 16.4 These are the technical designations for the glass types (Schott AG, Mainz, Ger- 1nm Heavy flint glass (SF4) Diamond Material Light crown glass (BK1) Light flint glass (F1) Table 16.2 are unimportant to us.additional observation Here, which is we ofplace are the fundamental first eye importance. by of a We all physicalIt re- indicator, concerned e.g. can a with be thermocouple (Fig. an moved 15.5). in Fig. to measure the radiantflection power of as the a indicator function does of not wavelength. go The to de- zero beyond the ends of the practical lab experiment). We thus find for a light beam in the violet spectral region, wavelengths of 400–440 nm many) 4 in the bluein the greenin the yellow + orangein spectralregions, the wavelengths red of 580–640nm spectral region, wavelengths ofThe spectral 440–495 region, process nm wavelengths of of 495–580 refraction nm thus spectral region, wavelengths of 640–750 nm. * sub-beams physically, i.e. by mean is the concept of the from the spectrum, which appears to(i.e. the to eye be to “monochromatic”), contain and, a usingdure single the of color now diffraction well-known proce- by a slit, we measure its wavelength (Fig. the radiation of (colorless) natural light,and which each are colored of to them the eye, can be attributed to a 400 and 800 nm.average In wavelength for the each coming of sections, thesemean regions. it a With whole will this, range. suffice however, The we tooften same quote is employ. true an of the red filter lightFor which we each such(average) wavelength, radiation we orrefraction can spectral determine range, separately characterized the by index of an Fig. called which are often usedgiveninTable in optics, we obtain the indices of refraction The dependence of the index of refraction disper- normal (bottom right) and The Simplest Optical Observations 27.1 . 16 27.2 sion. In other wavelength ranges, the relation between wavelength and index of refraction is more complex; see Fig. C16.20. The dispersion which occurs in the range of visible light, in whichindex the of refraction decreases with increasing wavelength, is termed 322 Part II 16.10 Radiation of Different Wavelengths. Dispersion 323 visible spectrum, that is beyond the border of the violet at one end and the red at the other. On the contrary, on both sides of the visible spectrum, we observe a considerable signal from invisible radiation. Refraction thus produces not only a visible spectrum, but also invisible light beams outside the visible range. These regions are given the collective names “ultraviolet”5 and “infrared”6. For our earlier demonstration experiments, we produced red light not by refraction, but instead by employing a red filter: We passed the natural light from an arc lamp through a piece of red glass. The word filter is based on an old custom: One describes the colorless radiation of an incandescent lamp as a mixture 7 of various colored radiations Part II having different wavelengths. The ‘filter’ allows only one of them to pass through. In a corresponding manner, we can also prepare filters for the invisible radiations. As a filter for the ultraviolet, we can most conveniently use a piece of glass with a high content of nickel. To the eye, it appears completely black and opaque; but in the language of the above description, it allows ultraviolet light from the mixture of radiation emitted by the arc lamp to pass through. In order to make the ultraviolet light beam visible, for demonstration experiments we can use excitation of fluorescent radiation. Numerous substances “fluo- resce”, i.e. they glow brightly with visible light when they are struck by ultraviolet light. Thus, in Fig. 16.3, instead of a red filter, we make use of an ultraviolet filter and paint the board which serves as screen with a fluorescent pigment, e.g. a layer of paint containing a powdered zinc salt. A bright, greenish fluorescence indicates the trace of the invisible ultraviolet collimated light beam. As a filter for infrared radiation, glass plates containing manganese oxide (MnO) are suitable. To detect the infrared radiation, one usually employs the warming of the irradiated object. Thus, in Fig. 16.32, we set up a searchlight with infrared light by using an arc lamp and an infrared filter, and we can ignite matches at a distance of 10 m using its invisible radiation.

CF V ca. 10 m St H

Figure 16.32 Igniting a match St by means of a beam of invisible infrared radiation (C is an auxiliary condenser lens, F is an infrared ﬁlter, V is the shutter and H is a concave mirror)

5 J.W. RITTER: Gilberts Ann. 7, 527 (1801). 6 F.W. H ERSCHEL: Philosophical Transactions, Part II, 284. London 1800. 7 This old-fashioned terminology is often convenient, but according to the current state of knowledge, it is at best rather imprecise (see Sects. 20.11 and 22.4). C revers- 1 2 γ γ (tilt axis between . This can ı = 2 σ ı δ B ). μ-γ μ σδ 2 180- Exercise 16.4 . A . But this setup is overly sensitive 2 1 16.33 and that permits A

between the incident and the doubly-reflected (the sextant ı C and B between two objects Angled Mirrors and Mirror Prisms Mirror prisms with right-angled triangular shapes. At the left: An angled mirror with The influence of ı , the angle can be used to invert images which are upside down, e.g. for the 16.34 with silvered cathetus facesa for silvered reversing beam hypotenuse directions; faceing at prism for the exchanging right: lightprojection with of beams physics 1 demonstration and experiments ( 2. A a measurable wedge angle tilting a mirror on the directiona of reflected light beam (onlycentral the axis of the beam is shown) a free-hand measurement of thegular an- spacing Figure 16.35 Figure 16.34 Figure 16.33 16.11 Some Technical Resources: Angled mirrors and mirrorto prisms experimentation. are In frequently-used addition, technical thein aids role mirror of prisms refraction provides and us dispersion with a usefulOften, subject one for wants contemplation. to deflect a light beam by a certain angle be most simply accomplished byto a the single scheme mirror shown reflection in according Fig. used by seamen and astronomers). Imagine that your eye is at in the directions the right face of the mirrortransparent, is e.g. partially only half silvered. to a sideways tilting of the mirror; tilting through an angle the incident and the reflected beam by 2 perpendicular to the plane of the page) changes the angle If we use aways double reflection tipping by of anFig. the angled mirror, mirror in has contrast, no side- effect, since, as can be seen in The Simplest Optical Observations 16 324 Part II Exercises 325

Figure 16.36 The optical path through a right- angled corner mirror: The principle of the “cat’s- eye mosaic”

beam depends only on the wedge angle between the two mirror faces. We ﬁnd

ı D 2 : (16.25) Part II For a beam deﬂection of 90ı, must be chosen to be 45ı. Two mirror faces which are perpendicular to each other ( D 90ı) yield ı D 180ı, that is they reﬂect the incident beam backwards, parallel but opposite to its original direction, etc. (Fig. 16.35). The same effect is produced by a corner mirror (Fig. 16.36).

Exercises

16.1 a) Derive Eq. (16.8) for a prism by application of the law of refraction (16.2). Note: Apply the law of refraction to each face of the prism, take the sum and the difference of the resulting expressions and rewrite them as products. b) A light beam is incident at ı ı an angle ˛1 D 30 onto a 60 -prism with an index of refraction of n D 1:5. Compute the angle of deﬂection ı, by which the light beam is refracted in total. Note: Calculate ˇ1 from Eq. (16.2)and insert its value into Eq. (16.8). (Sect. 16.6)

16.2 Show that for image formation using a concave mirror (Fig. 16.24), for near-axial light beams, the focal length f is equal to half the radius of curvature R of the mirror. (Sect. 16.7)

16.3 Show that the lens formula for near-axial light beams (Eq. (16.13)) is also valid for imaging by a concave mirror. (Sect. 16.7, 18.2)

16.4 In mirror prisms (Fig. 16.35), in addition to reﬂection, refraction also always occurs. If natural light (from an incandescent lamp) is used, one thus also observes dispersion (Sect. 16.10). Af- ter the light beam emerges from the prism, the different sub-beams, which are refracted through different angles, are shifted parallel to each other. Why can one still see the object without colored fringes when using a mirror prism? (Sect. 16.11)

Electronic supplementary material The online version of this chapter (https:// doi.org/10.1007/978-3-319-50269-4_16) contains supplementary material, which is available to authorized users. ial h ae ..frtery n nFig. in 2 and 1 rays the for prac- e.g. angle, opening same; large a the within tically even points, image the and object esi,i rn f rbhn h pruemksthe makes aperture the behind or of, length front in in, lens A ohn udmnal e;i ipymksi osbet s larger use to possible angles it is opening using makes This simply formation importance. it practical new; image their fundamentally to to nothing due turn mirrors, concave we and lenses question, this answering Before length ik hc aeu h mg. h nwrwl egvnltrin later given be will answer The image.) circular small the Sect. are up points make image which (The between plane? disks distance given image the a and for object “pixels”) the terms, modern in (or, points” hc pnn angle opening bulbs.Which light small with made letter or arrow nbt ae,a suitable a as cases, both In ih-emBoundaries Light-Beam and Formation Image formation B aperture the by played role The 17.1 Figure limited be must points” ful- spatial “object be small at to a originate needs to which condition rays one The only ﬁlled: basically formation, image For Formation Image of Fundamentals 17.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © (Sect. pnn pnoecmr!;i Fig. in camera!); (pinhole opening rapoorpi lt) nFig. In plate). photographic a or nimage in 21.8 hc sapoet fteradiation. the of property a is which , 16.1 (Sect. h eiieqatt hc eemnsi sthe is it determines which quantity decisive The . nterwyt h mg ln eg rjcinscreen projection a (e.g. plane image the to way their on ) 16.3 ! cd ab e loFig. also see , n hst emtteiaet ebte resolved. better be to image the permit to thus and pnn angle opening 2 Object, consisting of consisting Object, ω ! illuminated points olsItouto oPhysics to Introduction Pohl’s o h aito ie h hret“image sharpest the gives radiation the for 2 1 B B object 17.1 16.18 2 ! 2 ω' ob mgd ehv hsnan chosen have we imaged, be to ω 17.1 ,teaperture the a, ypsigtruhan through passing by o l h asbtenthe between rays the all for ) Apertures ,i sasalpaa mirror. planar small a is it b, Screens 2 , 2 ω 17.1 https://doi.org/10.1007/978-3-319-50269-4_17 B ω' sasalround small a is 2 ω Concave mirro mirror Planar .Aconcave A c. pia path optical B B aperture wave- r 17 327

Part II d). This 17.1 (the focal length f Screen 5 m = 70 cm f from the mount (i.e. the outer 1 L ay, using a good-quality telescope each 0.2 mm in diameter, at a spacing of C17.1 . There, we form the image of a lattice of 82 cm 17.3 . For demonstration in a large room, C Point lattice 17.4 , we remind the reader of an important result which K the intersection of two geometric straight lines (rays), Imaging of a small square point lattice using a telescope objec- A shallow-water lens for surface Patterns from the Lens Mount 17.2 ) not (an objective with a focal length of 70 cm). The point lattice 1 L boundary) of the lens orextension; it mirror. is This not a diffraction pattern “geometrical point”. has aFor finite light waves, this importantparatus fact sketched can in be Fig. demonstrated withpoints the on ap- a screen locatedlens 5 m aw tive. The lattice consists of 250.7 holes, mm; see Fig. of the lens) must be chosen to be shorter. Figure 17.2 waves on water. It imagesthe an wave “object center point”, located toure the on left the of optical the axis,in fig- as the an image “image plane. point” Thenite extension, image as point a has result a ofthe fi- diffraction opening by of the lensVideo mount. 12.2 (See Vol. 1, Figure 17.3 With Fig. we obtained by considering mechanicalalensis waves: An image point from but rather a diffraction pattern mirror has the sameis effect all owing to that its wedescribes curvature the can (Fig. light say beam withininto in the the terms subject of geometrical-optical of image rays. picture formationture only which We of by can considering the the penetrate radiation wave further na- Vol. (i.e. 1, its in wavelength). particular in Thisthat volume. Figs. is We 12.20, shown will in 12.21 refer detail and to those in 12.31 figures in in Chap. the 12 following section. of 17.2 Image Points as Diffraction (3 mm on a side)hole is of composed 0.2 of mm 25 diameter,red object and light. points, is The each light illuminated beam one from which a emerges behind round from by the lens intense is bounded by . .Amorede- (Vol. 1): http://tiny.cc/tfgvjy 16.9 Image Formation and Light-Beam Boundaries 17 tailed treatment will follow in Chap. 21. Video 12.2 “Experiments with water waves” C17.1. The diffraction of light was introduced already in Sect. 328 Part II 17.2 Image Points as Diffraction Patterns from the Lens Mount 329

Figure 17.4 The point lattice as imaged on the screen in Fig. 17.3 (photographic negative, 1=2 actual size)

the circular lens mount (diameter 5 cm)1. The image on the screen has been photographed in Fig. 17.4. The photo shows a lattice composed of 25 cleanly separated circular disks. They give an upper limit for the diameter of an “image point”. Now, we place an ancillary aperture B1 just in front of the object (Fig. 17.5), and let only the central hole be imaged, that is one single “object point”. Its image Part II remains on the screen, maintaining its sharpness. Now the decisive observation: We add a second aperture in the form of a rectangular slit B2 just behind the lens in Fig. 17.5. This produces a rectangular boundary for the light beam emerging from the lens, for example with a width of B D 0:3 mm. On the screen, we see the image which is reproduced as a photo in Fig. 17.6 (1=2 actual size): The object point is imaged in the image plane as a long “brushstroke”, with shorter replicas on each side. Using blue-ﬁlter light, we see the same “brushstrokes”, but they are somewhat shorter (Fig. 17.7).

C B1 B2 K

Ancillary Aperture slit, aperture width B = 0.3 mm

Figure 17.5 An ancillary aperture B1 covers 24 of the 25 openings in the point lattice (object). The remaining opening is imaged by the same objective lens as in Fig. 17.3; but this time, the light beam is bounded with a rectangular shape by a second aperture B2.

α = 0 5 10 15 20'

Figure 17.6 The image point formed by a lens whose beam has been limited by a narrow rectangular slit B2, 0.30 mm wide and perpendicular to the long direction of this ﬁgure; it has the form of a brushstroke. The image was photographed using red-ﬁlter light ( 660 nm) at a distance of 5 m. (Photographic negative, 1=2 actual size, ˛ D 200 corresponds to sin ˛ D 5:8 103 rad). For angular units, see the footnote at the end of this chapter.

1 The illumination lens or condenser C has to form an image of the crater K of the arc lamp (the source point of the illumination) in the plane of L1, and its diameter must be greater than that of the lens L1. ) –3 17.7 to be with 2 16.23 ( B and 17.6 This is, how- 17.6 , center). The . We can state C17.2 ition for the validity of = 0 2 4 6 ∙10 ; it is, as expected, ). α 16.29 2 ) sin 17.8 16.23 16.23 : B . Its first minimum appears, as D ). Quantitatively, this is not quite ˛ 17.6 sin ,but by a circular aperture (e.g. with a diameter An image point is in reality a diffraction pat- 17.6 ,asdefinedbyEq.( good approximation. Thus the cond 17.5 ˛ As in Fig. The image point cast by , we can consider the light beam which falls on the slit in Fig. , bottom). Therefore, we can explain Figs. 2 ) is fulfilled. The image point cast by a lens is a diffraction pattern of B 17.5 16.29 16.23 In Fig. the slit Normally, the boundary orrather mount circular; then of instead a ofthe lens lens a mount. slit, is Therefore, we in not have continuing our the rectangular, experiments, circular but we opening replace of using blue-filter light, with 470 nm Eq. ( ever, practically unimportant, given the widein range the of visible wavelengths spectral region (around 400–750 nm). Result: the opening which limits the effective lens diameter the diffraction pattern ofsay a that it circular is aperture.around formed by its Qualitatively, rotation center we of point can correct; the (Fig. in diffraction the pattern case from offactor a a of slit circular 1.22 on opening, the we right-hand side have of to Eq. add ( a numerical of 1.5 mm). The result can be seen in Fig. Figure 17.7 at a distance of 5 mtom (top picture picture 5 1 min. min., exposure bot- light, time; actual-size red-filter negatives) a telescope objective lens throughaperture a of circular 1.5 mm diameter, photographed 2 In both cases, the images resembleknown a horizontal diffraction section from pattern the well- cast by a slit (Fig. Figure 17.8 a collimated beam to a with some certainty: tern from the boundaryseen of from the the center lens ofimage the at lens, an on angle both sides of the midpoint of the minima are at theFig. same spacing as before (compare Fig. 1970)), Sect. 49 Principles of (Pergamon Press, Image Formation and Light-Beam Boundaries 17 (available online – see Com- ment C25.11). 1.22, see e.g. Max BornEmil and Wolf, Optics 4th edition ( C17.2. For the calculation of the value of this factor, 330 Part II 17.3 The Resolving Power of Lenses, Particularly in the Eye and in Astronomical Telescopes 331

Figure 17.9 Photographs of two images of the same letter ‘F’ ( a copper Part II stencil), formed on the left using a lens, and on the right using a pinhole camera. The lens mount and the aperture of the pinhole camera are both 3.5 mm in diameter (!) Object and image distances are each 17 m. The images were thus just the same size as the object. They are reproduced here actual size. The surprisingly large “image points” which compose the images are shown later in Fig. 21.19. without serious exaggeration that in imaging by lenses, the opening “. . . in imaging by lenses, which bounds the light beam is more important than the lens itself. the opening which bounds The role played by the lens is only secondary; it makes the incident the light beam is more plane or divergent wave train convergent and focusses it into a small important than the lens region. In this way, it places the diffraction pattern from its opening itself”. at a conveniently accessible distance. The image which is composed of these “diffraction-pattern image points” is then formed at a man- ageable size. If we remove the lens, the remaining aperture acts just like a pinhole camera (Fig. 18.35). In the latter, the diameter of the aperture must be adjusted to suit the desired distances of the object and the image plane. This topic, as mentioned at the beginning of this chapter, will be treated in Sect. 21.8. If this adjustment has been carried out correctly, the resulting sharpness (resolution) of the image cannot be further improved by inserting a lens into the aperture. This is demonstrated in Fig. 17.9 for an aperture of 3.5 mm diameter. There is no fundamental difference between the image points of a pinhole camera and those of a lens. Both are merely diffraction patterns of the aperture. To be continued in Sect. 21.8...

17.3 The Resolving Power of Lenses, Particularly in the Eye and in Astronomical Telescopes

The significance of the experiments just described will be explained by giving some examples. We return to Fig. 17.5, remove the ancillary aperture B1, and thereby form an image of all 25 object points in the point lattice. Then we again add a rectangular aperture to the ) ı D of the (17.1) 16.23 ( is verti- ' 2 calculated 600 nm B ˛ a. Instead of are drastically D : The central ). Seen from the 17.3 17.10 17.11 17.8 : B : ˛ B 2 actual size) = or min That is, the angular spacing 2 B ' 2 23 mm) is defined by the iris (pupil). Its D ), we see an image of five bright horizon- C17.3 D ˛ f 17.4 sin from the center of the diffraction disk. Then for , and with it the brushstrokes, by an angle of 45 b, etc. One inappropriate restriction of the light 2 ˛ B , which takes the form of a mosaic. The boundary , we find to a good approximation B 17.10 The images of the point lattice in Fig. retina ), initially with a horizontal orientation (the slit mm as the average wavelength of sunlight, we obtain from abc 4 17.6 10 opening diameter in daylight is about 3 mm. Taking Example: Our eyes are basically similar tostead photographic of cameras. the In- photographic filmcontain or the plate (or CCD detector array), they of the lens of the eye ( 6 cal). Then the lens forms the image shown in Fig. the point lattice (Fig. tal lines which result from theWe overlap then of rotate the slit horizontal brushstrokes. modified, depending on the(red-filter shape light, and photographic size negative, 1 of the boundary of the objective In order to separatehave to (resolve) be two roughly object as far points apart in as the shown in image, Fig. they object points should not be much smaller than the angle from Eq. (16.23). Then wespacing find for the smallest “resolvable” angular Figure 17.10 lens, that is we(Fig. again produce long “brushstrokes” as image points to the vertical. Now,shown instead in of Fig. a point lattice, we see the image as disk of the one image point(or must fall be in still the further first minimum away). of the other beam can thus makeit the is image supposed completely to different represent. from theWith object the usual shapewe of obtain circular the diffraction lensconcentric disks boundary, as rings a image of points, circular decreasing surrounded lens intensity by mount, (Fig. a lens diameter center of the lens, thebright in first the ring photographic of negative) in minimalangular this intensity spacing diffraction pattern of (which is appears at an criterion” in the Image Formation and Light-Beam Boundaries AYLEIGH 17 C17.3. This condition is known as the “R literature. 332 Part II 17.3 The Resolving Power of Lenses, Particularly in the Eye and in Astronomical Telescopes 333

Figure 17.11 The resolving power of a lens: Separating the two diffraction patterns (from a circular lens aperture of 1.5 mm diameter) which appear as neighboring image points. The object consisted of two holes of 0.2 mm diameter at a spacing of 0.3 mm (photo taken with red-ﬁlter light at a distance Part II of 5 m; photographic negative, actual size) (Video 17.1) . Video 17.1: “Resolving power” http://tiny.cc/9dggoy Eq. (17.1)3 The video shows that “image points” are simply diffraction 6 104 mm patterns from the lens mount. 2' D D 2 104 D 41 arc second : min 3mm The resolving power which is limited by the diameter This means that our eyes must just be able to distinguish two ob- of these diffraction patterns ject points at an angular spacing of around 1 arc minute; or, in other is demonstrated using two words: About 1 arc minute is the smallest “resolvable” angle of sight neighboring circular openings 2' for the eye (cf. Fig. 18.23). This rough estimate agrees with prac- at different spacings, illumi- tical experience. For a demonstration, we can use a black-and-white nated with red and with blue square lattice. For an observer at a distance of 10 m, the spacing of light. A preliminary experi- the lines on the lattice has to be about 3 mm in order to resolve them ment using a single opening visually. It then follows that: demonstrates the reduction of the effective lens diame- ' D 4 2 min 3 10 ter using a circular aperture. The initially sharp image is or gradually converted into the enlarged pattern of a diffrac- ' D : 2 min 1 arc minute tion disk as the lens diameter is decreased (see Sect. 21.3). With optimized illumination, we can approach roughly half this value. We thus do not have to make the spacing quite as great as shown in Fig. 17.11.

The astronomical telescope is also practically just a variant of a photographic camera: A lens or a concave mirror, with a photographic plate or photodetector in its focal plane. For a lens or mirror diameter of 300 mm, the smallest resolvable angle of vision is 100 times smaller than for the naked eye, i.e. about 0.4 arc seconds. With a diameter of 1.2 m, we can still resolve two fixed stars at a spacing of 0.1 arc second, etc. Each of the two stars is visible only as a diffraction disk from the lens or mirror mount. If the objective of a telescope has a triangular boundary, the diffraction pattern from a fixed star looks like that shown in Fig. 17.12. A true image of the disks of fixed stars, corresponding to an image of the Sun’s disk, cannot be obtained

3 Units: 1 degree (ı) D 1:745 102 rad, 1 minute of arc (0) D 2:91 104 rad, 1 second of arc (00) D 4:86 106 rad (cf. Vol. 1, Sect. 1.5). https:// for the min ' ) 250 nm with the 17.3 D ) 17.3 of two lighthouse beams so that d ) contains supplementary material, which The online version of this chapter ( are still much too large. criterion) in the camera image? (Use , not by details of the lens construction. C17.4 would a camera objective require at this altitude AYLEIGH B 047 arc second). (Sect. : 0 The image point D ˛ Space Telescope at a wavelength of under which the disk of the star Betelgeuse appears from ˛ Compare the smallest resolvable angular spacing 2 The International Space Station (ISS) circles the Earth at an UBBLE H 17.2 formed by a lens whichbounded is by a triangular aperture with a side length of(at 9 a cm distance of 5red-filter m, light, taken negative with shown actual size) the average wavelength of sunlight, 600 nm.) (Sect. Figure 17.12 angle 2 the Earth ( is available to authorized users. eye? b) What diameter in order to resolve two persons onresponding the to ground standing the 2 m R apart (cor- Electronic supplementary material doi.org/10.1007/978-3-319-50269-4_17 boundary of the lightThat beam is the important result of this section. Exercises 17.1 average altitude of ca. 350a) km. What is thethey minimum can spacing be seen as separate (resolved) by an astronaut with the naked with earthbound telescopes. The diameterminute, of the while Sun’s the disk diameters isthan 32 of 0.01 arc the arc disks second. of Todiffraction-limited image even form points nearby images of even of stars the the is largest telescopesdiameter disks less (mirror for of example fixed 5 stars, m) the The resolving power of the eye and of telescopes is determined by the ’s )(See IZEAU hnology (Cal- ). 20.16 Astrophysical Space Tele- Orionis (Betel- , L39 (1996)) Hale Telescope ˛ 463 UBBLE Image Formation and Light-Beam Boundaries 17 Exercise 17.2 C17.4. Here, Pohl is refer- diameter of the disk amounts to 0.047 arc second. Itdetermined was using F geuse). A large, bright spot was seen there. The angular method (Sect. e.g. R.L. Gilliland and A.K. Dupree, Journal sible to observe the surface of the star influence of the atmosphere, and which orbits the earthan as artificial satellite, it is pos- scope, which has a mirror diameter of only 2.4 m ataltitude of an 500 km – thatfar is, outside the perturbing is due to the Earth’s atmosphere. For example, using the H tech) since 1948 on Mount Palomar in the neighborhood of San Diego, CA/USA. A considerable perturbation ring to the operated by the California Institute of Tec ( 334 Part II h mg,tery utb xeddt n ftepicplplanes principal the of one to extended be must rays the image, the nweg.Tesxpgsi Sect. in detailed pages extensive, six rather The requires is contrast, knowledge. make and in to How lenses, learned of experience. quickly everyday use be effective with can compatible part leads experimental most wire the to for use aids to indispensable How are Both observations. magnetism. and tricity n emLimitation Beam and Formation Image of Details Technical and Fundamental xs h w rnia planes optical midplane: principal the single two to a the perpendicular than axis, planes more reference requires two rays introduce the must description We of the path objec- Then, optical (e.g. the photography). lenses of and compound microscopy for for lenses and to tive not lenses always it thicker nearly considering however, for lenses, is, admissible the approximation This of thickness insignificant. finite be the neglect thus We the (Fig. are in treatments which optics used high-school formation also in image popular is the of midplane of midplane constructions This the geometric quantities. distance from well-known image significant the the the lengths, to are distance focal lens the their and lenses, object the thin to simple with dealing In Nodes Planes, Principal 18.2 optics, In Remark Preliminary 18.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © npriua,tems motn seti akn hr:Temajor The there: lacking is by played aspect role important most the particular, In a egh n h betadiaedsacswt eeec othese to G reference F. with (C. distances image planes and object the and lengths cal g omto.W e pwt htapc nybifl nSect. in briefly only aspect that with up met We formation. age ntepeetcatr eofradtoa xmls neaande- observations. again individual once and examples, experiments additional from directly offer rived we chapter, present the In lenses ih-emdelimitation light-beam AUSS lyruhytesm oea ielasi elec- in leads wire as role same the roughly play .Lkws,frtegoerclcntuto of construction geometrical the for Likewise, ). olsItouto oPhysics to Introduction Pohl’s H and 16.7 H 0 n ems esr h fo- the measure must we and ; nalqetosrltn oim- to relating questions all in r yn en sufficient. means no by are 18.1 , ). https://doi.org/10.1007/978-3-319-50269-4_18 C18.1 17.2 . 1..P C18.1. h y.Seas Fig. also See eye. the of lens the with as example lens, for the of sides two the on located are refraction of indices differing with terials and figures. following and the here in case the is as distinct, not are they when even side the to referring those and prime, a without by letters 16) Chap. in seen already side the on quantities notes f ylteswt prime, a with letters by have we (as lens the of 0 r nqa hnma- when unequal are 18 OHL ossetyde- consistently image 18.24 object f 335 .

Part II : 0 F ' 0 This y (18.1) tan are shown D 0 f which corre- K 0 as an entrance y' P' P ' ' . . The physical y y F P' P' ' and are shown; they are K 0 18.2 F can be arbitrarily greater 1 3 2 and y 1 3 2 1 3 2 F F' ). F' φ' , while those that pass through f' and on the object side: . K' H' ; y 0 y' ' H K y . The object size 2 . The beam axes (rays) which pass through f ff' . The focal points 1 3 2 1 3 2 . If we want to measure the focal length using tan ff' P 1 3 2 φ D Exercises 18.1, 16.3 18.3 18.5 F 0 f F . With thick lenses and compound lenses, we measure the 0 y y P P H image-side telecentric also illustrates a general definition of the focal lengths, The definitions of the object-side and image-side principal Image Formation and Beam Limitation The geometric construction of the image point y P object-side telecentric and ), and therefore want to make e.g. ray 2 the central axis of a light 18.2 H 18.1 are called beam, we have to place an aperture around the focal point pupil. This makes ray 2optical a axis principal on ray, the and object thus its side angle is of conventionally inclination denoted to as the Eq. ( for comparison to Fig. sponds to the object point object distance on theas object well side as and the the corresponding image distance focal on lengths. the The image points side, properties of the lens. It suffices to trace any two of the three rays 1–3. construction is purely formal than the diameter of thethe lens, rays e.g. 1 in and the 2 casesect themselves its of no midplane. a longer photographic Nevertheless, intersect camera. theylower the are Then part actual ‘bent’ of lens, at the but the figure they midplane, ( inter- as seen in the planes meaning of this construction caniments be shown seen in in Fig. the demonstration exper- Figure 18.2 are termed Figure 18.1 and then ‘bent’ there. This is illustrated in Fig. Figure namely on the image side: F Fundamental and Technical Details of 18 336 Part II 18.2 Principal Planes, Nodes 337

H H' F' F

H' H

F' F Part II

Figure 18.3 Demonstration experiments for illustrating the schematic ray diagram in Fig. 18.2 (red-filter light). For clarity, only the light beams belonging to the rays1and2areshown(1=9 actual size). In the bottom part of the figure, the image-side principal plane H0 is closer to the object than the object-side plane H! (Drawings from photographs of the demonstration experiments. This applies also to later figures, such as Figs. 18.4, 18.10 and 18.13).

To locate the principal planes experimentally, one makes use of two telecentric light beams. They are incident ﬁrst from the right, parallel to the optical axis (Fig. 18.4, top) and then from the left (Fig. 18.4, bottom). One ﬁnds the positions of the focal points F and F0 by extrapolating the central rays (short-dashed lines) until they intersect. With this compound lens, the two principal planes H and H0 are no longer between the individual lenses (a large converging lens and a small diverging lens). Furthermore, we clearly see the very unequal distances of the two focal points from the central plane of the compound lens. In the usual applications, the same substance, namely air, is found on the object and the image side of the lens. In certain cases, however,

H F

f H H'

f' C18.2. With today’s tele- H' photo objectives, the focal length f ’ can be twice as long Figure 18.4 A demonstration experiment for locating the principal planes of as the whole lens system. a compound lens composed of a converging lens and a diverging lens. This In Fig. 18.4, the principal type of compound lens is often used in cameras as a telephoto objective for 0 taking large close-ups of distant objects, for example wild animals. This plane H would then be at the requires a long focal length (cf. Eq. (16.22)).C18.2 left margin of the figure. : 0 K , called relative and 0 K K in a pinhole , which are 00 0 a 4mmbehind ‘bent’ and K : . It can be most K 24 and and D a / K 6 : 1 C Frosted glass plate , which are 8 0 A' Image : ). The image formation of a nodal points 22 a'' . can be described in two ways: 0 . 18.5 accomodated) eye lie (for normal and a' K' A a through them are parallel to each ). The slider is in turn mounted on 18.2 K 18.6 . a N A as an image point . Object A The locations of the two nodal points Image Formation and Beam Limitation vitreous humor nodal points the crest of the cornea. This property of the nodalexperimentally. points can The be used compound totical lens locate axis the is principal (the mounted planes symmetry onthe axis a spindle or slider, of lens with the axis; its shown slider op- dot-dashed) (Fig. parallel to The two nodal pointsvision, of not the peripheral relaxed (un vision)(crest) 7.0 of the and cornea 7.3 (the mmcontrast, outer behind surface are the of only outermost the about eye).while point 1.35 the The focal and principal point lies 1.65 planes, within mm in the eyeball behind the crest of the cornea, a vertically-rotatable axle. Then, the lenstionary is used light to cast source the image ontoback of and a a forth. sta- This distant in screen, generalshifting causes the the and slider, image we the on can the find optical screencase, a to the axis position move; where by axis is this of motion swung ceases. rotationthe In axis is that of directly rotation under lies the in object-side the nodal object-side point, principal and plane. other on both the(denoted by object 3) are side drawn and in Fig. the image side. Two such rays This means that rays which pass parallel to each otherintersection on points the object with andaxis the the of optical image the axis imaging sides, aperture) (the and define whose dot-dashed the two symmetry points Either by referring to the two rays to each other by refraction, or by using the rays the In a correspondinga way, lens is we inserted into(Fig. define the 18.24). aperture. the As On nodal animage the side, example, points consider the object the interior even side, of eye called the when it the eyeball; it is contains bounded a by gelatinous liquid air, and on the camera filled with water.kept The from aperture leaking through by a which thin the glass image plate. is formed is simply explained for thefilled special with case water of in a its pinhole interior camera (Fig. which is In general, however, the samelens. material is Then the found intersection on points of bothplanes the sides (the optical of axis ‘principal with the points’) the become principal the “nodal points” the image side contains athe different eye!). substance, We then often require a the liquid concept (lens of of Figure 18.5 the object point Fundamental and Technical Details of 18 338 Part II 18.2 Principal Planes, Nodes 339

Figure 18.6 Illustration of the ex- Principal planes perimental determination of the object-side principal plane by locating the object-side nodal point. The lens can be rotated around a vertical Optical axis and can be slid with the slider axis relative to that axis.

Indexing Spindle of the drum micrometer slider Part II

Rotation axle

When a higher precision is required, one must determine both of the principal planes, even for simple lenses of moderate thickness; approximating them by the central plane of the lens is not exact. Fig- ures 18.7 and 18.8 show some examples.

Figure 18.7 The principal planes of three HHHH'H' H' thin lenses. Even in the case of the menis- abc cus lens a, they deviate only slightly from the central plane of the lens. (A “meniscus lens” is a lens with one concave and one convex curved surface). (1=6 actual size; fa D 28 cm, fb D 20 cm, fc D 21 cm)

H H' r1 = 10 cm = r2

F = 60 cm F' = 60 cm

Figure 18.8 A 10 cm-thick meniscus lens, which in spite of having the same curvature on both sides is a convergent lens, with its principal planes far outside the lens itself (also shown 1=6 actual size). See the references in Comment C16.14, and also e.g. http://ocw.mit.edu/courses/mechanical- engineering/2-71-optics-spring-2009/video-lectures/lecture-5-thick-lenses- the-composite-lens-the-eye/MIT2_71S09_lec05.pdf ' φ' 2 )and 0 .This 0 ! B , the lens The rays ). in front of the 16.21 B B' acts as exit pupil. 18.14 B aperture Exit pupil Image of the pupils. The entrance pupil y' ω' 2 Image 2 A pupil is (on both the object Object ght beams coming from the object y 2 ) and acts thus as an “entrance pupil”. ff' ! ). Follow the heavy lines from the lower 0 ). Behind the lens is its real image ω 2 ! are possible central axes or delimiting rays ! B 18.1 Aperture . But these light beams must by no means actually 0 , bottom, there is a circular aperture , 1840–1905). Entrance pupil F Mirror φ iting aperture for the field of view (Sect. 2 Image Formation and Beam Limitation . It is called the “entrance pupil” for light beams on the ω of Light Beams BBE 18.9 2 and A Delimiting of the imaging light beams by F RNST are the object- and image-side principal-ray angles of inclination. The lens therefore acts as anpupils “exit merge pupil”. into one. In this simplest2. example, the In two Fig. lens. It servesbeams as (opening angle entrance pupil and delimits the object-side light of light beams; theypoints are compatible with the positions of the focal aperture image acts asbeams exit (opening pupil angle and delimits the image-side light side and the imagelight side) beams a cross-sectionalobject area side, common to and the all(E “exit the pupil” for light beams on the image side be present in reality. Thelook very light different beams from the which rays are thatTheir we present draw form in in is fact geometric diagrams. often determined by pupils. Examples: 1. In the simplest application of a lens, e.g. as in Fig. It also delimits the image-side light beams (opening angle side (with an opening angle of mount (the “bezel”) delimits the li 18.3 Pupils and the Boundaries The contents ofsketched this in section Fig. are especially important. 0 ' y 2 Object and is an aperture inabove, both it is figures; a below, mirror it (reflection is aperture). a The transmission real image aperture of (opening), and bezel acts as a delim Figure 18.9 Fundamental and Technical Details of 18 340 Part II 18.3 Pupils and the Boundaries of Light Beams 341

S LBα B' α' Part II

Figure 18.10 A demonstration experiment for showing the locations of pupils. One half of an optical bench can be rotated around the center of the entrance pupil B and is hung from a spring S. Thus, the object point ˛, an opening which is illuminated from behind, can be swung up and down, and at the same time, its image point ˛0 moves. The light beam wanders up and down on the object and image sides in this process. Only two cross-sections of the beam remain at rest: The aperture B whichservesasentrance pupil, and its image B0, the exit pupil. To provide a marker, one can cover the upper edge of the entrance pupil with a red glass filter, and its lower edge with a green glass filter. Then the lower edge of the exit pupil appears red, and its upper edge appears green. We thus see how B0 arises as the image of B. In contrast, the image point ˛0, which is moving up and down, remains colorless; it contains both red and green parts of the beam, which are complementary colors, so that together, they give white light (for the demonstration, we use a cylindrical lens). edge of B to the upper edge of B0. They show that B0 is the image of B. Often, instead of a transmission aperture, a mirror is used (reflection aperture; Fig. 18.9, top). Example: The mirror is attached to the rotating coil of a sensitive galvanometer, and the lens as objective of a telescope is used for taking scale readings. The aperture B is imaged actual size (B0)inFig.18.9 by choosing the distance of the aperture from the lens in the drawing to be 2f (compare Eq. (16.13) and Fig. 18.1). When the aperture B is shifted towards the lens, the exit pupil shifts to the right, and at the same time, it increases in size. If the aperture B is placed in the object-side focal plane, then its image becomes virtual and lies at infinity to the right side of the figure; then the common cross-section of the image- side light beams, i.e. the exit pupil, is also at infinity on the right. Thus, the image-side principal-ray angle of inclination '0 D 0and the the optical path is telecentric on the image side.Examplesare showninFigs.23.1 and 24.20. The optical path of the light beams sketched in Fig. 18.9 and the positions of the two pupils there can be rather impressively demonstrated in the experiment. Details are giveninFig.18.10 and its caption. 3. In Fig. 18.11, there is an aperture Bbehindthe lens but within the image-side focal length f 0. B0 is its virtual image. This aperture image B0 acts as entrance pupil. Although it is behind the image, B0 delimits the usable light beams on the object side (opening angle !). The aperture B itself acts as exit pupil, it delimits the image-side light B φ con- 2 and which optical uniformly L 18.33 , called a whole . may be summa- B' , assume that the B Aperture 18.12 pupil 18.12 φ' Image – 2 ). , the area of the lamp (crater f' Exit Entrance B , likewise located on the image ). However, it is often diffi- ω' 0 18.9 2 B 16.1 Aperture the Image of and of the primary light source 18.12 16.2 L illumination lens C f (Sect. optical system, there is also a system for and some others (e.g. Figs. has to cast an image of the lamp onto C : The exit pupil here is a circular aperture and thus the maximum possible irradiation 18.12 ). Once again, some heavy black (dashed) 0 0 ω imaging 2 ! 18.9 ! aperture stop is a (virtual, upright) image of 0 B . In these cases, pupils are indicated for the As in Fig. Image Formation and Beam Limitation ), in addition to the , between the lamp and the opening. We demonstrate how illumination arrangement. Thus, for example, in Fig. forms the main image is theit exit is pupil. important When to severalothers. apertures distinguish are It the present, is one called the which limits the pupils from all the of the electric arc) is the entrance pupil, and its image on the lens In many cases, as in Fig. 24.20 the y 2 Object located on the image side. Its virtual image (e.g. the crater ofto image the the arc slit along lamp) its are whole small. length, so that Nevertheless, it we appears want bright. Then the condenser denser a condenser isserves used as with a an line-shaped example light in source. which In an Fig. illuminated slit diameters of both the imaging lens the imaging lens. Thisthe image of area the of lamp the may lens;trance itself in and be that exit smaller case, pupil than the forof lens the the bezel image lamp no formation, serves but longer thisto instead acts function. or the as If image larger en- we than makepossible the the opening lamp’s angle area image equal of the imaging lens, we obtain the largest 4. Frequently, for demonstrationsobject; we it use serves as an a illuminateda light opening well-defined source as size which and is sharply shape bounded (Sect. and with cult to place the lamp closeof enough the to lamp this opening, or andsuch the the cases, lens diameter we used can for make imaging use is of an also often too small. In rays indicate that side, acts as entrance pupil. intensity in the image. The facts which are elucidated in Figs. Figure 18.11 beams (opening angle rized as follows: Thefrom an light object point beams to the which lensimage actually and from point) propagate the are lens (going to determined the by corresponding the entrance and exit pupils. These Fundamental and Technical Details of 18 342 Part II 18.3 Pupils and the Boundaries of Light Beams 343

Condenser lens C α L 2ω 2ω' 2φ' Arc-lamp crater Image of the crater as entrance and exit pupil Illuminated α' slit

Figure 18.12 Delimiting of the light beam and the positions of the pupils when imaging with a condenser lens which uniformly illuminates an opening Part II (here, a wide slit); for the positions of the pupils, see also the above small- print paragraph) pupils are either physical apertures (e.g. openings, lens bezels, mirrors), or they are the light-emitting surfaces of a light source, or else they are images of an aperture or of the light source. These images can be real or virtual. The entrance pupil is the common cross-section of all the light beams coming from the object side, while the exit pupil is the common cross-section of all the light beams passing through the image side. The diameters of the pupils determine the usable opening angles ! and !0. The centers of the pupils practically always lie on the symmetry axis of the lenses (the optical axis). Then these center points are the intersections of the object-side and the image- side principal rays and are thus at the apex of the principal-ray angles of inclination ' and '0. In using lenses, one therefore needs to keep two things clearly sepa- “In using lenses, one there- rated: The rays used in the geometrical construction and drawn upon fore needs to keep two the patient sheet of paper (e.g. Fig. 18.1), and the real, usable light things clearly separated: beams, which are delimited by pupils. Of course, we could also con- The rays used in the geo- struct the images drawn in Figs. 18.9 ff. using the scheme shown in metrical construction and Fig. 18.1. The reader may even wish to check the validity of Fig. 18.9 drawn upon the patient or 18.11 in this manner. However, we must never confound the rays sheet of paper, and the real, drawn with the axes or the boundaries of the real light beams which usable light beams, which in fact are observed in the experiments. are delimited by pupils”. A precise insight into the delimiting of light beams by pupils is absolutely indispensable for all optical apparatus and experimental setups. The boundaries of the light beams play a decisive role in the construction of optical systems, e.g. the compound lenses with which we reduce the unavoidable imaging aberrations to an acceptable level. The bounding of the light beams determines the radiant power which can be transmitted by an optical system, and therefore (using the language of everyday life), all questions of brightness. In all optical instruments, e.g. microscopes and telescopes, the light- beam boundaries limit the field of view, the depth of focus, the perspective and the usable magnification. Finally, the light-beam boundaries also determine the resolving power of all optical instruments, not only that of the eye and of ,we 18.8 7, so that (left-hand : 0 the intersec- D . If at least one 18.13 ı presumed that the before 16.21 16.7 that lies on the lens axis. 4. This fact is the reason 1, sin 45 : 1 P D D ı simplest case by the lens bezel. 7 : till be distinguished; in a spectral 0 = were defined in Fig. 0 ! ). They limit e.g. in the microscope the smallest , we deal with the most important imaging aber- and 17 2, in contrast, 1 ! from a single object point D 18.9 0 – ı P Image Formation and Beam Limitation 45 (right-hand part) shows the corresponding experiment us- = 18.4 ı 18.13 while 90 part), we place the objectleft point on (the crater the of opticalscreen an arc axis. which lamp) is far Ina to grazed the the diaphragm by with plane the four ofparallel openings light light the near beams. beams the page to lens; is pass.page In it shows a an allows Their addition, outer four matte-white intersection and we an nearly with innerthrough place pair the of the plane beams. neighborhood of The of inner the pairpair the passes passes center through of a theintersection lens lens, point zone while of near the the its outer outer outer beam rim pair (lens occurs bezel). The ing a concave lens. The pair of beams from near the rim of the lens These image points noimage longer points overlap, along but instead thelens form thus optical have a axis. slightly series differingcalled of focal (The “spherical lengths.) aberration”. individual This zones imaging error of is the In order to demonstrate spherical aberration, in Fig. tion point of the more centralpropagation: pair, This as lens seen is in “spherically the undercorrected”. direction ofFig. the light of them is large,image then point each single zone of the lens produces its own 18.4 Spherical Aberration The derivation of the lens formulas in Sect. telescopes (Chap. length within the object which canapparatus s the smallest differencesseparated, in etc. wavelengths which can still be light beams were thin and near toused the was optical axis. that The the approximation ratiosby of the the ratio sines of ofFor two the larger angles angles angles, could themselves be thetheir (small-angle replaced ratio sines. approximation). of Thus, the for angles example, is sin 90 larger than the ratio of rations, giving a brief summary in each case. Up to Sect. for the occurrence of imaging aberrations, even whenlight monochromatic is used, asnear soon to as the the optical axis ideal (i.e. situation paraxial of rays)In no very longer Sects. thin applies. light beams The center point of thisaxis aperture of the is lens, presumed i.e. to the lie optical on axisWe the start of symmetry the with setup. spherical aberrations.opening angles The object-side and image-side assume that red-filter lightcontrary is is not being expressly stated, used.bounded by we a assume circular Furthermore, that aperture, in the when the light the beams are Fundamental and Technical Details of 18 344 Part II 18.5 Astigmatism and Arching of the Image Plane 345

F' F'

Figure 18.13 The demonstration of spherical aberration using cylindrical lenses (with their cylinder axes perpendicular to the plane of the page); at left: spherically undercorrected, at right: spherically overcorrected (Video 18.1) Video 18.1: Part II “Spherical aberration” http://tiny.cc/1eggoy intersects, again in the direction of light propagation, behind the pair In the video, the demonstra- which passes near the center of the lens. This lens is “spherically tion of spherical aberration overcorrected”. is carried out using a small glowing filament which is In order to correct for spherical aberration, we see that a suitable imaged onto the wall of the combination of convex and concave lenses can be used. Spherical lecture room. Various metal aberration can always be corrected only for certain values of the ob- panels placed in front of the ject and image distances. For telescope and camera objectives, one lens permit us to use either chooses an object at infinity. Microscope objectives are corrected for only the central portion of an object located close to the object-side focal point. the lens, or only the zone For many imaging applications (e.g. in the lecture room), the object near its outer rim. When the and image distances are very different. In such cases, it often suffices light passes only through its to use a simple plano-convex lens: We let the light beam with the central portion, the image greater opening angle be on the planar side of the lens. Then the rays is sharply focussed. When pass through the outer zones of the lens with approximately “mini- only the rim zone is used, the mum deflection” (Sect. 16.6) (since the object distance is much less image is completely unrec- than the image distance, the path of the rays through the rim zone of ognizable. Only when the the lens is more symmetric when the planar side of the lens is oriented experimenter moves the im- towards the object). This simple trick strongly reduces the spherical age plane closer to the lens aberration (compare Fig. 18.15, figure caption). by using a white cardboard screen can the image of the Not only convex lens surfaces, but also planar surfaces can give rise to filament be discerned, but it spherical aberration. As a result, microscope objectives can be corrected is still of a rather poor qual- only for a prescribed cover-glass thickness when light beams with a wide ity. opening angle are used (i.e. beams with a large aperture; cf. Sect. 18.12). This must be taken into consideration when using a particular objective.

18.5 Astigmatism and Arching of the Image Plane

Spherical aberration occurs even for object points on the optical axis. In general, however, a given object point P will be at some distance from the optical axis, for example when one wishes to photograph a landscape. In this case, simple lenses can produce an image point only with extended (planar) light beams or “light fascicles”. Such beams are obtained by using a very narrow slit in a diaphragm. The of the ' image surface in the object Cambered 0 are both of E of the screen at tangential D sagittal P'' b ' . The set of all the b b.) . P' 18.15 . They lie at different dis- 00 0, that is when the object and P the incident plane (tangential) astigmatism D , while the latter is shown below, Image distance in (Video 18.2) ' and 0 P 18.14 , both orientations of the slit each give ' for the planar light beams lying in the plane of 00 = 1.0 cm P d = 40 cm f Principal planes to the incident plane (sagittal). The former case is a and 0 P tures The demonstration of astigmatism and image-surface camber- Aper- Image Formation and Beam Limitation φ φ P P E E rather sharp image point, 0 0 Object distance 0.5 0 0.5 1.0 1.5 2.0 m plane Object

30°

30 20 10 20 10

perpendicular

–10 –20 –30 –10 –20 –30 (for example the crater of an arc lamp) along a rail Principal-ray angle of inclination inclination of angle Principal-ray φ tances from the lens.them Only found in at the the limiting samethese case distance. two of image The points difference of is the called distances the of image points plane of the page.of Below, the the page; planar we light see beamthe only is its slit perpendicular principal pass ray. to below The theby and lateral plane a rays above round which the emerge opening, plane from“image then of streaks”. we the In will page. ordera no lens to If longer of make we obtain large them replace diameter, image visible around the points, to 10 slit cm a but larger rather audience, we use ing (also known as “arching”),a using slit very (the thin, experimental planara setup light is beams side deﬁned shown view). by as In a the plan upper view part (from above), of not the as ﬁgure, the planar light beam is in the each other for the limiting case of incidence and perpendicular to itconcave each surface. form a rotationally-symmetric The two image surfaces are cambered and touch the image point are on the optical axis. P plane. At the same time,which we a determine sharply-focussed image the point distance appears.in (The distance, necessary often shifts several meters,using are a cart, most similar conveniently managed to that by shownFor in every Fig. angle of incidence one long axis of theor slit can lie either principal rays arriving from the object side, by sliding the object point Figure 18.14 shown in the upper part of Fig. as plan views. For the demonstration, we change the angle of inclination 0) of D ' size Fundamental and Technical Details of 18 (1:28 min), only the however, the distance to the lamp is varied while the lens is oriented normally ( moved closer to it. During its motion, both horizontal and vertical image streaks appear in turn (0:55 min). When, on long, stretched-out shapes (0:25 min). Finally, with the lens slanted, the lamp is which pass obliquely through the lens can contribute to image formation. The light disk in the image plane then takes sharp disk of light. Ifis the rotated lens around a vertical axis, then only light beams onto the wall of theroom lecture using a lens withfocal a length long and large diameter. We see a small, relatively image points caused by astigmatism, the crater of an arc lamp is projected in the video “Astigmatism” http://tiny.cc/reggoy In order to give an impres-sion of the distortions of the Video 18.2: the circular “image point” changes. 346 Part II 18.5 Astigmatism and Arching of the Image Plane 347

ab R L Part II Figure 18.15 A wheel with several rims drawn onto a matte glass plate (the spokes and rims are transparent, while the rest of the area is opaque) is an excellent object for testing a lens for astigmatism and image-surface cambering (about 1=2 actual size). Among others, a demonstration experiment using a plano-convex lens of around 13 cm focal length and 4 cm diameter is very impressive. If the planar surface is oriented towards the object, there is a strong image-surface cambering and astigmatism. The spokes and the rims are sharply focussed at different distances from the lens. It is expedient to put the optical bench onto a cart, since it must often be moved over several meters in order to sharply focus either the spokes or the rims, using beams either from near the center of the lens or from near its outer edge. If the convex surface of the lens is oriented towards the object, the image surface is surprisingly ﬂat, but now the strong spherical aberration makes the rims appear one-sidedly washed out towards the center of the image. (R D wheel ﬁgure (object), L D lens.)

If we replace the slit by a circular aperture, for example the lens bezel, then we observe an additional complication. At the locations of the two image points P0 and P00, we see at the same time two nearly linear formations which are perpendicular to each other: Each of the two image points has degenerated into an image streak.InP0, the streak is perpendicular to the plane of incidence, while at P00, it lies in the plane of incidence. To explain these image streaks, we can refer to Fig. 16.17. In each direction, a lens which is struck by light beams at a slanted angle can be replaced by two cylindrical lenses with different curvatures for light beams with a small opening angle. Therefore, with slanted incidence, the same effects must be observed as in Fig. 16.17b with incident beams parallel to the optical axis. When lenses in meniscus form (concavo-convex, e.g. Fig. 18.7 a) are used, at a suitable position of the aperture, the order of the concave- cambered image surfaces is exchanged. That means that the image surface nearer to the lens is due to the extended light beam which is perpendicular to the plane of incidence (that is, the sagittal beam). By combining convex and meniscus lenses, one can cause the two concave surfaces to very nearly merge, thus strongly reducing the astigmatism. In addition, the common concave surface can be more or less flattened out, so that the camber of the image surface is reduced to an acceptable level. Such compound lenses (objectives) with (18.2) ). With I circular disk on the image 18.15 const on the object side D 0 ! y y 0). It is no longer per- symmetry with respect to D '> 0 of the principal ray changes shape on the image side, so ! , to reduce the spherical aber- ! ' circular symmetry sin sin 18.4 . A coma can occur even when the elliptical of the principal ray is zero. The cross- ' , the object point lies outside the optical coma , the object point lies on the optical axis, on the image side. These angles must fulfill on the image side. The image points acquire const or 0 ! D . 0 : y y , the degeneration of the image points to image streaks astigmatism . D Image Formation and Beam Limitation 0 ! ! ! , a small angle of inclination on the optical axis. This can be accomplished by following ! anastigmatic sin sin 0 n not sine condition n spherical aberration To test an objectiveits for astigmatism image-surface camberingis employed: and for One the placesperpendicular and a degree symmetric drawing of to of theimperfect, a either optical wheel, only axis. the with spokes Whenfocus. spokes or the Often, and only the correction drawing the a is shows rim several rim, concentric can rims (Fig. be brought into sharp well-corrected objectives, even the outerbe rims projected as in well sharp as focus the onto spokes a should flat screen. the i.e. the angle of inclination 18.6 Coma and the SineIn Condition strongly reduced astigmatism and a roughlycalled planar image surface are sectional area of the beam retains its when the opening angle becomes too large. However, with side. Correspondingly, the image point degenerates to a axis, i.e. the principalaperture) has ray a (which finite passes angle of through inclination the ( center of the pendicular to the surface ofarea the of lens. the As beam a result, acquires the an cross-sectional Now, however, if at least onecross-section of of the opening the angles beam is large, retains then only the its the plane of incidence a “tail” which effectivelyties; broadens they them degenerate and to reduces a their intensi- that the two image pointsing become angles image streaks even at small open- spherical aberration has beenangles corrected. Even then, at large opening to the opening angle a certain rule for the ratio of the opening angle the focal lengths of thelens individual zone projects zones the of image the inthat lens. a is different Therefore, perpendicular size each to onto athe the surface use optical element of axis. light beams Thiscroscopes, with however a telescopes would large and make opening angle othersufficient, impossible as optical for shown mi- instruments. above in Sect. It is thus not ration for only oneRather, object this and must one be imageare done point for on other the object optical and axis. image points which Fundamental and Technical Details of 18 348 Part II 18.7 Geometric Distortion 349

B B' 2∆y ω ω' 2∆y' 1 F' λλ' 2 p = n'2∆y' sin ω' p = n2∆y sin ω

Figure 18.16 The derivation of ABBE’s sine condition. An aperture B whichservesasentrance pupil is imaged as the exit pupil at B0. For the illumination of B, we suppose that there is an extended light source some distance to the left. The indices of refraction n and n0 on the object and 0 the image sides are taken to be equal in this sketch; therefore, the wavelengths and are drawn Part II to be the same. the latter expression holds when the indices of refraction n and n0 on both sides of the lens are equal.

To derive the sine condition most simply, we refer to Fig. 18.10.There, the entrance pupil was formed by an aperture B, and its image served as exit pupil B0. The same can be seen in Fig. 18.16; however, here, the light source is not an illuminated opening, as in Fig. 18.10, but instead a light source with a large surface area, far to the left in Fig. 18.16.Wehavedrawn two collimated beams which each originate at a point on the distant light source. Some wavefronts are indicated in each beam. One of the beams passes through the center of the lens, while the other passes through a zone near the rim of the lens. The central axes of these two beams make an opening angle ! on the object side, and an angle !0 on the image side. Both beams intersect the image plane with the same cross-section (the diameter of the image is 2y0). The wavefronts are perpendicular to the rays which form the edges of the beams; they appear as straight lines on the object side. On the image side, we can see that their curvature just in front of the image plane is small enough to be neglected. The last wavefront which is drawn in on the right can be treated as ﬂat. Then from Fig. 18.16,we can read off the optical paths p D n2y sin ! and p0 D n02y0 sin !0. Since for image formation, the optical paths of all the rays that belong to C18.3 corresponding object and image points must be equal ,wemusthave C18.3. This follows from p D p0. This leads to the sine condition (18.2). FERMAT’s principle; see Comment C16.6. A detailed An optical arrangement for forming an image which obeys the sine derivation can be found condition is called aplanatic. Such a lens can thus image a certain for example in P. Drude, surface element which is perpendicular to the optical axis (and not “Lehrbuch der Optik”, Verlag just an object point) with wide-angle beams. However, a lens can de- S. Hirzel 1900, Chap. III.9. liver an aplanatic image only for certain object and image distances, This book, in its Chap. III.10, which must be predetermined during its fabrication. also contains a derivation of the tangent condition which is mentioned in the following 18.7 Geometric Distortion section.

Spherical aberration, astigmatism and coma impair the quality of the image points. They are aberrations in image definition. In addition, there are positional aberrations. They give rise to a camber of the b). . pupil as ,fulfill size Screen bout 10- 18.17 ;seeCom- 1) (compare position 18.11 n ca. 6 m of the lens material. The ca. ,e.g.inFig. n 0 50 cm ). Therefore, the lens has of the image and also its ' is the lens diameter tangent condition D 16.2 disortion and = 10 cm as pupil for = 37 cm f D ' position ca. c) or to a barrel shape (Fig. 30 cm Image of light source const ( . (In addition, the other aberrations acquire D 18.17 10 cm barrel-shaped pillow-shaped ' Object tan = 0 ' abc lens is proportional to the reciprocal of ( reproduction scale A square a centered on the optical axis has a barrel-shaped The dependence of the distortion on the position of the f Movable condenser Image Formation and Beam Limitation )). All materials which can be used to fabricate lenses, . 16.12 crater 18.18 Arc-lamp The focal length determines both the Therefore, there is a chromatic aberration both in the image well as in its a dependence on the wavelength which is of practical significance.) (light source) ment C18.3). Often, however,it the varies position for of different the zonesno pupil of longer is the be not objective. corrected. fixed; Fig. Then An the example distortion of can this situation can be seen in distortion at b and a pillow-shaped distortion at c (a is drawn with a fold magnification on a matte glassbackground) plate, preferably as a light figure on a dark (the object is for example a square lattice). the condition tan 18.8 Chromatic Aberration The focal length of asurfaces, lens but also depends on not the only index of on refraction the shape of the lens image surface and distortionther of to the a pillow image: shape A (Fig. square is distorted ei- Figure 18.18 Figure 18.17 focal length An image is free of distortionprincipal-ray at angles a fixed of position inclination of the pupil when the Eq. ( a different focal length for each wavelength. i.e. glasses, crystals and plastics,index exhibit of dispersion; refraction in increases general, withinwavelength their the decreases visible (cf. spectral e.g. region as Table the Fundamental and Technical Details of 18 350 Part II 18.8 Chromatic Aberration 351

Cf ~ 80 cm Video 18.3: K α violet “Chromatic aberration” http://tiny.cc/seggoy violet red The video shows a special 90 cm 6 m example: An arrow formed from holes illuminated from Figure 18.19 A demonstration of the chromatic aberrations of the image po- behind is projected onto the sition and its reproduction scale for image formation by thin lenses. Only wall of the lecture room us- with a thin lens is the position of the principal planes practically independent ing a glass lens with a strong of the wavelength. Therefore, only for a thin lens does the same position dispersion. The holes are illu- of the focal point imply equality of the focal lengths and of the reproduc- minated by an arc lamp with tion scale which they determine (the angle of inclination ˛ of the screen is

a condenser lens and an aux- Part II about 10ı) (Video 18.3). iliary lens, giving a weakly convergent beam, so that the slender light beams from the Both of these chromatic aberrations can be readily demonstrated using arrowhead and from its tail a simple eyeglass lens. We use the lens to image a slit onto a distant screen strike only the rim zone of which can be slid along the direction of the light beam, and put alternately the lens, while those from a red or a blue filter in front of the slit. In order to focus the blue image, the middle of the arrow strike the screen must be placed considerably closer to the lens than for the red only the center zone of the image: “chromatic aberration of the image position”. The blue image is lens. The image, which is also about seven-eighths the size of the red image: “chromatic aberration upside down, shows colored of the image size” (i.e. of the reproduction scale). If the screen is tilted borders for the arrowhead to the side (see Fig. 18.19), we see a broad, colored band instead of the and tail. The dispersion be- image of the slit. The non-physicist would be inclined to call this band tween the refraction of red a ‘spectrum’, just like a rainbow. The physicist can admit of only a remote and blue light, which gives similarity in both cases. rise to chromatic aberrations, increases near the rim As with all imaging errors, chromatic aberrations can be reduced but of the lens. Red light (with not eliminated. For this reduction, called “achromatization”, one uses a longer wavelength) is less in practice at least two lenses. For the achromatization of the image strongly refracted than blue position, convex and concave lenses made of different materials are light (with a shorter wave- required. To achromatize the reproduction scale, two convex lenses length). Slipping a red filter made of the same type of glass can be used with parallel light beams. into the optical path allows us Their axes must coincide and their spacing should be equal to half to compare monochromatic the sum of their focal lengths. Achromatic lenses consisting of two and polychromatic image formation. We can see that the lenses of the same type of glass can be found for example in the ocu- position of the red borders lar lenses of binoculars (Fig. 18.20): They allow parallel light beams fits onto those of the full red of different wavelengths to pass into the eye between the two focal light disks. points F0 with a parallel displacement relative to one another. A parallel displacement (just as in the mirror prisms in Sect. 16.11)leaves the image on the focal plane of the eye unchanged.

blue H' red

red F' blue blue 2y φ'

F' red

Figure 18.20 A demonstration of an achromatic ocular using two lenses made of the same type of glass. The second lens makes the initially divergent rays of different wavelengths parallel. This allows them to be focussed together on the focal plane of the eye. which A OLDEMAR V Mirror compensated this by mirror. Part a: At the is the aperture whose special profile is M ERNHARD , ounts relative to the image CHMIDT CHMIDT C18.4 CHMIDT mirror (B , they can change their focal lengths is large, there is an undercorrection of A . They have no chromatic aberrations, but C CHMIDT CC A and in Comment C18.4. rubber lenses of a concave spherical mirror 18.21 C SS ). The principle of the coma-free S the filming of a motion picture), with a focal ratio of nearly 1:1 Image Formation and Beam Limitation 18.15 Technology. The S M , 1879–1935). The principle of this important invention is ab during (see Sect. and reproduction scalesple continuously by up to a factor of 5 (for exam- plane. Jokingly called The development of objectives(“zoom lenses”) with is also continuously-variable remarkable.lenses, focal They which consist lengths can of be at displaced least by two differing individual am whose center of curvature is at CHMIDT S the spherical aberration of the mirror. Part b: S center of curvature serves as entrance and exit pupil.S The image points lie on the spherical surface when the diameter of the aperture a spherical overcorrection using a glass plate explained in Fig. sketched here (greatly exaggerated).corrected in After this the manner, spherical opening angle aberration ratios of had 1:1 been were readily attained. Figure 18.21 18.9 Achievements of Optical Optical technology has madesometimes it individually, possible sometimes to in reducefor lens combination. this aberrations, purpose The isa main series tool the of use individual ofoptical lenses axis. compound with lenses. Non-spherical sphericalexample surfaces surfaces These parabolic are and consist mirrors relatively a for of seldom common spherical telescopes used, lenses as and for condensers searchlights, for projection or apparatus. non- Every lens and every mirroruse. must be The precisely objectiverequirements adapted lens from to those of its of special a a telescopeused microscope objective. for A must magnifying meet reading lens a quite a magnifying different scale glass for haseral examining a methods photographs, for different and reducing so constructionbeen the forth. known, from individual but lens Gen- the that aberrations optimalsitates have of solution long of in each general special a problemof neces- numerical the calculation various which typesmirable makes progress of in skillful glass this use available. fieldsupport and for research. has Technology A thereby has particular provided madeopment success considerable of was ad- the for coma-free example the S devel- .His ,who 18.21 CHMIDT Fundamental and Technical Details of 18 C18.4. B.V. S lost his right hand atin age an 11 accident while experimenting with gunpowder, invented the telescope which bears his name at theburg Ham- Observatory in 1931. He developed the correction plate (lens) and at thetime same a process for grinding the lens. He used thethe lens lid as of an evacuated(the vessel vacuum is on theside right in Fig. 18.21, Partorder b), to in grind it toshape a while concave it was elastically distorted by air pressure, thus producing the “peculiar” profile shown in Fig. invention made it possible for the first time for astronomers to obtain wide-angle celestial photographs. 352 Part II 18.10 Increasing the Angle of Vision with a Magnifying Glass or a Telescope 353 18.10 Increasing the Angle of Vision with a Magnifying Glass or a Telescope

In considering the eye, we may continue to employ the analogy with a photographic camera for the time being. The eye can accommodate, i.e. it can produce sharply-focussed images of objects at various distances. In a camera, for this purpose the distance between the rigid

glass lens and the image plane (detector array, film or photographic Part II plate) is varied. The eye, in contrast, changes the curvature of its lens surfaces by muscular contraction, and thereby changes the focal length f 0 of its elastically-deformable lens. The range of accommodation for a normal eye stretches from an arbitrarily great distance (“infinity”) down to the “near-point distance”. This closest distance for sharp focus, requiring the strongest accommodation, can be below 10 cm in children. For adults between age 20 and 40, one finds near-point distances of around 20–25 cm, and so forth. – However, very strong accommodation is uncomfortable. For writing, reading and fine handwork, in general a distance of around 25 cm is preferred. This preferred working distance is called (rather unfortunately) the “distance of most distinct vision”. An important but complicated process is three-dimensional vision, whether directly, in a mirror, or through a water surface. Its essential aspects are dealt with in physiology. In describing even elementary optical observations, one fact must be kept firmly in mind: A single object point, alone in the field of view, can be physically localized only by two eyes.C18.5 One eye can C18.5. See also Sect. 1.4 in always determine only the direction in an unknown environment in Vol. 1: The stereoscope. which we see such an object point L, but never its distance (compare Fig. 18.22). With the aid of Fig. 18.23,wedefinetheangle of vision. The angle of vision must not fall below a certain minimum value (about 1 minute of arc) for well-known reasons (Sect. 17.3); otherwise, the eye can no longer resolve the point or separate it from its surroundings. How can we increase the angle of vision, how can we make previously indiscernible details in an object visible? Answer: We approach the object more closely. How closely can we approach it? Normally down to about 25 cm, the distance of distinct vision, with comfort. At still closer distances, persons with normal vision find accommodation difficult, and without accommodation, they would see only an unfocussed image. However, the curvature of the eyeball can be assisted by placing a convex lens in front of it (Fig. 18.24). Then the observer can approach the object more closely, e.g. to 12 cm, without any effort of accommodation, and still see a sharp image. This closer approach roughly doubles the angle of vision in comparison to the distance of distinct vision ('wi='w=o 2). Or, in other 0 L ,and N . That would ). Then the ) in front of the without any instru- 18.24 ' ) by using a magnify- loupe wi ' (or This cannot be understood in . 0 L o = w ' under water is viewed by two eyes and ap- f w/o wi L φ φ 2 2 to the point magnifying glass which appears to be shifted towards the observer. 00 L'' L L L' N y 2 Object y Object 2 The definition of the angle of vision 2 Increasing the angle of vision (to 2 An object point Image Formation and Beam Limitation raised vertically pears to be ing glass or loupe.opening Instead of the of eye, using thetrance the opening and itself image exit serves formed to pupil. by good Theare the approximation differences neglected lens as here. between the in the A en- the principal moreglass iris and detailed would nodal treatment go of points beyond the the action framework of of a this magnifying book. terms of this construction which is carried out with only one eye lead us to an object point ments or visual aids, denoted by 2 Instead, we have to carrythe out two the planes construction of separately incidence for both intersect eyes. in Then the vertical (normal) line eye. A magnifyingmit lens approaching with the object a5x, to still and 5 stronger cm, so curvature then forth.crease would its the per- The magnification angle is purpose of vision of about to by the a permitting object. magnifying a The glass closerthe magnification approach is of physical of thus the sense. the lens to It eye is in- gradually increases here decreasing with not ability the a of age constant the of in lens its of userExperienced the owing observers eye to always to use the accommodate. arelaxed, magnifying glass that with is their eye accommodatedin to the ‘infinity’. focal plane They of place the the magnifying object glass (Fig. words, we have placed a Figure 18.23 Figure 18.24 Figure 18.22 also lies on this normalbeams. as the point of intersection of the two axes of the light Fundamental and Technical Details of 18 354 Part II 18.10 Increasing the Angle of Vision with a Magnifying Glass or a Telescope 355

2φwi 2φw/o Image

fI a

Figure 18.25 Increasing the angle of vision by using a single-lens telescope, here as a simple sketch showing only the principal rays. One can imagine a frosted glass screen in the image plane; it is however not necessary. (Nu- merical example: fI D 4 m, distance of the eye from the image a 20 cm, magnification fI=a D 20x) Part II light beams coming from the individual object points enter the eye as parallel beams. The lens of the eye refracts the beams so that they are convergent and focusses their narrowest points of convergence as image points onto the retina of the eye. Often, it is not possible to approach the object more closely (an air- plane in the air, the moon, etc.). Then one can use an objective lens to project an image of the object. The image is of course much smaller than the object itself, but the observer can approach it to within about 25 cm (the distance of distinct vision) and thus increase the angle of vision. This is the principle of the single-lens telescope shown in Fig. 18.25. By adding a magnifying glass in front of the eye of the observer, the approach can be made much closer and the angle of vision still larger. This is a two-lens telescope as shown in Fig. 18.26. The objective and the magnifying glass (“ocular”) are connected by a tube (which holds them fixed and shields unwanted stray light from the surroundings). The telescope thus also simply serves to increase the angle of vision. The magnification or “power” x of a telescope is defined as the ratio “angle of vision with” over “angle of vision without” the instrument (its measurement procedure will be described in Sect. 18.13).

2φ EP Image wi 1 2φw/o AP

I II B' B 2 fI + fII ~fII

Figure 18.26 Adding a magnifying glass as ocular (lens II, focal length fII) allows the eye of the observer to be brought closer to the image and thereby increases the angle of vision still further. On the object side, we again show only the principal rays which come from the rim of the object; on the image side, however, the corresponding light beams are drawn in. The objective lens bezel serves as entrance pupil EP, and its real image B0, projected by the ocular, is the exit pupil AP. We can imagine this image to just fill the opening of the iris in the observer’s eye, as in Fig. 18.24. The eye should be relaxed when using a magnifying glass or ocular, that is the entering light beams should be parallel. (The rays 1 and 2 show that B0 is an image of B.) The magnification is roughly equal to the ratio of the focal lengths: x ' fI=fII (see Sect. 18.13). was suggested in 1611 ) between the objective 18.26 16.35 ly visible object points – called on the resolving power of lenses holds (1571–1630), and is called an “astronomi- 17.3 EPLER K Microscope. The Numerical Aperture with a Projector or a Microscope Image Formation and Beam Limitation OHANNES In order to measure the magnification ofruler a on microscope, we the lay object atrude millimeter table over of one side, the e.g. microscope towith and the the right. allow left a Then we eye, piecereadily look of while through bring it the the the to microscope right two pro- through fields eye the of looks microscope vision at over together. 130 thethe mm magnification We ruler on see is the directly. 130x. for directly-observed example ruler. We 1 Then can mm and the ocular. cal” telescope. it showsare an various methods upside-down for image obtaining of anditional the upright lenses image, object. or for There example mirror ad- prisms (Fig. equally well for the microscope asangular for spacing the of eye and two the still telescope. separate The by J 18.12 Resolving Power of the The discussion in Sect. 18.11 Increasing the Angle of Vision The generally well-known opticalcroscope’ instruments serve ‘projector’ – and like ‘mi- increase the the telescope and anglesimilar the of magnifying in vision. glass principle. –object-side The to focal In point two both, of instrumentsa an the are strongly objective object enlarged lens. essentially image is of Thison placed the lens a object. just screen then The (real projects outside image image). can the be observed When it is sufficiently large,a the sufficiently image large can angle bedistance clearly of away; observed this vision with is even thetures!). by function of observers a located projector some (slides, motion pic- The microscope, in contrast, isof intended small for objects. individual observations the upper The end image of projectedthe the by ocular microscope tube (a the (“tubus”). magnifying objectivethus glass) lens observes The it to observer is under uses approach at a“power” the large is image angle defined closely of vision. and forvision with” the The over magnification “angle microscope or of again vision without” as the the instrument. ratio “angle of The type of telescope sketched in Fig. Fundamental and Technical Details of 18 356 Part II 18.12 Resolving Power of the Microscope. The Numerical Aperture 357

Image Object Lens diameter B 2ω 2φ 2φ' 2y 2y'

2ω' ab

Figure 18.27 The resolving power of the microscope. Here, the same conditions apply as in Sect. 17.3 for the derivation of Eq. (17.1) for the eye and the telescope. On the right side of the lens, the light beams have in reality nearly parallel boundaries: The object points on the

left side are nearly in the focal plane of the objective. In the drawing, for clarity the object Part II distance a is shown overly large and the image distance b overly small. Furthermore, in the sketch, the indices of refraction n and n0 on the object side and the image side are shown as being equal. The drawing thus applies to a microscope without any immersion ﬂuid.

2' in Fig. 18.27 – must not be smaller than the angle ˛ calculated from the equation sin ˛ D : (16.23) B Then 2y0 sin 2' D or, from Fig. 18.27; D : (18.3) min B b B However, in the case of the microscope, the smallest resolvable angle 2'min is less interesting than the smallest separable spacing of two object points; that is, in Fig. 18.27 the distance 2ymin, measured in units of length. For its calculation, we read off from Fig. 18.27 for the image-side (small) opening angle !0 the following relation:

B sin !0 !0 : (18.4) 2b Furthermore, in the microscope, the sine condition must be fulﬁlled, i.e. the image-side opening angle !0 must be related to the object-side opening angle ! by the equation

n sin ! 2y0 D : (18.2) n0 sin !0 2y

Equations (18.2), (18.3)and(18.4) can be combined to yield 2y D ; (18.5) min 2n sin ! in the case that the space between the object and the microscope objective is filled with an “immersion fluid” (water or oil) with the index of refraction n,andn0 D 1. of ob- )we ,in . ), called 18.5 20.4 ! . This can sin ! bright-field il- : n m light-emitting D bright-field illumina- 21 The order of magni- NA : 0 ). For D of the light, and second by C18.6 is the index of refraction of . dark-field illumination 18.5 n is the opening angle of the light objects (secondary light sources), shows two types of construction. 5). The average wavelength 210 nm : ! 1 D 4 18.28 : ) to the coherence condition in Fig. n 1 using immersion fluids of up to about 1.4 18.5 2 94, 600 nm : illuminated ! 0 Figure D sin D n min C18.7 y ! 2 D 1 is assumed. This will give the equation an intuitively clear Image Formation and Beam Limitation . A fine Brussels lace, on the other hand, mounted on D ,sin ı NA n 70 . At the right, in contrast, the light for illumination of the object D interpretation! which Later, we compare Eq. ( ! We are already familiar withfrom bright-field everyday and life. dark-field illumination openings in We front use oflumination for a example bright window; lace this curtains provides with large At the left, theinto light the objective, passes and through thenceis the to viewed object the against eye a (a of bright thintion background, the section) or observer. with and The object is kept out ofcover the glass). objective Only (by the total(three light reflection small scattered at arrows!) or can the refracted enter surface bybright the of against the objective. a thin The the dark object section background, then or appears in tude corresponds toFigs. our 12.12 experience ff.), we in producedobjects mechanics. shadow using images water There of surfaceformation, waves. partially-immersed (Vol. we 1, found For that that the objects simplestthe must type wavelength be of of no the smaller image water than waves roughly (cf. Vol. 1, Fig. 12.20). a quantity which characterizes the objective, ( be seen from the denominator in Eq. ( Microscopic image formationobject-side at light high beams resolution have requires a that large the opening angle This shows that the resolving powertwo of quantities: a microscope First, is by determined the by wavelength The smallest spacingsolved of by two high-quality objecthalf microscopes points the is which thus wavelength only can of slightly still the less be light than re- used visible light is around 600 nm. With these values, from Eq. ( ( the “numerical aperture” . In it, jects (primary light sources)opening such angle as is a limited glowing onlyobject. filament, by the In the usable the structural characteristics case of of the in contrast, for example thebiomedical applications, usual the stained opening thin angle is sectionsof determined observed by illumination. for the type Itdenser is lenses”. provided by illumination lenses, called “con- find beams accepted by the objective and the material (air orobject (for immersion example fluid) stained thin between sections the of biological objectiveOptical material). and technology the has beenaperture able to attain values of the numerical . ), ELL 18.12 Fundamental and Technical Details of 18 with which resolving powers (smallest resolvable spacing) of typically less than 30can nm be attained. C18.7. The function of adenser con- lens was already explained in Fig. C18.6. There are more recent developments which permit the size of thepower resolving to be reduced below the diffraction limit. These include the optical scanning near-field microscope and the fluorescence microscope (e.g. the STED microscope developed by S. H 358 Part II 18.12 Resolving Power of the Microscope. The Numerical Aperture 359

To the Objective objective Cover glass Cover glass Slide J Slide

H Condenser Ring aperture Part II From the lamp

Figure 18.28 Two condenser lens systems. Both provide an extended light source in their focal planes, and this coincides with the plane of the object. The extended light source is obtained by using a convergent lens which images a (pointlike) light source in its entrance pupil. Left: A bright-field condenser. Right: A dark-field condenser with a twofold reflection at the inside surface of the cover glass. H is a cavity, J the immersion fluid (water or oil) for avoiding total reflection at the upper surface of the condenser.

Real images of the lamps B a' x' 2ω a'

s Free object distance

Figure 18.29 Measurement of the numerical aperture (sin !) of a microscope objective. A very fine circular opening, illuminated by two lamps, is laid on the object table of the microscope. The lines drawn through the small opening B are the axes of very narrow light beams. The table is moved towards the objective until one sees a sharp image of the opening with the ocular (that is, the free object distance as measured from the front surface of the objective is adjusted). Then one looks into the tubus without the ocular and increases the distance x of the two lamps until their real images a0, which lie practically in the focal plane of theÁ objective, vanish. Now we have 1=2 ! D x 2 C x2 x : sin 2 s 4 2s dark, non-reflecting velvet so that the light is kept from the eye of the observer, provides dark-field illumination.

Because of the fundamental importance of the numerical aperture of a mi- “Not the arrangement of croscope objective, a method for measuring it is described in Fig. 18.29. the lenses, but rather the bounding of the light beams Not the arrangement of the lenses, but rather the bounding of the leads us to a deeper under- light beams leads us to a deeper understanding of the microscope and standing of the microscope its resolving power. That is the essential content of this section. and its resolving power”. wi or φ lie at afor wi ' b). This 18.30 and o 18.30 = w ' terrestrial telescope II f telescope, the object distance Real images EPLER of the object point I . It permits us to periodically vary the angle of f 18.10 and indispensable for mariners. For this reason, or “afocal”: From an object point, a parallel, col- ). In this manner, an optical path is obtained which is , we refer to these angles of inclination as the angles Image Formation and Beam Limitation of the parallel light beam incident from the left. The position and telescope for distant object points (in Part a, on the optical axis; 18.26 o A demonstration experiment showing the ‘telescopic’ optical path 18.26 telescopic = w ' EPLER Entrance pupils Exit pupils limated light beam passes toemerges the from objective, and the a ocular, parallelshown however in beam with the again demonstration a experiment smaller illustrated diameter. by Fig. This is an object point lying far away on theTo optical continue axis. this demonstrationoscillate experiment, we from let above the to object below point the optical axis (Fig. motion allows us toclarity, discern the i.e. position the ofimage common the exit side. intersection pupil of withand The great all after beams the passing maintain light throughsive their beams the point parallel – telescope, on their boundaries but the anglesdifferent – before of before inclination and and relative this to after the is passingin optical through the Fig. axis the deci- are telescope. As before, termed is very large compared toimage the of focal length a of distant thetive. object objective. point Then The the lies object-side in(cf. focal the Fig. plane focal plane of of the the ocular objec- is coplanar with it 18.13 Telescope Systems In our description ofplace for optical a particularly instruments simple upand telescope an with to upright a now, image, modest known magnification there under the was name no Galilean telescope we add here a secondall description types. of telescope designs, applicable to In the usual applications of the K w/o a b inclination the centers ofsetup the is entrance similar to and that in exit Fig. pupils. The image is inverted. The optical in the K in Part b, belowlenses). it). The vertices (Threefold of the magnification principal-ray of angles of the inclination angle of vision, cylindrical φ Figure 18.30 the formation of theis exit advisable pupil to can place be aa clearly green red filter filter seen. in in front To front of mark of its the lower the rim. boundary upper rim rays, of it the entrance pupil, and Fundamental and Technical Details of 18 360 Part II 18.13 Telescope Systems 361

s= D·φw/o s' = D' · φwi

φ D w/o a' a b b' φwi D'

φwi φ w/o EP AP Entrance- Exit- pupil

Figure 18.31 The derivation of the relation between the angular magniﬁca- Part II tion and the change in the diameter D; D0 of the light beam

of vision 'wi and 'w=o (with and without the telescope), and obtain a quantitative expression for the magniﬁcation:

' Beam diameter before the telescope Magniﬁcation x D wi D 'w=o Beam diameter after the telescope f D I : (18.6) fII

The fact which we have obtained experimentally here can be readily understood: Fig. 18.31 repeats schematically the demonstration of Fig. 18.30b, but now only the beam boundary rays before and after the telescope are drawn in. Lines a and b, perpendicular to the rays, have been added to mark the wavefronts. Then we can imagine the incident beam to be tipped by a small angle into the position shown by dashed lines. a is converted to a0 and b to b0. In this process, the optical paths s and s0 must remain equal (sine condition (18.2), see also Fig. 18.16). Then we have 0 D 'wi D D 'w=o. From this, as can readily be seen in Figs. 18.30 and 18.32, we obtain 'wi='w=o D fI=fII.

According to this analysis, in order to construct a telescope, we need only set up a telescopic optical path. This can be accomplished with other optical arrangements, also, for example with one converging lens and one diverging lens. We thus can build a terrestrial telescope, also called the GALILEAN telescope (1609, GALILEO GALILEI, 1564–1642). The demonstration experiment in Fig. 18.32 shows the path of a light beam for one distant object point on the optical axis and one point below the axis. Knowledge of the telescopic optical path leads us to a simple procedure for measuring the telescopic magniﬁcation or “power”; we need only measure the diameter of a collimated light beam before and after it passes through the telescope and then we can apply Eq. (18.6).

The diameters of the light beams are the same as those of the entrance and the exit pupils. With a properly-constructed telescope, the entrance pupil is practically always the bezel of the objective lens. The exit pupil, the image of the objective bezel as projected by the ocular, is accessible only in the KEPLER telescope and its variants (e.g. prism binoculars). For the terrestrial telescope, it is a virtual image located inside the telescope tube, lting field eyes, EPLER EPLER wi steady φ 6 x, upright image). and determine the ). We hold a K 18.32 II f 18.32 I f , they carry out (unconscious) jerky individual regions within the field of fix glance telescope for one distant object point on the optical C18.8 :theeyes Entrance pupils Exit pupils Instruments A demonstration experiment showing the ‘telescopic’ optical ALILEAN Image Formation and Beam Limitation w/o φ moving is limited by some sort of obstacles, for example the frame between the objective and thetelescope ocular (cf. with Fig. its objectivelook pointed into to the the ocular sky frompupil a or as distance to of a a around small, bright 30Its cm. bright window, diameter and Then can disk, we be seemingly see measuredobjective suspended with the lens, a exit in divided millimeter front by ruler. this The ofof exit diameter the pupil of the ocular. diameter, the telescope. yields thethe magnification With demonstration a experiment terrestrial as shown telescope, in we Fig. must instead carry out diameters of the beams. axis and one point belowThe the exit axis pupil (magnification is of aocular. the virtual angle image Between of of vision the the 2.2x). objective objective-lens bezel, and projected the by ocular, the there is, unlike the K path in the G Its main advantage issmall the losses limited in light number intensity. This ofas type glass a of “night surfaces telescope glass”. is and today the still unmatched resu Preliminary remark: When the unaided eye is used, often the 18.14 The Field of View of Optical of view of a window. We view very small fields of view with Figure 18.32 telescope, no imagestructed of to give the only moderate object magnification point. (around 2 The terrestrial telescope is con- rotations in their sockets and but when the field of vieweyes encompasses are several degrees or more, our view during brief pauses. Theseand motions can shifts be assisted of by the rotations regions within entire the field head, of view butficult one to after in the maintain other. that an That overview. case, makesexample. Looking it through we dif- a see keyhole is the a separate good In optical instruments, the objective andthe the ocular essential are without lenses. doubt For practical construction of the instruments, Fundamental and Technical Details of 18 C18.8. However, among the modern “night glasses” there are also binoculars and telescopes with so-called residual light amplifiers, which make use of the infrared radiation present even on the darkest night. 362 Part II 18.14 The Field of View of Optical Instruments 363 however, they are not sufficient. Without additional lenses, the field of view is too restricted. The required auxiliary lenses are called condensers or field lenses.Examplesare more instructive than verbose “Examples are more in- explanations of the general kind. structive than verbose explanations of the general First of all, the projector, with which slides, transparencies, or film, kind”. that is typical “secondary light sources”, are imaged on a (usually large) screen1: Figure 18.33, top part, shows an incorrectly constructed projector with a light source (arc-lamp crater), slide and the objective which

forms the image. On the screen, we see only a small section from Part II the center of the slide. The field of view is much too small (and its boundaries are not focussed). The reason: Here, the bezel of the objective acts as a field of view aperture. It allows only the light from a limited angular range ˛ to pass from the lamp to the screen. The ray r has no physical significance, since no light beam is propagating in its direction. Therefore, the outer parts of the slide cannot be imaged on the screen. This can be easily corrected (Fig. 18.33, bottom): We place a large lens immediately in front of the slide, a condenser, and use it to image the light source onto the opening defined by the objective. Then all the light which passes through the slide will also pass through the objective.C18.9 The slide thus appears in its entirety C18.9. The essential char- on the screen; its edges are sharply focussed. Now, the mounting acteristic of a slide or trans- frame of the slide acts as the field of view aperture. Its image limits parency, which was presup- the field of view as an “exit window” and has the correct orientation, posed here, is that it allows i.e. in the plane of the image on the screen. light to pass through without deflecting (“scattering”) it. In general: Just as the aperture (or its image) limits the opening an- For an image which is ap- gles ! and !0 as a pupil, so do field-of-view diaphragms (or their plied to a plate of matte glass, images) limit the angles of inclination of the principal rays ' and '0 the field of view is not lim- as exit windows. ited by incorrect illumination, The condenser must be matched to the distance between the objective since the light is scattered in and the slide. For projection with varying image sizes and distances all directions. It is simply less to the screen, one requires objectives of different focal lengths. For bright. See the footnote in each one of them, a matching condenser must be available. this section. In a precisely corresponding manner, we can use condenser lenses in amicroscopeandinaKEPLER telescope. As a rule, they are combined within a short tubus with the eyepiece lens. This combination is also referred to as an ocular. In contrast to the usual descriptions, one seldom observes the image of a microscope or a telescope with a steady eye. Frequently, turning of the eyeball and shifts of the entire head are necessary. The reason is that the angular region of greatest visual acuity includes only a few degrees of arc. It is symmetric around the “fovea centralis”, the

1 For demonstration purposes, among other reasons, one can approximate a slide to a primary light source by putting it onto a surface which is emitting light in all directions. This could be e.g. a glass plate which is either itself fluorescent (primary light source) or a scatterer in the form of ‘frosted’ or ‘matte’ glass (secondary light source). ). lead to 18.12 ). – Bot- Screens max max . The bezel φ is exceeded, ' 2 2 18.12 max ˛ ' D 2 α = ˛ D ˛ , i.e. the largest usable pupil max ' Entrance , can be seen in Fig. 2 ! D Exit pupil Image of the of Image ˛ arc-lamp crater arc-lamp iting aperture for the field of view. From α Image of the of Image r and exit pupils α crater as entrance as crater for the field view of Lens bezel as aperture as bezel Lens At low magnifications, the exit pupil of a terrestrial telescope Top: An incorrectly constructed slide projector. The objective Condenser Image Formation and Beam Limitation ) has a larger diameter than the entrance pupil of a human eye. The Arc lampsArc Slides Objectives 18.32 crater as crater Arc-lamp entrance pupil entrance of the objective acts as diaphragm for the field of view. When the cross-section of theby beam two takes circular on segments the ofits shape different edges of radii). (vignetting). a The lune image (a fades figure away bounded towards the center of the entrancemation. pupil In the which example plays drawn,spot a this on decisive entrance the role objective pupil in lens covers which500 the only projects students, image a the a small for- image. 5-ampere central arc Inament) lecture lamp lamps, is rooms one quite for can up sufficient. in to Using factadd use incandescent an the (fil- entire unnecessary diameter complication ofthe the same when objective, applies projecting but to they physical condensers demonstrations; whose front surface is not freely accessible. its edges, principal rays with a large field-of-view angle The mount of the slide is the lim angle between two principallies rays as on always at the the object center side. of the The entrance apex pupil of (compare Fig. this angle tom: A correctly constructedof slide the light projector. source The (arc-lamp condenser crater)which at forms forms the an the objective image (the image, path and of its a partial opening beam angle bezel as aperture limits the angle of vision telescope and eye togetherthe make skull use of of the anthe observer. object-side entrance Its pupil center principal is, which rays.principal as lies rays The always, within determines largest the the point usable angle of of angle the intersection of field of inclination of view of the (Fig. Figure 18.34 Figure 18.33 Fundamental and Technical Details of 18 364 Part II 18.15 Imaging of Three-Dimensional Objects and Depth of Field 365 central region of the retina with the highest density of receptors. The visual acuity decreases within ˙ 2ı to half its maximum value and within ˙ 10ı to only one-fifth of its maximum value. These motions of the eye and the head have to be considered when determining the field of view.

In using a KEPLER telescope, one usually moves the eye in front of the exit pupil of the telescope as if looking through a keyhole (Figs. 18.26, 18.30). With a terrestrial telescope, the eye makes use of only a portion of the objective surface for each “instantaneous observation” (action or moment of time!). This is illustrated by Fig. 18.34 for two extreme positions of the eye. Part II

18.15 Imaging of Three-Dimensional Objects and Depth of Field

In our description of the process of image formation up to now, an image point was identified with the point where the light beam has its smallest cross-section (the ‘waist’). This corresponds to the usual practice, but is by no means always generally correct. Think for example of the pinhole camera, known to every child (Fig. 18.35). It makes use of narrow light beams without any constriction on the image side. Nevertheless, it yields good images (which are completely free of distortion and are planar). This is rather surprising. The image points, i.e. the diffraction pattern of the opening, is under similar circumstances 20 times larger in a pinhole camera with a 1 mm opening than with an objective lens of 20 mm diameter (Eq. (16.23)). But an artist can also paint very satisfying pictures using broad brush strokes. This is due to psychological processes and does not belong in this section (see also Sect. 15.2). For us at this point, the often-repeated experience suffices: High-quality images which are satisfactory for our eyes are by no means identical to images of the highest resolution. Even the most technically perfect lenses can form only an image of an object plane on an image plane. These two planes must be perpendicular to the optical axis. Nevertheless, in practice one often wants to image three-dimensional objects onto an image plane. As is well known, usually the resulting images are quite adequate: The eye, binoculars and cameras in general have a considerable depth of focus or “depth of field”. That is however due only to a peculiarity of our eyes as mentioned above; the eye does not always treat just the narrowest constriction of a light beam as an image point.

Figure 18.35 Apin- Matte glass plate hole camera Object 2y Object 2y' . 5 ,it The D min (18.7) f N C18.10 of the focal of an objective P' min a central projection square C18.11 of the lens. from the object point ). FD a 36 mm, the empirically- 6m.Thismeansthatall acceptable to the eye. It : : 18.7 1 near point distance a f b–f max D 2 f D m. At a focal ratio of f focal ratio min N a : The artist places a transparent objects always have a certain ge- 50 D D ]isthe B max fB 18.37 ; see also Comment C18.11.) D B on the image plane (in this case the focal max which is no longer . f D is thus proportional to the D N 18.36 max increases as the distance : 2cm,wefind ab min min D points a a D 1 a D ; that is, they exhibit a certain ratio between f lens diameter f 1 = f ). At the same time, all the object points which are closer three-dimensional D ) gives the physical rationale for two well-known facts: b 1 The calculation of the near-point distance 18.36 P 18.7 Image Formation and Beam Limitation ; perspective between the objects and one eye, and notes the points of f focal length b Œ W D b f N to the lens becomes smaller. Finally, at the plane; Fig. to the lens appear“disks”. Their in diameter the image plane not as points, but rather as small determined limiting value is Numerical example: For an image size of 24 mm ( length at a given focal ratio is given by derivation is shown in Fig. reaches a limiting value The objective lensesrected of for photographic an apparatusinfinity” “infinitely” and are distant imaged telescopes as object are cor- plane. Thus, object points “at the object points whichsimultaneously are with more than an 1.6 acuity mdifferently: which from The is the “depth lens satisfactory of focus” will fordistance in of be the this 1.6 imaged m eye. example to extends infinity. fromEquation Or a ( put minimum The near-point distance and a focal length of 1. Nature haswhales) to evolved be not the much eyes larger2. than of those The of the development humans. largest of35 mm photographic mammals cameras. technology (elephants led and to the production of D B D which is corrected to “infinity” (an “infinitely long” object distance). We have The combination yields the general form of Eq. ( the size and the distance ofartist objects may which reproduce are such behind a one perspective by another. An using a 18.16 Perspective Planar images of Figure 18.36 ometrical This is done as shown in Fig. screen 2 = D B f (as . of f N and b 16.7 b ). D , altitudes a 0 = P 18.7 ) of a general fb , is the “imaging )). and reordering the min 1 a , can be derived di- a P f fb 2, and widths relative aperture ); set the ratios of their b / = b f 1 f Fundamental and Technical Details of 16.13 or D D D B 18 = b B 1 b D . altitudes and of their widths equal. The second equation, and a lens is also calledstop”, the e.g. “f- of a cameraEquation lens. (18.7) is a special case (for the near-point distance f equation which applies to thin lenses. Compare Fig. 18.36 and Comment C18.11. rectly by comparing the two similar triangles on the right of the lens (with a common baseline C18.11. The first equation, C18.10. The focal ratio formula” which also holds generally for thin lenses. It was derived in Sect. terms gives Multiplying this second equation by Eq. ( Inserting this into the first equation yields the general form of Eq. ( 366 Part II 18.16 Perspective 367

Figure 18.37 A central projection for representing three-dimensional objects on a ﬂat image plane W (B is the eye of the artist) intersection of the lines of sight to various objects. The artist thus employs the pivotal point of the eye as the center of projection.

For image formation using a lens, the lens is placed between the ob- Part II jects and the screen. This is also a central projection, but with two centers of projection. They lie at the centers of the entrance and the exit pupils. Therefore, the boundaries of the light beams are decisive for perspective, also. We will document this with an impressive demonstration experiment. In Fig. 18.38, two brightly gleaming matte-glass windows of the same size are placed at different distances from the lens. One of these windows is in fact somewhat in front, the other somewhat behind the plane of the drawing. The back window is marked with H, the front window with V. The lens has a large diameter, but a narrow aperture and a thin light beam are used. As a result, both windows appear equally sharp on the screen, adjacent to each other. During the experiment, the setup remains unchanged (Fig. 18.38a); only the aperture is moved along the optical axis. The experiment is carried out in three steps: 1. The aperture is set up immediately in front of the lens (Fig. 18.38b). Both pupils are practically contiguous with the center of the lens; it serves as the center of projection. The more distant window H appears smaller on the screen than the closer window V. 2. The aperture is shifted to the image-side focal point F0 (Fig. 18.38c). This moves the object-side center of projection (the center of the entrance pupil) to the left towards ‘inﬁnity’: The two images of H and V have now become the same size on the screen.

Figure 18.38c shows the limiting case of an object-side telecentric optical path. This is often used; it is for example indispensable in a measuring microscope.

3. In Fig. 18.38d, the aperture is shifted on the image side to beyond the focal point F0. Then the object-side center of projection (the center of the entrance pupil) moves closer to the window H than to the window V. The result is that the image of H becomes larger (!) on the screen than the image of V; the perspective is inverted. We can thus vary the geometrical perspective of an image over a large range simply by shifting an aperture which limits the boundaries of a light beam. So much for the demonstration experiment. Pictures painted by an artist are supposed to be viewed from the same center of projection as that used by the artist. One should use only Therefore, when . Then, with high- Screen Screen Screen Screen 18.37 – A good free-hand attempt . in Fig. F' Aperture F' Exit pupils B Aperture V is changed by shifting the aperture which (Video 18.4) H V f' φ' = 0 for and aperture is shifted Range in which the Range in which φ' H V 2 and φ' Aperture Aperture, 0.5 cm dia Aperture, 0.5 H from this point. In Part c, for clarity only the beams . This presents no difficulties: In the objective are drawn. At intermediate positions between b and c, V V φ Lens, 7.5 cm dia. Lens, 7.5 cm H f = 38 cm f and Entrance and exit pupils H φ H Image Formation and Beam Limitation F F V V FF' F V eye and place it at the position of the distant entrance pupil Object Object Object one looking at a photo, thecenter pivot point of of projection the eye should be placed at this quality paintings, one hasdimensional view. the impression of seeing aIn natural a three- photographiccenter camera, of the the exit pupil principalpupil to serves rays the as film the propagate or image-side detector. from center The of the center projection. of the exit H H ca. 15 cm cm 30 ca. cm 65 ca. ca. 7 m H windows Matte glass HV and from the bottom of V each 3 cm high V Object Object V and Object H ) is to be thought of as being some distance in front of the plane of the page, the other and To the upper edge To the center To V )! Note also H The influence of the light-beam boundaries on perspective. Part a: The experimental setup. One = 0 for φ 2 Image of the aperture as entrance pupil 18.15 . d c b a Fundamental and Technical Details of 18.38 ) somewhat behind it. Parts b–d: The size ratio between 18 H at the inverted perspectiveglass) in about Part 30 d: cm in Holdbox front a larger of lens than your the of eye closer and around edge. observe 10 a cm box diameter of and matches. 20 You cm will focal see the length more (a distant reading edge of the Video 18.4: “Perspective” http://tiny.cc/5eggoy The two letters the entrance pupil lies as a virtual image to the right of the aperture are somewhat fuzzy due to the limited depth of focus (Sect. the ‘free-hand experiment’ described in the caption of Fig. coming from the top of ( limits the light beam boundaries.of On projection. the object The side, the lens center “looks of at” the entrance the pupil objects always serves as the center of the windows ( Figure 18.38 368 Part II 18.16 Perspective 369

C Part II

Figure 18.39 Objects of the same size at different depths are projected from the centers A, B,andC onto the same image plane W. The points of intersection of the lines of sight with the image plane are the same for all three examples shown here. – With these figures, we indicate the distortion of perspective which results when a picture is viewed from the incorrect distance: An image cast from the center B appears foreshortened when seen from C, and stretched out when viewed from A. lenses common today, the entrance and the exit pupils nearly coincide with the center of the lens. One thus needs only one center of projection, as in the schematic of Fig. 18.38b. Furthermore, the film is always placed near the focal plane of the objective. Then we find the following rule:C18.12 One should always view a photograph with C18.12. We make special one eye and at a distance between the pivot point of the eyeball and mention of the rule described the photo which is equal to the focal length of the camera lens that here for viewing photos. As was used to take it. For focal lengths of around 25 cm and upwards, POHL says further on, we this can be readily done. However, the usual 35 mm cameras have have forgotten how to see the considerably shorter focal lengths (not to mention the electronic cam- three-dimensional structure of photos, and treat them eras in smartphones, tablet computers etc.), at most a few centimeter. as if they were only two- In this case, one has to use a magnifying lens between the photo and dimensional images. the eye; then the correct distance can be maintained. When this rule is observed, every photo shows a surprisingly good sculptural effect and lifelike perspective.

Good-quality magnifying lenses should be designed for the gazing eye, and the distance between the pivot point of the eyeball and the lens should be fixed by a suitable shape of the lens mount (bezel). For an x-fold linear magnification of the photographic print relative to the negative, the eye-to- photo distance should be equal to xf . Unfortunately, this condition can be fulfilled for only a few members of the audience in a large motion-picture theater, and the seats where it is met vary with the magnification of the film.

All pictures when seen with one eye, both those painted by artists and those taken by a camera, should give a three-dimensional impression, even when viewed from the incorrect distance, although the perspective may then be distorted. The depth of field appears too shallow when the viewing distance is too short, and too deep when it is too long (Fig. 18.39). But today, our visual senses have all been dulled by the flood of images in magazines, television and digital media. ; the 18.2 short https:// D a . )) ( , but the depth 21.11 and should there- 16.13 f . Use this drawing to (Eq. ( P 0 would however make the f = f 1 es again emerge. For example, D three-dimensional b ) contains supplementary material, which = . Only when the circumstances are The online version of this chapter ( 1 C images in the focal plane of a telescope the image – lens distance). (Sect. a = D b ). flat surfaces shows the graphical construction of the image 18.1 Image Formation and Beam Limitation two-dimensional which belongs to the object point Exercise 16.3 0 Figure P . An ocular lens with a focal length of object – lens distance, see also derive the lens formula 1 fore be viewed with the correctf depth of field from this same distance Electronic supplementary material Exercises 18.1 doi.org/10.1007/978-3-319-50269-4_18 is available to authorized users. We have given up on seeingrience pictures them as only three-dimensional and as expe- unusual does the true abilitywe of see our ey the of field of allor the boulevard objects is is especially foreshortened. suitable forprojected The by seeing an long this objective effect; axis with the of a image long a is focal street length as viewed through the ocular lens as magnification of the angle of visionfeat equal the to purpose one, and of thus the would telescope. de- Only with an ocular lens of focal length can the anglesmagnified. of vision But be then increased itshort, and makes so the that the image we thus viewing see distance all unavoidably depths too inStill the image more as foreshortened. impressivethrough is the telescope a ‘backwards’ and useThen reversal we its see of objective the as depth this an ofThe ocular. field experiment: stretched ocular, out with in a We itsimage, very comical short look and manner. focal we length, viewa projects distance. it a through two-dimensional the objective fromMore details much on too image formation, great inible particular images the of production invisible of objects, can vis- be found in Sect. point Fundamental and Technical Details of 18 370 Part II ufc mt aito naldrcin ihnahmshr,s that so hemisphere, a d interval within time directions the within all in radiation emits surface itiue nsae nodrt nwrti usin edtc the detect we question, this (Sect. answer radiometer a to with order radiation In space? in distributed e t resraeae ed be area surface free its Let nudrtnigo nrytasotb aito,wehro o it not to formation. or us image whether lead by radiation, also accompanied by will is transport point energy decisive of understanding This an principle. explanatory an as ea iha with metal emLimitation Beam and Energy Radiation nFig. In without points op- object an to the corresponding be presumed in to have elements, comment points” we surface “image but or the fact; regions experimental treated small always as have image we tical now, to Up Angle: Opening and Radiation 19.2 the used have empha- been we not have instead, diagrams ray sized; instruments, and fabrication optical lens and of formation details image the of discussion whole our In Remark Preliminary 19.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © distance hsfr u esol ecrflt orc t nraiy radiation reality, In it. emitte correct always d to is careful energy be non-zero should of we but far, thus utemr,bt h iesoso h mte,d emitter, the incidence”). of (“normal dimensions radiation the the both of Furthermore, propagation of direction the to h inldtce yterdoee ie t eeto)corresponds deflection) the its (i.e. to radiometer the by detected signal The ftercie,d receiver, the of ihteui at locle h nrycret enwvr the vary now We current. energy the called also watt, 1 unit the with A . C19.2 ain flux radiant 19.1 R . Law Definitions. ttelf,w hwd show we left, the at , at black matte ahmtclpoints mathematical A 0 d r hsnt esalcmae otermutual their to compared small be to chosen are , W P hc al ntercie,ta senergy/time is that receiver, the on falls which olsItouto oPhysics to Introduction Pohl’s t teisteeeg d energy the emits it , ufc.I csa an as acts It surface. A 0 C19.1 hssraei retdperpendicular oriented is surface this ; iiigboundar limiting 15.3 hthscue sn problems no us caused has That . rmafiiesraeelement surface finite a from d A L .I evsa small a as serves It ). AMBERT sasalgoigpeeof piece glowing small a as W o sti energy this is How . , A emitter e flgtbeams light of ies https://doi.org/10.1007/978-3-319-50269-4_19 sCosine ’s ,aswellasthose t front Its . receiver . ain u d flux radiant The role. no plays surface the of shape overall the that uniisd quantities The C19.2. for especially defined are which quantities analogous the of detailed treatment A 15. Chap. in duced intro- already were radiation, by transport energy to lated re- are which here, discussed quantities the of Some C19.1. fd of derivative (second meant is quotient differential true a d h ufc lmnsd elements surface The values. nonzero but small d d area emitter the from ample ln the along nto fteradiance the of inition def- the (e.g. cases some In ilb ie nCa.29. Chap. in given be will account, into eye human the by brightness of perception Measuring%20Light.pdf down, Ash- Ian by article the e.g. see quantities, radiometric expos general a For ˝ A A 0 otercie rad area receiver the to W r hsnt es small so be to chosen are nti hpe represent chapter this in P www.helios32.com/ ...d w.r.t. 19 # W infinitesimal ieto o ex- for direction P W # P light A ,d n d and propagates A ,takingthe ,d to of ition A ˝ L 0 A e A and ). ), and 0 371 .

Part II . - V EIN 19.2 (19.1) H is varied refers to which is M 2 #

R ϑ = 0

OHANN

), under different I ~ ϑ

Ẇ d is the surface area of is as expected from : ). 0 ϑ 0 A 2 R A (d R d 28.4 ˝ and # which is emitted from a surface 0 A A P d W B The proportionality of the radi- cos in contrast, can be derived only ,d , A # A d )). At the left is a schematic of the emission e M L D , and find (where we denote the proportion- # A' # into the solid angle d P d W . It is an open cone. Its apex is at the center of ’s co- d # and C19.3 ˝ . It is generally obeyed only approximately. (It is ’s cosine law, described in 1760 by J in the direction ): , 1728–1777). An example is shown in Fig. d R R e 1 , # L 0 ϑ A Measuring the radiant flux d AMBERT ), which was found empirically, the ratio d The angular Ω cos ’s law is however exact for a small opening d , for example from a tungsten-ribbon lamp ( ,d . (The points 0 A A AMBERT )(L , right). The large AMBERT A 19.1 L 19.1 19.1 A d AMBERT The unit of a solid angle is, like the unit of every angle, simply the number 1. L RICH here only within the horizontal plane. The influence of the quantities d angles of inclination geometry, at the right the experimental arrangement. The angle circles are calculated from Eq. ( the solid angle element d a radiometer, e.g. a thermopile ( simple geometrical considerations. ant flux to from experiments called L distribution of the radiant flux reaching a detector of area d were measured as shown in Fig. sine law). ality factor by It is often expedient to(sr). give See the also number the footnote 1Sect. at 1.5. in the this end connection of the Chap. name 17. ‘steradian’ More details are given in Vol. 1, 1 size of d Figure 19.1 Figure 19.2 Solid angle d the source of the emissiontemperature, a through “black the body wall radiator” of (Sect. a cavity at a uniform In Eq. ( . 19.1 is is the can R .Inthe and the A R # A , whereby R Radiation Energy and Beam Limitation 19 range over the whole hemisphere in front of d following examples, however often allowed to rotate only within the horizontal plane, as in Fig. angle between the surface normal vector of d C19.3. In general, vector 372 Part II 19.2 Radiation and Opening Angle: Definitions. LAMBERT’s Cosine Law 373

Figure 19.3 The “projected emitter dA' 2 0 0 area” dAp D R d˝ . Here, d˝ is the Solid angle dΩ' solid angle under which an arbitrarily- shaped emitter is seen from the position ϑ of the receiver. In the special case of R dA a planar emitter dA as sketched here, the projected emitter area is dAp D Projected emitter area R2d˝0 D dA cos #. 2 dAp = R dΩ' the surface element dA, and thus at the center of the emitting surface. Its base is the irradiated surface element dA0, and thus the area of the Part II 2 0 receiver. Furthermore, dA cos # D dAp D R d˝ can be seen from Fig. 19.3 to be the projected or apparent emitter area,andd˝0 is the solid angle under which the emitter is seen from the location of the receiver.

The proportionality factor Le in Eq. (19.1) (in differential form) is then

2 P d W# Le D I d˝ dAp it characterizes the emitter. The quantity Le is called the radiance of the emitter. Its unit is 1 watt/(steradian m2) D 1watt=m2.

The experimental introduction of the radiance Le is by no means limited to the special case of a planar emitter for which LAMBERT’s cosine law holds. Imagine for example in Figs. 19.1 and 19.2 that the emitter is a glowing cylinder which is perpendicular to the plane of the page. Then the radiant ﬂux dWP that it emits is independent of #. The angle # is varied only within the horizontal plane (as in Fig. 19.1, right). Instead of Fig. 19.2, we would then have a circle with the emitter at its center. We P would ﬁnd dW D Led˝dA, and thus the proportionality factor (the radi- 2 P ance) would be Le D d W=d˝dA.

The quantity P dW# Radiant ﬂux in the direction # I# D D (19.2) d˝ Solid angle or

I# D Le dAp D Radiance times projected emitter area (19.3) characterizes the radiation of the emitter in the direction #, and therefore, I# is called the radiant intensity in the direction #. Its unit is 1 watt/steradian (W/sr).

The same radiant intensity I# can be produced by emitters of very different sizes. At white heat, a small surface area is sufficient; at a dull red glow, a large area would be necessary. . A emits ,reg- ,pre- 0 (19.4) (19.5) 0 2 ). The A A R ϑ ˝ 19.5 d must first d emitted from R 0 hichgoesfromtheemitter . Its unit is ! A D P w r W 0 to its area. The A 2 #: is small compared to ϑ d is irradiated with the d 0 ϑ R 0 A from the receiver # A ! R (right), a spherical surface sin 2 2 of the emitter R sin was supposed to be practically 19.4 # I 0 2 A A perpendicular A d d to the emitter) irradiation intensity D e ’s cosine law, receives the radiant L R # (receiver) according to Eq. ( till supposed to have a small area d or d 0 0 axis, dashed in the figure). Each has an R A' A D ). It is sent out by the emitter, which has . D r arriving e A . At the right: a spherical surface construction ! L e # d P 2 AMBERT P L W . All of the radiation which reaches W 0 e d Receiver area (Distance D A L at normal incidence). Then this receiver irradiance Radiant intensity Incident radiant flux 0 ring ω D (i.e. the A d R , and, apart from its center, it receives the radiation ˝ not which is located at a distance d ! D D = ; the surface element d A ! R , at the left, a large circular area At the left: The calculation of the radiant flux d 0 and a radiance P C19.4 W P . A W A d , we use a construction as in Fig. A 2 0 d d d 19.4 A D D ! I (emitter) and transmitted to area of pass through this virtualof sphere. narrow circular zones We which decomposedirection vector are its ring-shaped surface and into concentric a around the series Derivation: To calculate the radiant fluxarea which is received by the receiver in front of the receiver e E A is given the name 1 watt/m suming the validity of L radiation with the radiance perpendicular to the radiation direction.however, We the now emitter relax is this initially limitation; s (or at an angle (i.e. opening angle small emitter d flux d isters the radiant flux d the distance to aid in the calculation. an area d In Fig. Thus far, we have assumed that the receiver d Figure 19.4 The receiver, the small irradiated surface element d quotient , E energy flux density . Radiation Energy and Beam Limitation e E 19 C19.4. The irradiance is thus an i.e. the energy per time interval and surface area arriving at a receiver. It isthe denoted literature in by the letter or 374 Part II 19.2 Radiation and Opening Angle: Definitions. LAMBERT’s Cosine Law 375

Figure 19.5 A large emitter A which emits at 0 the radiance Le irradiates a small receiver dA ω' (Eq. (19.8)). For this light beam, we cannot A dA' draw any sort of simple wave picture.

Each of these ring-shaped circular zones receives a radiant ﬂux (from Eq. (19.1)) equal to

dA0 P ring Part II dW# D L dA cos # D 2L dA sin # cos # d# D 2L dA sin # d.sin #/: e R2 e e Integration of this ﬂux over angles between # D 0 and the full opening P angle # D ! yields the overall power dW! which arrives at the circular receiver area A0 (i.e. Eq. (19.5); cf. Fig. 19.4).

The radiant ﬂux in Fig. 19.4 which is radiated out from the emitter dA and is incident on the circular area A0 of the receiver attains its P ı maximum value dWmax for the limiting case of ! D 90 . This value is used to deﬁne the radiant exitance of the emitter by the equation

dWP Radiant ﬂux emitted to one side M D max D : (19.6) e dA Emitter area

When LAMBERT’s cosine law holds, we ﬁnd from Eq. (19.5)theradiant exitance of the emitter to be:

dWP M D max D L : (19.7) e dA e When emission to both sides is considered, a factor of 2 must be included. If we reverse the direction of light propagation (Fig. 19.5), the large area A acts as emitter and the small area dA0 as receiver. We again denote the radiance of the emitter as Le. Then the radiant ﬂux arriving at dA0 is

P 0 2 0 dW!0 D LedA sin ! (19.8)

.Derivation as above/:

P Equation (19.8), i.e. the dependence of the flux dW!0 on the opening angle !0, can be illustrated by a demonstration experiment. We use a secondary source as emitter, e.g. a circular area on a high- quality matte white projection screen which is irradiated by an arc lamp (compare Sect. 26.10 and Fig. 26.14). Then, for varying open- 0 P 0 ing angles ! , we measure the flux dW!0 . ! can be varied in two different ways, namely by changing the diameter of the circular area, or by changing the distance between the emitter and the receiver. In Sect. 19.3, an application of this important equation (19.8) will be discussed. ) kW. 19.8 6 (19.9) .The is equal 10 A 5 : .Fromthe 0 19.5 A into Eq. ( bout 200 km thick, 0 . We insert these .) 3 ! in Fig. / 10 0 ! of an emitter d 7 : wi : 0 4 ! 2 : A 2 m 2 D kW and of sin with a lens 0 and the 0 4 sin m kW solar constant ! A averaged over the surface of the e A 10 d ght source (the emitter) and the e d = e L in Image Formation 367 M P 3 L : W : 1 e d 6 . The usable value of the radiance can at the receiver, but never the available e D E D D D E e e e wi E L E P of the sun’s surface delivers about the same W 2 d , we find the radiant flux The index ‘wi’ means . 19.4 An image of the emitter can never radiate at a higher , a lens projects an image d . The latter is a characteristic quantity which describes of the Sun Irradiance e L 19.6 : 2 The radiation which is emitted by the sun comes from a layer a power in this image is collected by a receiver of area d values of the irradiance and compute the radiant exitance sun radiance schematic in Fig. to 16 minutes of arc, and we have sin power as a large AC turbogenerator with a rated power of 1 The sun’s disk has anfrom angular the diameter earth. of Therefore, 32 the minutes opening of angle arc as seen (Astronomers call this irradiance the in the most favorablebe case conserved in (absorption-free an lenses imagingderived thermodynamically process. or from mirrors) the This Second result, only in Law, which more will can detail be also in discussed the be following. In Fig. the emitter. radiance than the emitter itself For comparison: 25 m 2 which is increasingly cooler towardsthe the paths outside. of Near the theAt edges radiation the of through very the edge, the solar one cooler disk, thereforearound measures regions 60 % (of of lower course the values depending of on layer the the become wavelength) radiance longer. than at the center of the solar disk. 19.4 The Radiance In numerous cases, between the li irradiated surface (the receiver),With there these is lenses, a lens orchange or with only a the any series irradiance kind of of lenses. imaging apparatus, one can 19.3 Radiation from the Surface The sun irradiates the surfaceand of (neglecting the absorption earth losses at inance perpendicular the of incidence atmosphere) with an irradi- Radiation Energy and Beam Limitation 19 376 Part II 19.4 The Radiance Le and the Irradiance Ee in Image Formation 377

Figure 19.6 Ir- radiation of the B ω ω' receiver dA0 dA wi wi dA' with and without a lens. The lens increases the ω'w/o opening angle !0. dA which goes from the emitter dA to the lens, passes through it and 0 produces the image dA . Here, the lens itself acts as an emitter with Part II a still-unknown radiance Le;x. Its exit pupil sends out the radiant fluxC19.5 C19.5. Since the radiant flux reaches the lens with P D 0 2 !0 dWwi Le;xdA sin wi (19.10) a distribution corresponding to LAMBERT’s cosine law, 0 onto the image area dA , as in the schematic in Fig. 19.5. Here, we the lens surface on its other have implicitly idealized a limiting case: We have neglected radiation side again radiates with this losses by reflection at the lens surfaces and by absorption in the glass same distribution, only that – of the lens, and have taken the radiant flux to be the same in front of owing to refraction – the di- and behind the lens. In this limiting case, we can combine the two rections are changed and the equations (19.9)and(19.10) to obtain radiation is again focussed. The result is that the radia- 2 ! D 0 2 !0 : LedA sin wi Le;xdA sin wi (19.11) tion emitted from the lens can be described by a constant We make use of light beams with a wide opening angle for the imag- radiance Le;x. ing of dA onto dA0. Therefore, the sine condition (18.2)mustbe fulfilled:

2 ! D 0 2 !0 : dA sin wi dA sin wi (19.12)

C19.6 0 Equations (19.11)and(19.12) , when combined, yield Le;x D Le, C19.6. Since dA and dA 0 an important result: For the image dA , the disk of the lens radiates are areas, the sine functions with the same radiance Le as the surface of the emitter; both surfaces occur here as squares. appear to have the same “brightness” (see Sect. 29.7). This fact will first be verified in a demonstration experiment (Fig. 19.7). What, however, changes energetically as a result of image formation? It is the irradiance Ee.InFig.19.6, top, the lens (with a sufficient di- !0 ameter) can irradiate the receiver with a larger opening angle wi and thus produce a higher radiant intensity at the position of the image than would be possible for the emitter dA without a lens (think of a “burning glass” !). We use Eq. (19.8) to calculate the irradiance at the receiver in both cases in Fig. 19.6, that is the quantity

dWP E D D L sin2 !0 : (19.13) e dA0 e !0 D !0 !0 D !0 With the lens, we have to set wi; without the lens, w=o. Then we obtain the ratio of the two irradiances with and without 2 A D m), o (that = 11 ı 0 w (19.14) 10 ! )forthe 5 : kilowatt/m 1 4 (both appear to 19.14 D (surface density) 10 of the lens is not R 3 f ). Therefore, for : 3 6 D 10 emitter A radiance of up to about 50 e : L . In spite of its triviality, this 0 wi 7 To a distantTo eye : A 0 ! wi 0 4 w/o ! 1 A ! 2 d A 2 D in the focal plane of a lens of 10 cm consist of two identical frosted-glass sin sin B B 0 w/o C )). The focal length D ! and o = 29.7 wi (for normal incidence and neglecting the A ; w 2 ; e 2 e B E E on the surface of the earth, the irradiance is is, the smaller the angular region from which the ra- 0 f A 37 kW/m to 5 mm, and place : 77). As a result, we find from Eq. ( : 1 B 0 , for example from phosphorescent materials, clearly visible A A comparison of the radiance of a small emitter with the ra- D D which passes through the lens, but the ). Then the large opening of the parabolic headlight mirror radi- ). Due to its great distance from the earth ( o = radiating with the same radiance as the C w 0 wi ; flux e ! 3 19.3 E Here, as always, we assume the same material in front of and behind the lens, for be equally “bright” (see Sect. at the lens surface remainsinstead constant). of An a analogous lens experiment ising with shown surfaces a in mirror d the bottom figure. In order to make small glow- to a large audience, wereflector put them ( at the focal point of an automobile headlight ates with the same radiance as the small area d experiment often surprises even professionals. is sin the earth is irradiated with the very small opening angle 16 minutes of arc (that is sin diameter. We then observe the emitters fromlens a great area distance and see the large diation that emerges from theradiant lens surface originates (this indeed reduces the important. The longer only a surface element d losses of about 50mirrors, % we can in produce the opening angles atmosphere). With lenses or concave The sun radiates with a radiant exitance of example air. 3 (Sect. alens Figure 19.7 disks which are illuminated from behind.ified, After we their use similarity two has circular been iris ver- diaphragms 1 and 2 to limit the diameter of to 10 cm and of diance of thefigures): large Two area similar emitters of a lens which forms its image (top and middle Radiation Energy and Beam Limitation 19 378 Part II 19.5 Emitters with Direction-Independent Radiant Flux 379 irradiance of the solar image: Â Ã : 2 kW 0 77 4 kW E ; D 1:37 D 3:7 10 : e wi m2 4:7 103 m2

In order to attain a similar irradiance without a lens or a concave mirror, we would have to approach the sun so closely that the solar disk would reach from the horizon up to 10ı past the zenith!

With a focal length of 1 m, we can project a solar image with an area of 2 !0 D ı 0.6 cm . With the opening angle wi 50 , the radiant ﬂux in the solar Part II image4 is thus 0:6cm2 3:7kW=cm2 2 kW. This power is the same as the power of an electric arc lamp carrying a current of 40 A at a voltage of 50 V.

19.5 Emitters with Direction- Independent Radiant Flux

LAMBERT’s cosine law (Eq. (19.1))is,aswehaveemphasized,an empirical limiting case. It holds exactly for a small opening into a “black-body” radiator, as we mentioned above. Planar, matte black surfaces with strong scattering or diffuse reﬂection are a good approximation to a black body, no matter whether their radiation is excited thermally or by some other means, for example as ﬂuores- cence.

A black body and a planar, matte surface have a common property: For both, the extinction constant, i.e. the ratio of the radiant flux that is not reflected or re-emitted, to the incident radiant flux, is independent of the angle of incidence.

A very different limiting-case law is found for the radiation from the interior of a flat, transparent body.Fortheradiant flux in the # direction, we obtain 0 P dA dW# D L dA D L dA d˝; e R2 e i.e. the radiant intensity in the # direction, P dW# I# D D L dA ; (19.15) d˝ e is independent of the angle of emission #. A graph of the emitted radiant flux (Fig. 19.8)showsone circle with the emitter dA at its center and not, as in LAMBERT’s cosine law, two circles located symmetrically to either side of the emitter (Fig. 19.2). This

4 E. W. TSCHIRNHAUS, 1651–1708, mathematician, owner of a farm at Kies- lingswalde, near Görlitz, and member of the Paris Academy from 1682 on, constructed a ‘burning mirror’ in 1686 with an opening diameter of 2 m and a focal length of 1.3 m, made of polished copper, and used it to melt materials. r in- e L ϑ has an A' d # Aperture I Radiometer I Ω II Fluorescent laye Fluorescent d ; C19.7 ϑ e L D # : If we look from a graz- :In A # # d cos # A I ’s cosine law holds; instead, , there are equal : Perpendicular to the emitter sheet light II # Ultraviolet is the emitter. The two volumes are of AMBERT acts as emitter, and under the angle of in- II

I ϑ Radiant intensity

~ I ~

Ẇ d Projected emitter area d A d and in the rhombus shows a suitable arrangement which minimizes disturb- illustrates how, with this arrangement, the radiant inten- Demonstration of a directionally-independent emission inten- I The production of a radiant intensity ,thevolume # with increasing angle of emission 19.8 19.9 0), the volume light D Ultraviolet # important consequence: The radiancethat is of the the quantity planar emitter surface, ing angle at the surfacecent layer of with the an emitter almost sheet, dazzling luminous we density. see the thin fluores- creases is not constant, as when L numbers of fluorescent molecules the rectangle which is independent of the emission direction sity. The emitter is ain sheet of the uranium visible glass whichflections range is at by excited to the strongly-absorbed fluorescence surfacesand ultraviolet of carbon the light. disulfide, sheet, whose it index (Tosame of is as prevent refraction that immersed re- of for in the the a glass) fluorescent mixture light of is the benzene ( sity becomes independent of ing reflections. Figure limiting case of direction-independentplemented radiant in intensity various can waysfluorescence radiation be for from a im- a clearin planar glass sheet. Fig. emitter, The most right-hand image simply with Figure 19.8 Figure 19.9 equal size, and thus containting an molecules, equal number indicated of as independently dotsfluxes emit- in simply the add, figure. sincethe the Their fluorescence glass emitted radiation. radiant sheet is completely transparentThis to independence of the direction of the radiant intensity clination ), see )and L 29.12 (second footnote), . luminance L 29.2 29.3 Radiation Energy and Beam Limitation 19 Table and also Eq. ( Sect. C19.7. For the concept of “luminous density” (now called 380 Part II 19.6 Parallel Light Beams as an Unattainable Limiting Case 381

Figure 19.10 One of the many possible forms of an X-ray source with a hot cathode. The cone C concentrates the electrons onto a small spot on the anticathode A.(M is a metal tube, and F is a glass radiation window)

A X-rays F C M K M Part II

A direction-independent radiant intensity can also be found at the planar anticathode of X-ray tubes (Fig. 19.10). The reason: the accelerated electrons can penetrate only into a thin surface layer of the anticathode, while the X-rays themselves, in contrast, can pass through the material almost unimpeded. A practical application: One can use the X-rays which are emitted nearly parallel to the surface of the anticathode to obtain a sharply-deﬁned focal spot of high radiance (a “streak focal line”) through perspective foreshortening (W.C. RÖNTGEN, 1896).

19.6 Parallel Light Beams as an Unattainable Limiting Case

According to all experimental evidence, “parallel light beams” or collimated beams can be only approximately obtained in practice. The reasons for this are already well known: 1. Every light source has a finite extension, however small it may be. ! Such a source can emit only beams with a nonzero opening angle , C19.8. Even the light beams no matter which of all possible arrangements of pupils and lenses is from high-quality lasers have used. a very small but finite open- 2. Every light beam exceeds its geometrically-constructed bounding angle owing to diffraction aries due to diffraction.C19.8 at their exit pupils. For example, the laser beam which We can now add to this list: A light beam with mathematically strictly ! D was used in 1969 to measure parallel boundaries would have an opening angle of 0°. As the distance from the earth a result, its radiant flux as calculated from Eq. (19.5) would be zero. to the moon had a diame- For all these reasons, we should, strictly speaking, refer only to quasi- ter on the lunar surface of parallel light beams when discussing experiments. around 1.6 km. uny hnrfr nyt h ipito htrange. that of midpoint the to trains wave only monochromatic refers then quency’ hyawy aeacrepnigfrequency corresponding a have always They aetan flmtdlnt r eeial called generically are length limited of trains on Wave section the why to is measures that such particular; of optics. in treatment can light, the for we postponed taking have case by fulfilled, the we is only well This patterns sufficiently measures. interference special not fixed spatially are clear-cut, conditions obtain two these If two between interferences dent produce we can trains wave such 2. radiation. the of wavelength the to odtos hc a sal eflle oago approximation: good 1. a to fulfilled be usually can which conditions, aegop rpgt otepito bevto,a mu as observation, of point the to propagate groups wave ih sal nldsabodrneo ih rqece,u ohl ftevisible the of half to up frequencies, light of range broad spectrum! a includes usually light hti iha with is that 01PeiiayRemark Preliminary 20.1 Interference 2 1 Figure Groups Wave of Interference The 20.2 framework the (Chap. within waves detail of in topic general treated the is of interference 1, Volume In .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © ment. n ruh nti example, this In trough. a and of sist centers between interference lustrates aegop ie ipersl:Teitreec aihswhen vanishes interference The difference path result: the simple a gives groups wave ,then foemeasures one If called be preferably should They onlk aecenters wave Pointlike aetan fulmtdlnt n igefrequency. single a and length unlimited of trains Wave mteswt h aefeuny ..towhistles. two e.g. frequency, same the with emitters N m 20.1 I idvda ae”o wvlt” ahoewt aecrest wave a with one each “wavelets”, or waves” “individual (Sect. and scle their called is rmPitieWv Centers Wave Pointlike from igecolor single hw oe xeiet(o.1 et 21) til- It 12.12). Sect. 1, (Vol. experiment model a shows II 29.10 htoclaea h aefeuny h ruscon- groups the frequency; same the at oscillate that s h ifrnei h egh ftoptsaogwihtwo which along paths two of lengths the in difference the , 2 ). m ahdifference path sa nap hieo wording: of choice unhappy an is , olsItouto oPhysics to Introduction Pohl’s ftewv rusbcmslre hnthe than larger becomes groups wave the of ..terdaeesms esalcompared small be must diameters their i.e. , 1 antb produced be cannot N aegop hc r mte ytwo by emitted are which groups wave single-frequency D . .Tesproiino hs two these of superposition The 5. 12 .Tee ecniee two considered we There, ). .Theterm tpe fterwavelength their of ltiples range , https://doi.org/10.1007/978-3-319-50269-4_20 nafiieexperi- finite a in h em‘fre- term the ; aegroups wave monochromatic Monochromatic nywith Only indepen- Strictly . , 20 383

Part II 0 at the wave center, . M recognized as 3 N b wavelengths : Along the line B D Physikalische Blätter same ω 2 m (violet). He also pho- m 20.1 OUNG a 0 Y I II ttingen and lived in London where HOMAS To accomplish this redirection, group from the , each other without a fixed phase m (red) and 0.4 the wave groups have a phase dif- II a was the first to determine the wavelengths of one wave centers (emitters), but rather two and yone has I ). He found for example the OUNG ). A mechan- 20.7 ı , for case suggested that reflection by mirrors, diffraction, O 180 ; wave crests meet wave crests and troughs meet (a “beam splitter”) can , the phase difference between the two wave groups ı , 1773–1829, studied in Gö independent b 0 , so that wave crests from one group fall on wave D M ı O ); at the right (case b), they D ı OUNG Specular reflection by a semi- The interference of two wave ' OUNG 0 180 iting. In 1802, Y of the wave groups themselves. Or, put differently, we can Y Y D ' D N ' HOMAS ' HOMAS T , 208 (1961)). T and were then split and redirected. frequency from groups which were emitted as early as 1807 that one should not superpose wave groups of the same surface and produce short groupsillary waves, of like cap- those that ever seen when watching raindrops falla onto puddle individual water droplets fall onto a water ical example: At the locations troughs. In case are shifted relative to eachwavelength other ( by a half Figure 20.1 individual spectral regions, byshaped making glass use plates of (Sect. interferenceends fringes of the in visible thin spectrum to wedge- tographed be the 0.7 interferencepaper fringes dipped from into ultraviolet a5 silver light nitrate as solution! early (See as R.W. Pohl, 1803 using transparent plate be used to split onegroups wave group into two groups: At the left (casewere emitted a), simultaneously the (phase groups difference 3 observe interference fringes only up to an order In general, waverelation. groups follow Then thedomly direction between the of extremes theof sketched interference symmetry in fringes Fig. variesference ran- Figure 20.2 length troughs from the other. Averaged overof time, maxima there and are minima equal along numbers aaverage no given direction, maxima so and that minima we (i.e. see no on interference the fringes). In order to get around this problem, T he had ainterests, medical and he practice. also maderoglyphic an important wr He contribution to was deciphering Egyptian a hie- natural scientist with unusually broad is Interference 20 384 Part II 20.3 Replacing Pointlike Wave Centers by Extended Centers. The Coherence Condition 385 and refraction, or some arbitrary combination of these, should all be equally applicable.InFig.20.2, for example, a wave group which is incident on a glass plate is “split up” into “transmitted” and “reflected” wave groups. When the reflection occurs at normal incidence, the front and the back sections of the same wave group can be superposed to give standing waves. This special case of interference was demonstrated in Vol. 1, in Fig. 12.10 for transverse surface waves on water, and in Fig. 12.47 for longitudinal sound waves in air. A standing electromagnetic wave was demonstrated in this volume in Fig. 12.28 (see also Video 12.1). Standing light waves are shown in

Fig. 20.25. Part II

20.3 Replacing Pointlike Wave Centers by Extended Centers. The Coherence Condition

Arbitrary phase differences can always be rendered harmless by using YOUNG’s method when the wave groups are emitted from a pointlike wave center as a statistically distributed sequence over time (Sect. 20.1!). Such a wave center can be either a single emitter (Fig. 20.3, top), or many small, neighboring emitters which are independent of each other (Fig. 20.3, bottom). In both cases, there is no difference between the wave groups which propagate along the directions 1, 2, or 3. This lack of dependence of the wave groups on the direction of propagation of the radiation is lost, however, when the diameter 2y of the region over which the numerous emitters are distributed is no longer small compared to the wavelength. Then, an extended wave center of diameter 2y can replace a pointlike center only for the radiation emitted within a limited angular range 2! (Fig. 20.4). Its size is determined by the inequality called the coherence condition: 2y sin ! =2 : (20.1)

Figure 20.3 Point-like wave centers, 2 i.e. 2y . The upper image shows just one center, the lower image shows 1 a number of independent emitters of the same frequency, e.g. the excited atoms in a ﬂame which emit light. 3 2

2y 1

3 )): ce”) nguish 18.5 3 2 1 oduce can be used y unknown phases ω ω 1. This second inequality, how- . coherence condition to disti shows a radiating surface area ω ω ) t ω ω 20.5 sin sin spatial y y then it is called ‘coherent radiation’ 2 2 ), . ), Fig. magnitudes than those at the point in direc- It simply limits the permissible path difference Video 16.3 )asthe ( 20.2 20.1 ). 20.1 4 y 20.1 2 20.1 Eq. ( different y 2 coherence condition, (an optical emitter, e.g. a piece of glowing metal, the win- 2. At this point, we should mention the connection between emitted by the bottommost surface element along direction 2 is y The radiation from a light source of diameter 2 The = light through the objective and into the eye of the observer. . This path difference must be small compared to the lengths of the 20.5 temporal ! For the derivation ofof Eq. width ( 2 plane waves from a distantis source). divided into We individual imagine surface thatwhen elements this there marked emitter are by surface irregular dividing and lines.plitudes unknown among Even variations the of the surface phases elements,plane and the waves the resulting am- radiation at will avariations pr distant in point their along phasesdistant and the point along amplitudes. direction the direction 2. 1, The Butand with same there, amplitudes the is unknown have resulting true time oftion an 1: equally Theelements paths in traversed by direction the 2elements rays depend and are emitted on not by the the thein locations same Fig. as different of those surface the in individual direction 1. surface The path of the ray dow of a gas-discharge lamp, or a slit which is irradiated from the left by sin Some authors refer to Eq. ( y for the occurrence of interferencesuperposed fringes between the wavewave groups, groups as that shown are in to Sect. be it from a ever, does not characterize aangular property of range, the as radiation which in is Eq. limited ( to a certain only if the opening2 angle of thethe light coherence beam condition and the obeys resolving the power ofOne a coherence can microscope condition (Eq. distinguish ( anonly object if it from (when made its inincoherent surroundings some manner in to act a as microscope a primary image light source) projects instead of the radiation from a single pointlike wave center (“point sour herence condition derivation of the co- 4 Figure 20.5 It plays an important role,If the in radiation obeys particular inwithin interference this experiments. angular range Figure 20.4 )by y spatial co- is demonstrated by http://tiny.cc/xdggoy Interference 20 Video 16.3: of the slit which serves as light source (of width 2 rotating it around the optical axis (5:20 minutes). The meaning of herence changing the effective width “Diffraction and coherence” 386 Part II 20.4 General Remarks on the Interference of Light Waves 387

longer by 2y sin ! than that of the ray emitted by the topmost surface element. This path difference changes the phase ' of the waves in direction 2 by ' as compared to those emitted along direction 1. A phase difference of ' between directions 2 and 1 can be neglected only if 2y sin ! =2. This is just Eq. (20.1), the coherence condition.

20.4 General Remarks on the Interference of Light Waves

All of what we have discussed in Sects. 20.2 and 20.3 is purely for- Part II mal geometry; it holds for any kind of waves. With our knowledge of these conditions, we can produce and understand the many forms of interference phenomena with light waves. Although interference phenomena in optics indeed offer nothing fundamentally new, we must nevertheless treat them in detail for three reasons: 1. Interference effects involving light waves play an important role in science and technology. 2. They give rise to long-known phenomena, for example the lively coloration of soap bubbles and thin oil films on water. 3. Using the interference fields of light waves, we can obtain cross- sections on a plane perpendicular to their propagation direction (e.g. a projection screen, a frosted-glass plate, or the object plane of a lens) and can recognize the form of such cross-sections immediately. It is expedient to distinguish among longitudinal, transverse, and oblique observations. These terms are defined in Fig. 20.6.

Oblique observation ( Plane of view for longitudinal observation (L)

L L O

O ) ) T

I D T 0 II 2 6 8

4 observationTransverse (

10 13 12 12 10 8 6

Figure 20.6 Model experiments for the definition of longitudinal (L), transverse (T), and oblique (O) observations of the interference of two wave trains. Two wave trains drawn onto glass plates are projected together. At the right, the spacing D of the two wave centers is an even multiple of =2; at the left, it is an odd multiple of =2. The image at the right was first drawn by THOMAS YOUNG in 1801/02. The numbers on the axes are the orders, for minima at the left, and for maxima at the right. . - in HO 2 20.8 S OUNG 25 mm). ): Their . Here, : (which 0 ). Then and ! ). Y 20.9 17.2 D 1 can be con- 20.5 S y ! and 20.16 just a few meters .2 Screen 16.9 20.4 : One must often make mm is still smaller than 20.7 . One thus obtains two 4 0 S , dashed), these two beams 10 ) 5m )!). When the opening is irra- onal Interference ). Such an opening may however 20.7 20.1 2 1 , and also Comment C12.3 in Vol. 1. 20.1 S S ! 20.2 sin important tip for the experimental demon- 2 Slits of y ’s interference experiment from 1807 (red 0.1 mm0.1 width, spacing mm 0.7 2ω with Two Openings . From that point on, one can capture the 2 ), are not only of historical significance, but S OUNG 1m Y Videos 16.3 and 20.1 20.7 and 0 ,sothat2 C20.1 S 4 relative to its surface normal (compare Fig. 1 S arc lamp; see the end of Sect. ! mm. ( K width Slit of (Fig. 10 4 0.25 mm D HOMAS 5 T 5 as Wave Centers: Transverse Observation Field 10 K : 3 , serve as wave centers or point sources. They are illu- 5 : OUNG 3 Y 20.7 sidered to be the pointlike wave center or “point source”. To conclude, we give onestration more of interference and diffractionuse of phenomena irradiated openings (circularstead apertures of or pointlike wave slits) centers as (Sect. waveradiate sources like in- a point source only within a limited angular range 2 fulfills the coherence condition of Eq.diated from ( all directions, theby axes of some these angle angular rangesthe may surface be of inclined the opening perpendicular to the inclined axis ! 2 See the footnote at the end of Sect. MAS Fig. minated from the left byare approximately emitted planar by waves. ait These light waves is source a which lamp is that emits at light a through distance a of slit around 1 m; = separate beams ofconstruction light. (the beam Using axes a in strictly Fig. geometrically-drawn ray do not overlap, so that theybeams cannot diverge interfere. as In a reality, however, result both of diffraction (Sects. Thus, the two light beams overlap in Fig. true distribution is illustrated by the model experimentbeyond in the Fig. slits filter light, interference fringes on a screen from any point within the spatial 5 used two openings (circular holesfrom or slits) one to select original two wave group. groups These openings, called 20.5 A Three-Dimensi The classical interference experiments, described in 1807 by T Figure 20.7 are also still of practical importance today (Sect. The interference patternsin is shown as a photograph in Fig. OHL HOMAS .P (beginning HOMAS spatial ’s interference exper- ’s interference http://tiny.cc/xdggoy Interference OUNG OUNG 20 Y Y ence” The video shows T Video 16.3: “Diffraction and coher- ple experimental setup used here.) ference is still seen in one of the secondary maxima, is unavoidable with the sim- is observed at the principal maximum of a single slit. (The fact that even withsecond the slit covered, inter- the interference pattern onto a wall for demonstration to a large audience. Interference also shows T iment, but now using a laser as light source and projecting “Interference” http://tiny.cc/9eggoy The first part of the video the video, the experimental setup is explained. Video 20.1: era for registering the image. Both red and blue filteris light used. At the beginning of demonstration at 4.00 min.), using a slitminated illu- by a halogen lamplight as source and a CCD cam- a three-dimensional region and not only on thetion observa- screen. superfluous, since fields are always wants to emphasize that the interference occurs in tial” in the characterization of a three-dimensional interference field is in fact C20.1. The concept “spa- 388 Part II 20.5 A Three-Dimensional Interference Field with Two Openings as Wave Centers 389

S2 Part II Figure 20.8 Two model experiments illustrating YOUNG’s interference demonstration. At the left: The divergent light beam coming from one slit; at the right: The superposition of the two beams which emerge from two slits. The left image was obtained in the same way as the one in Vol. 1, Fig. 12.29, right side. In order to produce the image on the right, two glass plates, each containing the left image, were placed one above the other with an appropriate shift. interference ﬁeld.C20.1 The fringes shown in Fig. 20.9 (actual size) were photographed at a distance of 5 m in transverse observation. For the angular spacing ˛m of the maximum of m-th order, we ﬁnd:

m sin ˛ D (20.2) m D

.D is the spacing of the two slits S1 and S2/:

There are numerous variations on this experiment: For example, in Fig. 20.7 we could leave out slit S0 and replace slit S2 by a mirror image of slit S1 (with a mirror in the dot-dashed plane of symmetry in Fig. 20.7. (H. LLOYD, 1837)). Or we could deﬂect the waves emerging from the slits S1 and S2 using ﬂat prisms (“FRESNEL’s biprism”), so that they pass along the line of symmetry, simplifying their superposition (A. FRESNEL, ca. 1820). Examples will be described below.

Figure 20.9 A section of the inter- Order m of the maxima ference pattern (from m D6to –6 –4 –2 0 +2 +4 +6 m DC6) which is observed with YOUNG’s experimental arrangement at a distance of 5 m on a screen at normal incidence (actual size, red ﬁlter light, a photographic positive)

–6 –4 –2 +2 +4 +6 Order m of the minima . (20.3) .This partial .When ! becomes . It covers : ! of the light . The light ˇ layer of air R 20.10 y C 20.10 C is a lamp which sin 2 A d K , and furthermore, ). ! D in Fig. 2 20.3 ˇ ! mthick,sin ! as wave centers. We pro- ) with planar, parallel sur- cos C d : ˇ C . Then the light source can ˇ A 6 sin d A 2 2 sin 2 D d ˇ ,wehavesin2 ! can be made still smaller than the thickness , thus obtaining Eq. ( mirror images cos A ! d is too large, the fringes disappear. A large- /= z C 0 S ; circular interference fringes on the screen are C from an observation screen. )); that is, it acts as a “point source” of light. We A II A . ’s interference experiment has a disadvantage: The relative to : D C 20.1 of slit and ! y I shows a plate (of thickness OUNG 20.10 in Front of a FlatMirror Plate Images with as Two Wave Centers: Longitudinal Observation sin 2 Y 20.10 , whose opening angle is denoted as 2 For small values of the angle Of course, the experiment could also be carried out using a thin Derivation: For sufficiently thin platesfrom and Fig. neglecting refraction, we find we can neglect This has the advantage that HOMAS T beams the result. The coherence condition need be fulfilled only for the plate is thin, we find could for example use afor small the Hg photograph of discharge the lamp; interference this pattern was in the Fig. case the whole wall of arequires large no lecture adjustments hall. at This all. impressive demonstration have a diameter ofcondition several (Eq. centimeter ( and still fulfill the coherence is shielded on its sides and behind by the small box beam is strongly divergent. It isback reflected surfaces from of both the the plate; frontthe therefore, and plate two the light to beams the propagatewave screen. from centers The two mirror images of the lamp serve as extremely small, of the order of 10 source must be kept smallIf in the order to width fulfill 2 the coherence condition. visibility of thebe interference demonstrated fringes to is a not large sufficient audience. to The allow diameter it 2 to can be obtained using two duce them in asurfaces. first experiment by usingFigure a flat plate with parallel diameter light source requires a very small opening angle faces at a distance For thin plates, e.g. a mica sheet about 40 20.6 The Spatial Interference Field Interference 20 390 Part II 20.6 The Spatial Interference Field in Front of a Flat Plate with Two Mirror Images 391

Screen x C20.2. This simple interference experiment, whose pattern covers the whole wall of a large lecture room, is among the most impressive AA 2ω demonstrations of the inter- 2ω β β ference of light waves. POHL Lamp R published it for the ﬁrst time K in 1940 in Die Naturwis- C senschaften 28 2ω , p. 585. The artful inclusion of his assis- Part II d tant in the picture makes the C impressive size of the inter- z ≈ 2d sinβ d I ference pattern clear even 2

Mirror images

= in a book illustration. Later, of the lamp II D other textbooks contained references to this experiment, often calling it “POHL’s interferometer”. For clarity, the thickness d D 40 m of the plate is drawn much too large in comparison to the diameter of the lamp ( 1cm).The mirror images of the lamp are in fact shifted by only a very small distance relative to one another. The diameter 2y of the lamp, which enters into the coherence condition, is nevertheless sufficiently small so that one can observe Figure 20.10 The interference experiment illustrated here produces a spa- the interference rings quite tial interference field by making use of a plate with planar, parallel surfaces close to the surface of the and divergent light beams. The picture shows a cross-section of the result- plate. The opening angle 2! ing interference field where it intersects the screen (the distance between the of two rays emerging from lamp and the plate is a few centimeters, while the distance between the lamp and the screen is several meters. Longitudinal observation as illustrated in the lamp becomes greater Fig. 20.6). One can capture “segments” of the interference rings with a matte when the screen or a matte glass screen close to the surface of the plate if the diameter 2y of the lamp is glass disk is brought closer to sufficiently smallC20.2 (Video 20.2, Exercise 20.1). the plate.

Video 20.2: “POHL’s interference ex- of a mica sheet. Then we can use even a carbon-arc lamp as light source periment” (incandescent light!) Furthermore, with the air layer, the minor disturbance due to double refraction in the mica is absent. (It can be seen in Fig. 20.10 http://tiny.cc/kfggoy . below the arrows at the top of the lower image).

The angular spacing ˇ which belongs to an interference ring of order m, that is the path difference D m, is denoted by ˇm. Then for . . 2 # for we ˇ 6 (20.4) after very small . The inner- ! = d doubly-reflected 2 was produced and . # D = . It forms a wedge due around 10 years d N 2 20.11 D ) can be obtained from the : ; and finally also the phase jump ). The light source is at the m d RESNEL 2 m 20.12 till begin with an interference ex- . 20.32 D 20.11 by putting the two reflecting surfaces soap-bubble film . m , and in addition refraction is neglected, as C20.3 y and ˇ ! 20.11 cos , to a sufficiently good approximation 20.13 d ’s setup (Fig. , ’s interference demonstration (see below), the 20.12 RESNEL of rings is limited. We obtain RESNEL .F N Front of a Wedge PlateMirror with Images Two as Wave Centers: Oblique Observation OUNG .Y H T to gravity, with its thicker base at the lower end. Strangely enough, manyperiment textbooks which s was described by A. F The air wedge can be replaced by a One can observe interferencebut also not with only transmitted with beams. singly-reflected Then the light direct beams, and the beams interfere. The amplitudes of theirresulting wave minima groups are are however thereforemode rather not of unequal, observation as and can dark also the in be as the used with following in sections. most reflected of light. the experiments described This beside each other rather than behind each other. This arrangement is arrangement sketched in Fig. This approximation neglects small differences in the angles of inclination find: It is hardly necessary totion. darken the lecture room for this demonstra- angular extension of the interference field is here independent of It is determinedthe by wedge the arrangement, we diameter can of make the the angle plates. As a result, with and obtain large, widelysources visible with interference a large patterns diameter by 2 using light also in the later Figs. rays separated only by the angle 2 of the waves uponunimportant in reflection the from above connection, an optically denser material, which is quite most ring has the largest order, namely The number 6 The air layer (orduced “plate”) described with above a can wedge be shape conveniently (Fig. pro- In contrast to F 20.7 The Spatial Interference Field in an air layer of thickness photographed using this setup. Its area on the wall was around 1 m upper left. The interference pattern in Fig. , or 25 . 25.8 and 25.7 Interference . This phase jump plays ı 20 Sects. a role in several placesthe in quantitative treatment of interference phenomena involving reflections in the following. The effect is treated in detail in Chap. C20.3. When a light wave is reflected at the boundary of an optically denser material, i.e. at the boundary of a material with aindex larger of refraction, it experiences a phase jump of 180 392 Part II 20.7 The Spatial Interference Field in Front of a Wedge Plate 393

1 Meter K

2ω= 2ϑ Part II

I II

Figure 20.11 An interference experiment with two glass plates placed one behind the other (an air wedge). The wave centers I and II are represented here by mirror images of a large light source K (e.g. the crater of an arc lamp with a red ﬁlter). No adjustments whatever need be made. One places two thick glass plates, for example squares 7 cm on a side (or the bases of two right-angle prisms) on top of each other, with a thin metal-foil strip clamped between them on one side to produce the wedge shape.

Figure 20.12 An interference ex- Screen periment with two adjacent glass plates (FRESNEL’s mirror experiment, 1816). The wave centers I and II are represented by the mirror images of a light source in the form of a narrow slit S0.Theadjust- K ment is difﬁcult: The surfaces of the 2ϑ ca. 5 m two mirrors must not form a step S0 at their point of intersection. This 2ω=2ϑ would give an additional path difference and would make the setup usable only with long wave groups, ϑ such as those from a sodium-vapor discharge lamp, but not with incandescent light. A section along the length of the image on the screen is

similar to the image shown actual ca. 0.5 m size in Fig. 20.9. II I . ˇ A The aper- illumi- ition C act as “point ,wethenuse ω 2 Plate with plane-parallel surfaces K Then the large K β B Focal plane ’s experiment. The angular of the lamp . As a result, the wedge angle II the rays which are reflected by # 2 RESNEL and , at the left, we see the opening of the aperture the An illuminated screen as light source d I Bsorts . This however sets an upper limit on the , in turn reducing the visibility (intensity) hought experiments. Here, it is an 0 # Mirror images of images Mirror S 20.13 D at the image plane. according to their angles of inclination e ! E . itter for t ). In Fig. 20.10 d ω 17.1 2 of the light source , the mirror images y β of a Pinhole Camera Plate with plane-parallel surfaces A ) must be obeyed for 20.10 (Sect. ω cannot be reduced too far; at the same time, the coherence cond 2 20.1 I diameter 2 however rather disadvantageous for F of the interference pattern. range of its interference field is only ( # II camera Pinhole A 20.8 Interference in the Image Plane In Fig. an accessory which is indispensable for image formation, an ture B sources” of spherical waves.these In spherical the waves figure, are on sketcheddirection each in of side as the two rays light radii (light-beamscreen, in of axes). illuminated the for example rays by could Hg-vaporact discharge be as lamps, the reversed. would light source. Instead of the “pointlike” lamp a pinhole camera. The aperture the plate (the beam axes) These in turn determine theflections, path which differences then produced give by risethe the to irradiation two interference re- intensity maxima and minima in C The production of circular interference fringes in the image plane behind an β . The shape and position of the extended light source are not important. It could ω 2 B the aperture the An illuminated screen as light source Mirror images of images Mirror always be replaced by a plane-wave em nated screen which is setrecognize up the parallel connection to with the Fig. reflection plate. This setup allows us to clearly aperture Figure 20.13 Interference 20 394 Part II 20.9 Interference in the Focal Plane of a Lens: Longitudinal Observation 395 20.9 Interference in the Focal Plane of a Lens: Longitudinal Observation, Curves of Equal Inclination

Behind the small aperture of a pinhole camera, the luminous density (see Comment C19.7.) of the circular interference fringes is small. Therefore, we replace it by the large aperture of a lens and use its focal plane as our image plane. In this type of arrangement,

the aperture (independently of the thickness of the plate) defines the Part II boundaries of two parallel light beams with the angle 2! D 0. In Fig. 20.13, on the right, besides the axes of the two beams, we see four wavefronts sketched in at two points along the beams. By far the most convenient arrangement is to use the relaxed lens of an eye. The walls of the room, its furniture etc. are irradiated with the light from several Hg-vapor discharge lamps and the observer looks at a mica plate (e.g. about 0.15 mm thick) from an arbitrary position. The eye of the observer may approach the plate very closely. The interference fringes are formed not on or in the plate, but rather on the retina of the observer’s eye as the image of an infinitely distant plane. They are completely absent without the observation instrument, in this case the eye of the observer, in contrast to the three-dimensional interference fields in Figs. 20.10 and 20.11. An objective observer sees the dark interference fringes as a pattern on a luminous surface, and they seem to be localized on the surface of the reflecting plate. This phenomenon is not explicable in terms of physics, similarly to e.g. the inverted vision in Fig. 15.3.

An analogous example: We see the sky within a small mirror at a distance of ca. 1 m as a bright area. If then for example a wire grid is held midway between the mirror and the eye, we will see the grid as a dark pattern which subdivides the bright area.

To conclude this section, we add a quantitative supplement: The path difference D m D 2d cos ˇm of each pair of rays is given for an air layer by Eq. (20.4). For a layer with an index of refraction n ¤ 1, we ﬁnd q 2 2 D m D 2d n sin ˇm (20.5)

(m D order of the interference maximum D integer. D m holds for D 1 maxima, m 2 for minima; derivation in Fig. 20.14).

The path difference for a given layer (d D const) is determined only by the angle of inclination ˇm. Therefore, we refer to interference , γ cos d lines : III = are de- Ã l

UMMER β γ sin

β γ

γ by an amount tan

β β ’s rings

cos

sin sin ˇ:

d d

2 2 = . ˇ a n n sin sin Â , 1849, O. L d EWTON D 2 2 n

curves of constant thick- D q d 2 . D ). If the AIDINGER ; and sin , the upper surface of the air wedge 20.5 tan ;

ˇ 2 (W. H ˇ ˇ 2 2 0, (and is therefore observed only with ) and is thus not sin 20.11 sin n 2 by a considerable amount according to This is only rarely n ) The explanation becomes simple only for d sin 2 sin ˇ 1 . We then refer to 21.12 20.5 C20.2 aperture!), we can consider the interference ˇ q 1 n

s D colors of thin ﬁlms nd d 2

cos 2 small D D cos a than the material in front of the The derivation of Eq. ( These play an important role in research and technol- n and thus pass from one interference fringe to its neigh- nl ), in order to change the path difference 2 D C20.4 20.5 . Then we set to obtain reflecting surface, then the reflectionadditional produces path an difference. 1884). scribed. Imagine that in Fig. were replaced bylower a surface weakly at its convex center. the surface case which that the just lightand touches falls the on the plate the is plate viewed or perpendicular layer to at its normal surface incidence, material in the layer orof plate refraction has a larger index Figure 20.14 ogy. of importance (e.g. in Fig. taken into account in Eq. ( When the incident and reflectedlar monochromatic light to is the perpendicu- plate, that is fringes from layers whoseof surfaces equal are layer not thickness plane-parallelness to be With extremely thin layers, e.g. soap-bubbleetc., films, oil the films interference on water fringeswe become must very vary wide. the To angle observe them, curves of constant inclination Eq. ( equal to a lens behind a boring fringe. For this reason,example when daylight, observing with we natural often light, see for We large then surfaces refer in to a single the bright color. The colors of thinbook. films are Their correct includedany explanation in other is every interference phenomenon. however school Often, more physics N difficult text- than for - UM plate in . EHRCKE 22.7 Interference -G 20 MER Sect. C20.4. See also the L 396 Part II 20.10 Sharpening the Interference Fringes. Interference Microscopy, MÜLLER’s Stripes 397 20.10 Sharpening the Interference Fringes. Interference Microscopy, MÜLLER’s Stripes

In Volume 1, we ﬁrst discussed the interference of waves from two wave centers (or “point sources”). Then we treated interference of waves from three and more wave centers, and ﬁnally from many wave centers arranged on a lattice (Vol. 1, Sect. 12.15). In this process

of increasing the number of wave centers, we found that the inter- Part II ference fringes became sharper and sharper, without changing their positions (Vol. 1, Fig. 12.40). Experimentally, this was demonstrated using two setups (Vol. 1, Figs. 12.65 and 12.66). For light waves,we initially show just one experiment. A detailed discussion will follow in Chap. 22. Figure 20.15 is an extension of Fig. 20.13. The two surfaces of the plane-parallel plate are made into semi-transparent mirrors by adding a layer of evaporated metal. Then, collimated light beams (that is, 2! D 0 from infinitely distant sources) can reach the aperture B after not just two, but a greater number of reflections. Here also, this aperture determines the diameters of the light beams and their angles of incidence ˇ. The essential aspect of this setup is the lattice-like series of images of the aperture, one behind the other. In Fig. 20.15, we see the resulting dark fringes on a light background. C20.5. To explain this com- For practical applications, particularly for spectral apparatus, one car- plementarity, remember ries out the observations with transmitted light. This yields light that looking from above, interference fringes on a dark background. one of two interfering light waves experiences a phase shift of 180ı (at the bound- As usual, the interference patterns for incident and transmitted light are complementary to each other, i.e. when they are superposed, the ary of the optically denser fringes cancel each other to give a featureless illuminated areaC20.5 material), while for the trans- (cf. Sect. 20.12). mitted light, there is no phase shift (see Comment C20.2). How we can carry out the observation using transmitted light is de- How such a phase shift can scribed in practical terms in the caption of Fig. 20.15. exchange light and dark within an interference pattern Sharp interference curves of this kind are used for example for incan be very clearly seen in terference microscopy (J.A. SIRKS, 1893). With this technique, one YOUNG’s two-slit experiment can investigate samples in which the optical path (Sect. 16.3), that (Fig. 20.8, right): Imagine is the product of the index of refraction n and the layer thickness d, that the waves emerging from varies somewhat from one microscopic region to another. A simple D one of the slits are delayed example (for the special case of n const) is the measurement of the by a phase of 180ı; then the thickness of thin films (Fig. 20.16). A shift of an interference fringe = light and dark regions are by 1 100 of the fringe spacing corresponds to a height difference of exchanged. 3 109 m D 3nm(usingmonochromatic illumination, 600 nm). For the application of interference microscopy, an old technique has again become interesting, namely the spectral decomposition of interference fringes which were produced using incandescent light and lie transverse to the long axis of the entrance slit of the ) with ghbor- A Plane- parallel air layer C d ). The sketch applies which was described in ˇ cos

d 1+2+3+4+5+... S III mirror surfaces of two glass plates curves of constant inclination . The thick air layer is located between the β B , Focal plane ˇ light is used, one exchanges the lens for the m. For ABRY ,these 1 : 0 semi-transparent B 20.13 54321 D and A. F S transmitted high-resolution spectral apparatus An illuminated wall screen as light source ) The sharpening of the interference fringes through the use of ÉROT Interference mi- .P H the aperture aperture the . light. If Mirror images of images Mirror 22.7 Exercise 20.2 observed with reflected light.dark The interference fringes on a light background are segments of large circles (use a ruler to check( this!). are curves of constant inclination, with a large angle of inclination clarity, the silvering on thenot plates drawn. is As in Fig. two mutually parallel, croscopy. At left we seeprofile the of stepped a thin airduced layer the which interference pro- pattern seenat the right. The thicknessevaporated layer of the is ing beams (for an air layer, this difference is 2 many collimated light beams with equal path differences between nei mirror image of theof aperture a closest plate to in thethe the scheme plate. form of of the In an this air way, layer for several the centimeters thick, case we arrive at 1897 by C reflected to the observation of concentric circles ( (which are slightly wedge-shaped tominated avoid screen disturbing is reflections). usually Thethat replaced illu- is by a under condenser investigationSect. lens in with its the focal light source plane. More details can be found in Figure 20.16 Figure 20.15 Interference 20 398 Part II 20.11 The Lengths of Wave Groups 399

Figure 20.17 On the right: red violet MÜLLER’s stripes in a continuous spectrum; left: The proﬁles of the thin air layers which produced the interference patterns. (First published in 1807 by Th. YOUNG as hand-colored drawings) Part II

spectral apparatus. In this method, the continuous spectrum of these “MÜLLER’s stripes” is scanned. These are colored curves of the same fringe order m. Each profile transverse to the slit axis corresponds to a particular shape of MÜLLER’s stripes. Figure 20.17 shows two examples. Stepwise changes in the layer profile produce steps in the stripes. When the surfaces are silvered, MÜLLER’s stripes also become extremely sharp. Using these stripes, S. TOLANSKY (1907–1973) was able to measure steps on single-crystal surfaces of 1 nm, i.e. of molecular dimensions, with an incandescent light source. He achieved the power of electron microscopy by using simple interference microscopy.C20.6 C20.6. For measurements of layer thickness in the nanometer range, in addition to interference methods, 20.11 The Lengths of Wave Groups various other techniques are used, among others the Using red-filter light, we can observe interference fringes out to an quartz-oscillator balance order of m D 10, i.e. with a path difference of D 10.From method and scanning tun- Sect. 20.2, this allows us to draw conclusions about the lengths of nelling microscopy (STM). the wave groups: The wave groups of red-filter light must consist of Each of these techniques has about N D 10 individual waves (i.e. wavelets, each with “one crest its specific advantages and and one trough”). disadvantages, depending on the application intended. Interference fringes of much higher orders m, with path differences of up to many thousands, sometimes even more than 106, can be obtained with the radiation emitted by some metal vapors excited electrically or thermally to light emission. This can be rather conveniently seen using the light from technical Na-vapor lamps (an electric arc between electrodes made not of carbon, but rather of sodium). We can attribute wave groups of considerably greater length to such light sources; in the visible, they are in the range of 0.1 mm up to 1 m. They consist of around 1:5 102 to 1:5 106 individual wavelets. Light with long wave groups is called “monochromatic”. Wave trains of practically unlimited length are emitted by the light sources called “lasers”.C16.4 How should we imagine the wave groups of incandescent light, that is the light from a glowing solid body (an arc lamp, an incandes- 20.19 ). ), and is , the unit of 20.18 15.5 , observed us- mm. Both plates 3 20.18 steradian andVol.1,Sect.1.5) 19.2 can be used; however, the plates are . The zero point of the abscissa marks . The results of the measurements are 8 20.11 20.19 ), and to some regions not at all. Interference minima in red-filter light minima red-filter in Interference sr for NaCl, the best-known of the alkali halides. Interference minima with incandescent light watt

0 2 4 6 8 cm

29.9 5 10 intensity Radiant 27.1 Position the of thermoelement in the observation plane The same interference pattern as in Fig. Interference fringes, produced using red-filter light with an air . At the sharp edge of the air wedge, the two plates are in 7 (power/solid angle) of the radiation reflected from the two # I Compare Fig. This radiometer is therefore not selective for radiations in different wavelength now made not ofis glass, superficially but similar rather to of glass,ation, lithium but but fluoride. it also transmits the This not neighboringultraviolet material only spectral visible regions, radi- the infrared and the the plates remain in “optical contact”of at the the minima thin is edge reduced of by the multiple wedge. reflections. The width are rectangular and are groundreflections. to In be addition, flat a wedges, linear in order facette to is avoid ground disturbing into the upper plate so that shown graphically in Fig. the position of the “opticalsity contact”. Starting there, the radiant inten- wedge 28 mm long, whose thickness increases up to 10 wedge surfaces increases untilvalue: it reaches Instead an of approximatelyonly many constant two interference flat fringes, maxima. we find The in result: Fig. An interference field produced To test the setup, wefringes on first a employ light red-filter background appear light. on the Dark screen interference (Fig. thus a physical radiometer “optical contact”; there, no reflectionof occurs the light and into there two is partial no beams. splitting For the observations, instead of themoelement eye of which an has observer, we been use a blackened ther- with soot (Fig. regions like the human eye,matic which hues! reacts Sect. to some regions in varying ways (chro- 7 8 cent filament lamp, theof tiny a carbon candle, particles etc.)? insetup the described To hot answer in in flame this Fig. question, gases the simple interference Figure 20.19 Figure 20.18 a solid angle; see the footnote in Sect. which are in “optical contact” at this point. (sr means ing incandescent lightradiometer) and a in thermoelement theby (as plane shading. a of non-selective At the 0, receiver observation the or radiation screen. passes without Its reflection width through is the plates, indicated Interference 20 400 Part II 20.12 Redirection of the Radiant Power by Interference 401

Figure 20.20 Examples of two short, nearly aperiodic wave groups. An un- structured series of such groups could be used to describe the behavior of incandescent light in interference experiments.

with incandescent light shows only a weakly-developed structure. Part II Incandescent light behaves as though it could be described as an un- structured series of short, nearly aperiodic wave groups, as shown for example in Fig. 20.20. The true wave picture of incandescent light can be imagined as being similar to a noise spectrum in acoustics. More details will be given in Sect. 22.4.

20.12 Redirection of the Radiant Power by Interference

Interference cannot create or destroy the radiant power (or energy current), but instead just redirects it. What is lacking in the direction of the interference minima is added in the directions of the interference maxima. We offer two technically important examples of this:

1. Eliminating reflections, non-reflective coatings. A single glass surface reflects about 4 % of the normally-incident radiant power, while about 96 % passes through the surface and into the glass. Imagine a thin, evaporated crystalline film on the glass block G in Fig. 20.21. Its material is chosen so that the indices of refraction nair-film and nfilm-glass are approximately equal in some spectral region. Then the same fraction of normally-incident radiant power is reflected at the boundaries 1 and 2. In addition, the thickness d of the crystalline film is adjusted so that the two wave trains which are reflected upwards have a path difference of D m=2 for an average wavelength of m within that spectral region. Then they will cancel each other through interference. The surface is thus strictly non-reflecting for m, and all the radiant power which falls at normal incidence on the boundaries 1 and 2 passes without reflection losses into the glass block G. For the neighboring spectral regions on each side of m, the thickness d is only approximately equal to =2, so that the sup- pression of reflection is incomplete, but still, for many practical purposes, it is sufficient (compare Sect. 25.8). 2. Layered mirrors with nearly loss-free reflection; reflection filters.In- stead of providing a non-reflective coating, we can redirect the large fraction of light that penetrates into the glass block in spite of reflection losses,

Figure 20.21 Non-reﬂective coating 1

2 G 2 2.66 9 TiO = m ÉROT G n ound -P ABRY Wavelength λ for the average wave- visible infrared ; these are due to reﬂection 0.4 0.6 0.8 1.0 μm 1.2 0.4 0.6 0.8 1.0 μm 1.2 2 0 0

1.45 % %

20 20 80 80 60 60 40 40 which remove certain spec- = Transmissivity SiO n ace, as an ancillary mirror surface, re- m. C20.2 reflection filters 1 D , bottom). mand 20.23 , the amplitudes of the reflected wave trains all add with m 8 : The structure of a reflec- Top: A reflection filter ). Each boundary surf of the spectral region that is to be reflected is equal to , top); or no infrared light from the spectral region adjacent to 0 2 are included in the path differences D m D = 20.22 20.23 superior to metallic mirrors.20 For to broad 30 spectral layers regions, we by can using obtain ar the same phase. Infor this selected, manner, narrow we spectral can regions; in fabricate the nearly visible loss-free region, mirrors these are vastly tral regions notfor through example absorption, reflection bur filters(Fig. which rather allow by no reflection. visiblethe light visible (Fig. to There pass are through that is roughly 96 %this, of the we radiant need power a canbe also large obtained be number by reflected. of alternately To evaporating(Fig. reflecting achieve two ancillary types surfaces. of thinflects They crystalline the films can same fraction ofthicknesses are the chosen normally-incident so that radiant thelength power. path The difference layer With Jumps of visible spectral region to pass, inviolet the and the red. Bottom:tion filter A which reflec- passes aroundthe 80 light % in of the visible, butbetween no infrared tion filter which allows only small portions of the Figure 20.22 9 by a more optically dense material. 20.13 Interference Filters Like all spectralétalons) can apparatus, also be plane-parallelincandescent used or to plates natural separate light. narrow (F spectral In regions this out way, of we arrive at “interference Figure 20.23 Interference 20 402 Part II 20.14 Standing Light Waves 403

Figure 20.24 An interference fil- 4 a ter. A shows the light from the 3 interference plate alone; in B, an m = 2 absorption filter (e.g. colored glass) for short wavelengths dotted,and λ λ λ a reflection filter for long wave- 4 3 2 lengths dashed. In part C, we see 80% b the combination of A and B (the half-width of the transmitted spec- δ tral region around D 600 nm is H D 10 nm, so that the spectral

= Transmissivity selectivity H is equal to 60). Part II

40% c H

0.4 0.6 0.8 1.0 μm visible λ

filters” (usable only for light at normal incidence). Their principle: In Fig. 20.15, we replace the plane-parallel air layer by a very thin (d < 103 mm), non-absorbing crystalline film, e.g. a film of MgF2, partially silvered on both surfaces. For incandescent light at normal incidence (ˇ D 0ı), it permits only narrow spectral regions to pass through, whose wavelengths 1:2:3: ::: have the ratios 1 1 ::: 1: 2 : 3 : They emerge as bright interference maxima on a dark background, with the orders m D 1; 2; 3;:::(Fig. 20.24 A). These plane-parallel crystalline films can be combined with suitable filter layers so that only one of the transmission regions remains, e.g. at 3,asinFig.20.24 C.

20.14 Standing Light Waves

(OTTO HEINRICH WIENER,C20.7 1890). The wavelengths of vis- C20.7. See H. Jäger, Annalen ible light are only some few 10 4 mm. Nevertheless, we can ob- der Physik,5thseries,Vol.34 tain standing light waves – although not readily in simple demon- (1939), p. 280. stration experiments – by using the technique described in Vol. 1 (Sect. 12.5). For example, one can press a liquid mercury mirror against an extremely fine-grained photographic plate. The light which arrives at this mirror at normal incidence and is then reflected by the mercury darkens the photographic emulsion in equidistant, separated layers, with a spacing of =2. Figure 20.25 shows a thin slice from such an emulsion, cut perpendicular to the plane of the plate, at a high magnification. D and thus ooking at the in 1935; the image reflection is a small planar mirror H AGUN – Arc lamp Arc H bout 2 m in front of the mirror, + to it, we can see concentric circular . A redirection by small diffracting or ’s rings” (but they were described already in great detail). scattering , each ray coming from the light source was redi- UETELET , we have allowed ourselves a certain luxury: The perpendicular or 20.13 ’s “Opticks” 20.26 and The photographic detection of standing light waves ( which Redirect the Light Subjective observation of the interference rings which are pro- diffraction Dusty mirror EWTON 20.10 onto the dusted surface of the large mirror. This allows us to place H In Figs. rected at the two surfaces of a plane-parallel plate by split into two partial rays.rected Instead by of reflection, the rays can also be redi- interference rings on its surface.common They center, are surprisingly we clear. see Behind therings their image varies of with the the light distance source. of the The observer diameter from of the the mirror. In red-filter scattering particles can for examplearranged also directly be in front used, of whenOne or these could, on particles the for are surface example,plate, of the by take plate. no an means ordinaryaround plane-parallel, 30 household cm which mirror diameter. is (i.e.elling silvered The clay. a on glass A glass its surface small light rear isand source dusted face) is the set or of eye up rubbed of a source. with the mod- observer In is Fig. atarc some lamp arbitrary is distance placedror behind at the the light side andour eyes sends practically its “at light themirror via position in a of a small the direction metal light mir- source”. L makes several hundred wave crests and nodes visible (cf. Vol 1, Fig. 12.47). in 1704 in N which acts as light source. duced on the surface ofThey a are thick often household called mirror “Q after dusting or clouding it. Figure 20.26 Figure 20.25 20.15 Interference due to Particles 546 nm). A magnified viewto of swell a up section to through 10 aa gelatine times small emulsion, its section allowed original thickness from by an immersing image in water. produced This by is S. M Interference 20 404 Part II 20.15 Interference due to Particles which Redirect the Light 405

Scattering particle B

1 β' l4 l1 2 y β β' l2 γ' 1 x 2 l 3 γ 1* g γ' 2 2* 1

γ β drs Lamp Eye Part II Figure 20.27 The occurrence of small path differences in thick mirror-glass plates (refraction was neglected here), and the derivation of Eq. (20.6). The decisive angle for the coherence condition, 2! (denoted here as ), is quite small. For clarity, the angles ˇ and are exaggerated in the drawing. How- ever, for the calculation, we assume (as in the actual experiment) that the angles are small. We thus set sin ˇ D tan ˇ,etc.

light, we can easily count the usual 10–15 orders. When we view the mirror obliquely, the center of the rings appears shifted. In incandescent light, we see a bright, colorless ring of zeroth order; behind it is the image of the lamp. The rings bordering this central maximum appear deep black to the eye. They are followed by the usual series of higher-order rings, in gradually fading colors. Interference fringes of low order can be produced only by small path differences. How can they occur here in spite of the thick mirror-glass plate? Answer: As small differences between two large path differences (THOMAS YOUNG, 1802). In Fig. 20.27, B is one of the large number of particles which redirect the light rays reaching the mirror. Two paths lead both from the light source to the particle B and also from the particle to the eye of the observer. Along path 1, the light from the source arrives at the particle B via a ‘detour’. From B, however, after being redirected by diffraction or scattering, it reaches the eye directly along the path 1 . Along path 2, the light from the source arrives directly at B, but from B, after redirection by diffraction or scattering, it again takes a ‘detour’ along path 2 to arrive at the eye of the observer. The path difference between these two wave trains is thus rather small. We deﬁne the ratio Distance r to the eye D q : Distance s to the lamp

Then, for small values of ˇ and , and in the limiting cases q 1or q 1, we have d D q2 1 sin2 ˇ (20.6) n .Derivation see below; nomenclature as in Fig. 20.27/:

At the m-th maximum, D˙m; then for its angular spacing, we have

m n sin2 ˇ D˙ (20.7) d .q2 1/

(ˇ is the angle of inclination as in Fig. 20.27, d the thickness, and n the index of refraction of the mirror-glass plate. The minus sign holds for D . q observa- ; (20.8) (20.9) r .Thisis ! 2 ˇ S 2 2 that we care 2 to be used for L n Ocula y sin D ; and 2 1 Ã 0 . Their spacing 1 ˇ;

2 S C Image plane 2 ), the wave centers 2 ; RAUNHOFER S ˇ: 1 2 sin sin sin 1 m d Limit sin 1 2 20.7 n and n d dn ~ 2 / 2 1 1 b C ) determines the path differ- D S D D 1 0 1 1 4 2 Â L

l redirects the two lightas beams divergent beams; they now 16.3 q nl . 2 1 dn sin S n d L 2 ,weﬁnd D 2 1 d

S S D D behind the slits ). The innermost ring has the smallest and D 0 and 0 1 a D /; and

3 L ). 2 = ˇ S ! nl 20.26 ; ω tan

sin q , it had the largest. 2 dn ˇ/: therefore appears for two different values of C 2 2 d RAUNHOFER 2 2 n 1 D . The lens ˇ l ’s Interference Experiment 1 20 m 20 D sin . and analogously, s r sin permit readily-measurable widths 2 20.10 ~ x 1 2 p b ’s interference setup with the F a D /

4 C y

D 20.28 2

ˇ ; 2 1 nl 0 and

is a metal-vapor discharge lamp (Na or Hg)). The lengths sin C OUNG sin tan sin tan dn a OUNG . L 2 d 2 x dn n 0 l in the F Y n d 2 cos . 2 2 S 2 Width D ’s interference experiment (Fig. D D D D D D

ˇ Videos 16.3 and 20.1 , while in Fig. ( 2 2 3 3 l l m L 0 sin sin 1). The same angle nl nl S OUNG < q In one case, the eyebehind is the in source front (as oforder in the Fig. light source, inThe the difference other in case, the it opticalence is between paths the (Sect. two wave trains: For small values of the angles of the distances the slit tion condition ( can be chosen tootherwise, be at the most two a light fewslit beams spacing millimeter is in will often the no annoying. simplestthis longer However, setups; limitation we overlap. can and free use This ourselves any small from values of the slit spacing illustrated in Fig. which emerge from the slits converge towards the optical axis. They then intersect at the image are two openings (holes, or better, slits), Figure 20.28 20.16 Y In Y to: We need only set a lens ’s ob- ,which http://tiny. Diffraction RESNEL ” ’s interference ,“ , after 4:00 min.; ). OUNG Video 20.1, “Interfer- tiny.cc/9eggoy 20.5 Interference Video 16.3 20 shows Y and in ence” In and coherence cc/xdggoy however with F experiment in its ﬁrst part, servation condition (see also Sect. 406 Part II 20.16 YOUNG’s Interference Experiment in the FRAUNHOFER Limit 407

Figure 20.29 A single- slit diffraction pattern, observed using YOUNG’s setup

plane with practically planar wavefronts, but the wavefronts are more strongly tilted relative to each other than without the lens. Therefore, the resulting interference fringes are more closely spaced than they

were without the lens. We can observe the fringes either through Part II a magnifying lens L2 (ocular or TV camera), or else project them as an enlarged image onto a frosted glass screen using an objective lens. The lens L1 and the magnifying glass L2 (ocular) together form a telescope. Indeed, one usually makes use of a telescope with two slits S1 and S2 in front of its objective. In this form, YOUNG’s experimental arrangement is especially important. We will thus treat it in some detail.

Initially, we set the slit S0 to a narrow width and cover either slit S1 or slit S2. In both cases, we obtain the diffraction pattern as photographed in Fig. 20.29, and it is at the same position in the image plane, symmetric to the optical axis. It is the central maximum of the diffraction pattern that we have seen in Figs. 17.6 and 17.7 (its secondary maxima are too faint to see).

Next, both of the slits S1 and S2 are opened at the same time. The wave trains coming from S1 and from S2 interfere with each other: The diffraction pattern is sliced through by sharp interference fringes (Fig. 20.30). Here, an important precondition was met: The slit S0 acted in spite of its ﬁnite width 2y like a “point” (or rather a line- shaped) light source. The slit width 2y thus meets the coherence condition 2y sin ! =2 : (20.1) Now we describe something new: the Measurement of the diameter of a distant light source .A.H.L. FIZEAU; 1868/:10

We gradually increase the width of slit S0 and thereby violate the coherence condition. Nevertheless, we still see the interference fringes, however more faintly, i.e. with poor contrast between a maximum and its neighboring minimum. This decrease in the contrast is easy to understand: Imagine that the slit S0 were sliced up into small segments along its length. Each one gives rise to an interference pattern as in Fig. 20.30, but the individual interference patterns are shifted along the axis of the slit relative to each other. Their superposition gives rise to washed-out fringes.

10 Comptes rendus, Paris, 66, 934 (1868), and J.M. Stephan, ibid., 78, 1008 (1873). D . ! (20.11) (20.10) but only tem- '; partial coherence was varied in a mea- 2 D y a 2 20.28 ,weﬁnd ’s technique to determine the Tauri (Aldebaran), for which a ; D 2 ˛ = D D of a distant light source to known 2 D in Fig. IZEAU ! 2 ' could be made even greater than the D S is increased still further, they return, sin or y D ! . With this relation, we can refer the y a 2 and ,etc.).Thisistermed 1 y a S D 2 2 or angle 2 = y of several nearby ﬁxed stars at known distances D 3 made use of F 020 seconds of arc. To achieve this, the distance : y ' 0 2 D ), setting sin : Single-slit diffraction and two-slit interference are super- ! D The interference fringes observed within the diffraction pattern ' 20.10 is the angle of vision subtended by an object of diameter 20.29 : As the slit width 2 ' ICHELSON , they again disappear, and so on through several repetitions y 2 seen from the distance between the two slits from the earth, for example that of y = angles of vision 2 a he found 2 diameter of the telescope objective. surable way. Using mirrors, where 2 2 The simplest but already quiteof transverse serviceable observation. interferometers They employ make the use basic experimental setup 20.17 Optical Interferometers Optical interferometers are used to accomplish two1. tasks: For highly precise comparisons ofmeasurement lengths standards) or on distances (e.g. the length onelight hand, on and the the other wavelength (cf. of Vol. the 1, Sect.2. 1.3). For comparisons of twories, coherent e.g. light after beams they with have different passed histo- through different materials. When the interference fringes have completelyporarily disappeared, even more washed-out or lacking in contrast than before. For sin Figure 20.30 from Fig. posed D unknown diameter 2 quantities, and thus measure it. A.A. M 2 (that is for sin From Eq. ( Interference 20 408 Part II 20.17 Optical Interferometers 409

To the Manometer G

CKS1 L1DL2

Figure 20.31 An interferometer for determining the index of refraction of gases at various densities (schematic in Fig. 20.28; G is a rubber bulb for changing the gas density %). The two slits D immediately behind the lens L1 (f D 2m)are2mmwide. Part II Their spacing is 10 mm. The distances S1L1 and L1L2 are about 4 m. Both light beams pass through the glass window K which closes off the end of the gas container and is sufﬁciently wide to intercept the collimated reference beam in the air beside the gas vessel at K.

Figure 20.32 An interferometer with two α x collimated light beams which are shifted sideways and parallel to each other. The II effective thickness x can be varied by tipping the plates relative to each other. When β they are parallel, x and thus also the path 1 difference between the two beams 1 and 2 2 1 β 2 are equal to zero. All unnecessary reﬂec- tions, which in reality are eliminated by I apertures, have been left out of the draw- α ing.

of THOMAS YOUNG in the arrangement shown in Fig. 20.28.There, the two light beams are separated transversally by several centimeter just behind the lens. It is thus easy to allow one beam to pass through the air and the other through some other gas, in order to compare the wavelengths in the two gases. Experiments of this kind are discussed below; Fig. 20.31 shows a practical example. All of the other interferometer designs also make use of interference fringes in the image plane of a lens (often the lens of the eye), but they employ longitudinal observation (Fig. 20.6). They implement a plane-parallel plate of thickness x as the difference between two plates of unequal thicknesses (TH.YOUNG, 1817). This is illustrated for example in Fig. 20.32, where the axes of two light beams which are shifted parallel to each other are drawn. Often, one replaces the plates completely or partially by mirrors (e.g. at ˛), or partially- silvered mirrors (at ˇ). One thus arrives at the interferometer design of ALBERT A. MICHELSON (Fig. 20.33), with two mutually perpendicular light beams (cf. Vol. 1, Fig. 12.67). The path difference is denoted by x. By tipping the mirror II, one can also produce a wedge plate. The plate III is in fact not necessary in principle, but it allows one to achieve equal path lengths through glass for both light beams. That simplifies the observations. I . . each S Amer- unified uniform x within each can initially A can change the ities; rather, the piece of frosted only in a A must be on the screen lling the coherence ;::: II III ;::: II ; moving II I ; . This must be taken into I )) is fulfilled. Some of them, point wave source fluctuations 20.1 11 is a sheet of white paper which is granulation oduce A onochromatic light beam from a laser. , 919 (1964)) . The whole solid angle from its center to the y 32 . Within each of these elements, the radiant intensity C20.8 itting surface represents an aperture which radiates in all directions ;::: variations of the phase distribution are absent, and with cannot be irradiated with different intens II A ; The phase distribution in the emitting aperture I , ;::: in the Wave Field The interferometer of (only the axes of the II ;::: can be subdivided into small solid-angle elements, within ; I 20.34 II S ; transverse I radiant intensity in each of the surface elements can vary between zerowithin and the a emitting area maximum therefore value. pr Statistical phase changes manner. Such changes mayface be elements of different strengths for the different sur- glass at a single, distant light sourceican (W. Journal Martienssen and of E. Physics Spiller, In the wave fieldto of be the observable with light,successfully simple these techniques. in fluctuations model But occur they experiments,observation: much can in too be We the simulated quickly look simplest quite through case a using red subjective filter and a irradiation of each individual surface element Phase changes of the individual emitting elements within of the individual angularregions elements 1, 2, . . . As a result, sections of the be thought of as arbitrary, but constant over time. Fulfi condition means that the aperture acts as a them the fluctuations: Theirof spots a on fluctuation, the the screen screen are shows “frozen a in”; instead chosen randomly, are marked inpropagates the within figure these as solid-angle 1, elements 2,gions . strikes . the . screen The in radiation the which re- illuminated with the extremelyThen m screen of which the coherence condition (Eq. ( account in making photographic imagesOur (e.g. understanding for holography). ofconsidering fluctuations Fig. and 12.49 granulation inproduces a Vol. can granulation, 1. while be a There, hand increasedproduces which a fluctuations. is by hand changing Both its at are shapetifice made rest randomly of visible in the in acoustic the the replica wave figure method field by (Vol. using 1, Sect. the 12.18). ar- Suppose that the em In Fig. and has a diameter of 2 ICHELSON To show this demonstration to a large audience, the surface of a glass plate is M light beams are drawn). Inexamples the of largest this design, theally mutu- perpendicular light paths (“armlengths”) are 30 m long. Figure 20.33 covered with a layerperpendicular of to the glued-on beam glassscreen. axis powder of and a beam then of moved light back which and is forth projected onto a wall 11 20.18 Coherence and Fluctuations ). https://en. onger arm lengths Interference 20 C20.8. Today, similar interferometers with arm lengths of up to several kilome- ter are in use, for example in the gravitational-wave detector system LIGO- Virgo. (See e.g. wikipedia.org/wiki/LIGO Still much l are planned for the space version of the experiment, “Cosmic Explorer”. 410 Part II Exercises 411

Figure 20.34 The origin of ﬂuctua- S tions in the wave ﬁeld I 2ω 1 A 2 y II Lamp 2ω 2 Screen S

Exercises Part II

20.1 The interference experiment shown in Fig. 20.10 is carried out with an air layer of thickness d D 40 m. The wavelength of the light used is D 600 nm. a) What is the value of the largest order mmax, and how large is the radius x of the corresponding interference ring, when the distance of the plate from the wall of the lecture room is A D 3m? b) Given this distance, how large is the radius of the tenth interference ring x10 (counting from inside to outside)? (Sect. 20.6)

20.2 The application of interference microscopy to the determination of the thickness S of an evaporated metal ﬁlm gives a shift in the interference fringes of 1/3 of the spacing between two fringes whose orders m differ by 1 (Fig. 20.16). The wavelength of the light used is D 600 nm. How can we calculate the ﬁlm thickness S from this, and what is its value? (Sect. 20.10)

Electronic supplementary material The online version of this chapter (https:// doi.org/10.1007/978-3-319-50269-4_20) contains supplementary material, which is available to authorized users. Fg 22 nVl ) ups htpaewvswt ra aern are wavefront broad opening a the with on waves left plane the that from Suppose incident 1). Vol. in 12.29 (Fig. o.1 et 21 ntrso F of terms in 12.14 Sect. 1, Vol. ftebokdotcn.Ti a hw yamdlexperiment, Fig. model in a again by here shown reproduced was This is axis which cone. the blocked-out along alternate the opening, zones of circular wave-free a and Behind waves an for containing behind 12.13). are zones Fig. cone shadow 1, There (Vol. the disk of circular an structure. axis opaque behind complicated the and along a waves disk, 1 has always be opaque example field Vol. first an wave will in behind the there Both depth opening, discussed observations in here. the repeated treated of briefly Some was waves 12). (Chap. mechanical of Diffraction detail. more some in tion 17.3 11CsigShadows Casting 21.1 Diffraction iue21.1 Figure Sects. in geomet- already the treated of been softening has and boundaries extension shadow an rical as light of diffraction The .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © h emai sepandi h etuigF using text the best in structure explained this is see axis can diffraction. beam we the of beam, structure. complex the result a of a exhibits direction field as the wave the boundaries opening, shadow the Near geometric parallel its beyond to Fig. In repeat: briefly We h point the w nems ( innermost two aclec te oagetetn ttepoint the at extent great a to other each cancel nti hpe,w ilivsiaetepeoeo fdiffrac- of phenomenon the investigate will we chapter, this In . P 3 hc eiaiet i ntesmer xso h aefil.At field. wave the of axis symmetry the on lie to imagine we which B P oe xeietsoigtesao atb nopening an by cast shadow the showing experiment model A 2 h pnn evsan leaves opening the , m D 1and olsItouto oPhysics to Introduction Pohl’s 21.1 m D P h rosso h bevto points observation the show arrows the , 3 ) h lmnaywvswihte emit they which waves elementary The 2). RESNEL even B P h ctot aeba extends beam wave “cut-out” The . 2 ubro oe re aeythe namely free, zones of number szn construction. zone ’s RESNEL P 21.1 2 tpoint At . twsepandin explained was It . zones. h tutr along structure The . , https://doi.org/10.1007/978-3-319-50269-4_21 P 3 ,however,the P okn along Looking 1 16.9 P 1 and 21 413

Part II b 21.2 are and of the (21.1) .Itis which : 3 a b m P M r with that of S are varied. m in Fig. ,wealways b ’s spot D M L 2 m , a to f) exhibit r ). However, the 1, 2, 3 ...) are and ; a are of the same or- 21.3 D 21.1 OISSON openings m D1 21.2 . The obstacle a are varied. In every case, ; b ’s spot could readily be ob- b 20.4 and C and for ab ). In the shadow of the disk, the a a I ; it is called P 2 OISSON = m ab b D have values of nearly 20 m. The shadow m UYGENS b 2 m , there is no longer any such structure. 1 r D H persists P 2 m r and 3). The elementary waves emitted to point number of zones free, namely the three innermost LM a , the number of rings increases when must take the same values for the light waves for light waves in Fig. D , only a few centimeters away from the obsta- ; b b odd m m r D disks 21.1 a HRISTIAN 1to C A comparison of the shadow of a circular disk D and the observation point. Then for the radius K m L , the lengths that is, for 21.2 conserved. Beyond point opening leaves an zones ( -th zone, we ﬁnd from Eq. (12.21) in Vol. 1: m are decreased, but theilluminated center (C of the pattern always remains brightly Behind opaque see only a fewnating rings. maxima At and the minima center when of the the picture, distances we can see alter- or diffraction photos made with this setup (Fig. however, they show noticeable differencescircular for openings of circular the disks same and size. for Behind the rather complex diffraction patterns instead of sharpchange boundaries. continually They as the distances a point in the shadowof of a a circular rectangular disk, obstacle, a etc. straight line P in the shadow served using water waves (Vol. 1,discussed Figs. there 12.13 in and Sect. 12.15. 12.14, Its point origin 4). is bright spot at the center a circular opening of the same size at the same position If the zone radii With light waves, the situation is similar. Assume that Figure 21.2 is an opening which radiatescenter light waves. as It described replaces at a pointlike the wave end of Sect. casts a shadow, or a circular openingbetween which limits the beam, is located wavelength of visible light iswavelength more of than the 1000 times waves smaller inobservation than the points the model for experiment. the innermost As zones a ( result, the cle; instead, theyFig. are at distances of many meters. Therefore, in der of magnitude asthen those the for product the waves in the model experiment, not, as in Fig. as for the waves in the model experiment (Fig. Diffraction 21 414 Part II 21.1 Casting Shadows 415

Shadows of circular disks abc Ø = 4,3 mm Ø = 6,1 mm Ø = 7,4 mm =^ 1 zone =^ 2 zones =^ 3 zones Part II

Shadows of the circular openings def

g hi along a diameter a along

Illumination intensity Illumination 8440 8 mm 8440 8 mm 8440 8 mm

Figure 21.3 The shadows of circular disks and circular openings of the same diameter exhibit very different diffraction patterns. For demonstration experiments, we employ red-filter light. For the photographs (positives), green light with a wavelength of 546 nm was used. The distances a and b were each Video 16.3: 17.5 m. The images in the lower series show the distribution of the radiant intensity along a diameter of the patterns in the middle series. Images e and “Diffraction and coher- h correspond in Fig. 21.1 to a cross-section perpendicular to the beam axis at ence” http://tiny.cc/xdggoy. the observation point P2 (Video 16.3). The diffraction patterns shown in the images a–c from circular disks are demon- At a distance of a D b D 11 km, for red-filter light ( 650 nm), strated using thin wires of different diameters (1.7, the diameter of the central zone is 2r1 D 12 cm. That is the size of a small saucer. Such a saucer would thus block out only the central 1.0, and 0.2 mm) (from zone from the free wave field. As a result, the shadow pattern of the 14:30 min.). FRESNEL’s saucer would look like that in Fig. 21.3a, but its brightest ring would observation mode is em- have a diameter of around 50 cm. ployed, and both red- and blue-filter light is used. An For a D1and b D 1 m, with red-filter light, the diameter of the first zone explanation of the experimen- is 2r1 D 0:6 mm. It is thus rather simple (e.g. as in Fig. 16.29) to block out tal arrangement is given at fractions of the first zone with apertures or slits. the beginning of the video. 0 DD as an either Screen Screen x :From D' D D D' D' D D' D P P 21.5 , both diffraction A x B A B x . It can represent without x x of a semi-plane which blocks all a b is used . The reason for this is the follow- AB Semi-plane , and illuminate the observation screen a small opening of exactly the same size or opening ’s Theorem ’s ’s theas an x . RESNEL ABINET ’s observation ABINET previously dark AB . Every linear shadow boundary looks like this, if the 18 m, B The ) ’s theorem is useful for the treatment of diffraction phenom- represents an opaque and for D is sufficiently small, the angular deflections of the diffracted x 21.4 x b which is several centimeters wide. A light beam emerges to the D ABINET RAUNHOFER Video 16.3 a B F photographic positive, red-filter light) ( orem. Above: F mode. If obstacle, only the free surface area outside it radiates light waves Figure 21.4 observation mode. Below: diffraction stripes at the boundary of the shadow of a semi-plane ( Figure 21.5 which were As the diameter increases,lead both to circular the disks same and limitingthe circular light case, on openings that one side. Thein diffraction pattern is Fig. shown as a photograph 21.2 B obstacle there. The diffraction pattern should have the same form for ing: When the free aperture patterns occur simultaneously. Therefore, the wave amplitudes of ena. A relevant thought experiment is illustrated in Fig. the left, a weaklyAB divergent light beam is incidentright. upon Its boundaries an are aperture somewhat fuzzyis as indicated a by the result shading of at diffraction, as the edgesNow of the we beam. draw in a small line segment a small, opaque obstacle and shape as the obstacle, in anthe opaque aperture screen (not shown) that covers When waves are large, and the light can penetrate into the regions diameter of the light source is sufficiently small. . are shown in http://tiny.cc/xdggoy Diffraction 21 Video 16.3: “Diffraction and coherence” both red- and blue-filter light. An explanation of the experimental arrangement is given at the beginning of the video. Beginning at 12:20 min., the diffraction stripes behind asemi-plane 416 Part II 21.2 BABINET’s Theorem 417

Figure 21.6 a: The diffraction pattern from a wire (obstacle) of 0.5 mm diameter; and b: The diffraction pattern from an equal-sized slit (FRAUNHOFER’s observation mode, as in Fig. 21.5b; photographic negative. The center of the image is overexposed in spite of the blocking. The distance to the screen was about 5 m). Part II the two diffraction patterns must cancel each other exactly at every point in the dark regions DD0 at every time. The amplitudes must be equally strong for the obstacle and the opening, and they must have opposite phases (ı D 180ı). This thought experiment leads us to BABINET’s theorem. It states that if we insert one after the other an obstacle and an opening of the same size and shape into a wide light beam, and limit our observations to the regions which were completely dark with only the free light beam (that is, outside the fuzzy, diffraction-broadened edges of the beam), then in these regions, we will find the same diffraction pattern for the obstacle and for the opening. BABINET’s theorem holds both with FRESNEL’s and with FRAUN- HOFER’s observation modes (Vol. 1, Sect. 12.12). With FRESNEL’s mode, the diameter of x must usually be made smaller than 0.01 mm. Only then does the diffracted light have a sufficiently large angular deflection, only then can it penetrate into the previously dark regions1 DD0. One single such small obstacle or opening however gives rise only to an extremely faint diffraction pattern. It requires several thousand of these obstacles or openings x to produce a clearly-visible pattern. With FRAUNHOFER’s observation mode, we consider Fig. 21.5bin- stead of Fig. 21.5a; then, the free light rays from the aperture AB at an “image point” are concentrated into a narrow band. The dark regions DD0 occur on both sides of this band, rather close to the dashed optical axis. As a result, even the few diffraction stripes from large obstacles or openings x will fall in the dark regions DD0.Theneven just one opening will produce a readily-visible diffraction pattern. Figure 21.6a shows, as an example for the validity of BABINET’s theorem, the FRAUNHOFER diffraction pattern from a wire. It is the same as that from a slit of the same width, as seen in Fig. 21.6b. For these observations, the center of the diffraction pattern is blocked out by a small screen.

1 In Fig. 21.3a–c, all the diffraction processes took place within the initially- present free light beam. As a result, the decisive precondition for BABINET’s theorem was not met, and thus the diffraction patterns from the disk and the opening were quite different. ! small .The .The b), we ! one oduction. It 21.5 , we obtain prac- 21.7 )) for the angle 2 20.1 The position of the diffraction pattern is (Video 17.1). circular opening by a large number (around how it will look for a small, circular opening ’s observation mode (e.g. as in Fig. 17.8 single ’s observation mode, photographic negative image). A small The diffraction pattern from a large number of equal-sized, and Randomly-Arranged Openings or Particles RAUNHOFER RAUNHOFER F (e.g. of 1.5 mm diameter). independent of sideways shiftsdifferent in segments the of position thesymmetric of lens to always the the produce optical opening axis. apractical importance: This diffraction leads pattern to a conclusion whichWe is replace of the 2000) of similara openings random of arrangement. the Then, same as seen size in (0.3 Fig. mm diameter) in subtended by the lens surface image of the pointlike lightoccurs source when the at incident the radiation center is was not lost coherent within in the repr entire angle 2 randomly-distributed circular openings (aboutlar 2000 area distributed of over 5 cm a diameter; circu- the diameter of the individual openings is 0.3 mm. Figure 21.7 21.3 Diffraction by Many Equal-Sized With F opening, but it isa now large audience. visible The from diffraction a patternsnearly no of considerable mutual all disturbances. distance the The openings and reason: add Theor for with light more beams from openings two can indeedinterference mutually fringes, interfere if and they form liedifferences additional within a are coherence different angle, but formaxima the and all path minima such of the combinations.so additional that fringes Therefore, superpose on and the the cancel, average, everything remains unchanged, apart from make use of aaxis. small The light diffracting source openingfulﬁlling at the is a coherence placed large condition directly (Eq. distancediffraction in ( pattern on front appears the of in optical the theknow lens, focal from plane Fig. of the lens. We already tically the same diffraction pattern as before with only . . 17.3 Diffraction 21 equal-sized and randomly- arranged holes is placed in front of the lens.Sect. See http://tiny.cc/9dggoy In this video, a masktaining con- a large number of Video 17.1: “Resolving power” 418 Part II 21.4 The Rainbow 419

Screen

Iris Dusted diaphragm glass plate ca. 5 m

Figure 21.8 Demonstration of the diffraction pattern from many randomly distributed spheres of uniform size using FRESNEL’s observation mode. With Part II the dimensions used here, this gives the same result as FRAUNHOFER’s observation mode using a lens and convergent beams: The light-wave beams which are deﬂected to the sides by diffraction are clearly separated from the original beam (the zeroth order diffraction maximum) even without a lens (Sect. 16.8). a weak granulation structure (which is radial in non-monochromatic light). This results from the random arrangement of the openings. Granulation always occurs in diffraction patterns when coherently- illuminated diffracting objects are arranged randomly. Generally, one observes such granulation subjectively in the diffraction patterns from semi-transparent structures (Sects. 21.4 ff.), for example looking through a frosted glass plate or a “cloudy” windowpane at a small, distant light source. (In both cases, the disordered particles do not have a uniform shape and size. Therefore, the rings are missing.)

In the range of validity of BABINET’s theorem, small disks give the same diffraction patterns as openings of the same size. Therefore, we can replace the randomly-distributed openings by randomly- distributed circular disks, and the latter by small spheres of the same diameter: We dust a glass plate with lycopodium seeds, tiny spheres of around 30 m diameter. With light of wavelength 650 nm (red-ﬁlter light), the ﬁrst diffraction maximum has an angular spacing of about 1.3ı from the normal to the plate (the optical axis; cf. Eq. (16.23)). We could thus use FRESNEL’s observation mode and project the diffraction rings onto a wall screen. Figure 21.8 shows a suitable experimental arrangement.

21.4 The Rainbow

The little spheres of lycopodium seeds mentioned in the previous section were randomly distributed on a glass plate. Instead of this two-dimensional object, we could use a three-dimensional, random distribution of diffraction centers (spheres). Nature provides such a distribution in the form of the fine water droplets in fog and clouds or in a rainstorm. Artificial fog can be readily prepared: We put a little water into a glass bulb and reduce the pressure in the bulb rapidly from ), has SR Secondary ( ): rainbow 51° structures. We 21.9 Primary 42° combined effects SR PR takes the place of the S 51° 42° transparent secondary rainbow . With incandescent light, the SR From the Sun droplets Water is phenomenon, we of course cannot the spheres. We thus arrive at our first ) occurs only when the sun is not too and above the horizon. PR ı ( PR around the axis of symmetry. It exhibits the ). Here, the water droplets are represented ı through 21.10 ; the diameter of the rings varies with the diameter 21.8 Schematic of the ). primary rainbow is seen; as a rule, it is red on its outer edge, then tinted yellow, ı C21.1 21.9 high in the sky, at most 42 1. The start with the facts that are relevant to rainbows (Fig. example of diffraction phenomena from 2. Thepasses central from the axis sun through of theFig. eye the of the rainbow observer (bottom lies arrow in on a straight line which eye of the observer.patterns, i.e. On this two screen, bands, we see two typical diffraction of diffraction, interference, refraction, dispersion and reflection and within the randomly-distributedessential spherical features water can droplets. mostexperiment readily (Fig. The be seen with the aid of a model same colors as thebut primary in rainbow the (although reverse it order:etc. is going red outwards. usually is fainter), at its inside edge,The then elucidation of yellow, green these phenomena is found in the by a thin jeta funnel. of water, Instead aboutilluminated of 1 slit the mm with sun, in a a red diameter, line-shaped filter). which light The flows source screen from is used (an Figure 21.9 primary and the secondary rainbow an opening angle of 51 green and blue on goingtinct inwards. rings Inside which the main gradually arcThe become are sequence fainter of several (“supernumerary” colors dis- arcs). in the rainbow reminds4. us of A a second spectrum. system of rings (arcs), the consider the droplets as opaque disks;the we waves must which also take pass into account with an air pump. Thissaturation leads of to the cooling water of vaporput the and a air formation in glass of the bulb floating bulb,plate filled super- droplets. in with Fig. We artificial fog in place of the dusted glass In a quantitative treatment of th of the droplets, andcan the be latter easily increases trackedrings. in by the the course decrease of in time. diameter of This the diffraction 3. Around thisca. axis 42 of symmetry, an arc with an opening angle of ). (1977) https:// http:// and http://www.ams.org/ Diffraction 21 samplings/feature-column/ fcarc-rainbows (available online at www.phys.uwosh.edu/rioux/ genphysII/pdf/rainbows.pdf See also C21.1. For more details on the role of diffraction in rainbows, see for example the article by H.M. Nussenzveig, Scientific American en.m.wikipedia.org/wiki/ Rainbow 420 Part II 21.4 The Rainbow 421

58° K C Jet of water B 42° 51° 42° Jet of water PR S PR 51° 69° SR SR

Figure 21.10 A model experiment demonstrating the formation of the rain- Part II bow (red-ﬁlter light). The jet of water ﬂows downwards through the plane of the page. The screen S is set up perpendicular to the plane of the page. It shows the two diffraction patterns, PR and SR. For subjective observation, we would need a whole “cloud” of parallel jets of water; only then can the interference fringes of various orders from both “rainbows” pass simultaneously into the pupils of the eyes of the observer.

1 x Figure 21.11 Changes in the 2 3 4 wavefront by reﬂection and refrac- 5 6 tion in a water droplet (calculated 7 x for monochromatic light; xx be- R δ fore, yy0 after passing through the droplet). The beam marked with y ‘R’ is reﬂected back onto itself. y J y'

1 7 6 2 5 4 5 well-known superposition of a series of colors is seen. By varying the diameter of the water jet, we can produce many different color series. We can simulate all of the optical phenomena observed in the atmosphere, including the nearly colorless rainbow from extremely fine fog droplets. We can complement this model experiment, starting with the primary rainbow PR, by an elementary calculation. In Fig. 21.11, we consider a parallel light beam which is incident on a water droplet. We first draw several parallel rays 1–7 within this beam, and then, perpendicular to them, a planar wavefront xx. We then calculate the paths of the individual rays through the water droplet, applying the law of refraction twice and the law of reflection once. Now comes the essential point: The emerging rays are concentrated near a certain angle (the angle of minimum deflection, see Sect. 16.6) at the edge of the diffraction pattern. This results in an overall angle of deflection ı between the incident and the emerging rays of 42ı for red light. We calculate the optical path length for one of the rays between the points x and y (Sect. 16.3). That is, we decompose the segment xy of the beam into individual path elements SW which pass through ) / in 1 and SR (see is an x PR wave- . Con- n lies on . L 0 d y .Thenwe 21.12 ,totheleft ,givingthe L for a single ). Then the J and curved. D n ı 0 D y When the max- A . These diffrac- . We can change S one planar 33 (for red light; : 1 . Its diffraction pat- C ,lowerleft: W D step nS n 21.12 order-one position C21.2 . This angle is 42 ı and refractive index glass plate, for example a micro- . Their lines of intersection yield / 0 order-two position d yy . ). Now, we cover half of the slit parallel wavefronts, intersecting at 2; this is the which pass through the air, multiplying the = 16.9 A transparent two S determined in this way gives the new wavefront 2; this is the y for two reflections. , upper left: The path difference ı = along the other rays so that between the points in the interior of the droplets. The point (Sect. y ),andthentakethesumtobe B 21.12 an Amplitude Structure . The patterns observed in the secondary rainbow (at 16.1 twice 21.10 most conveniently by a slight tipping of the step. , the optical path lengths of the other rays are also equal to front, we now have after passing through the water droplet. Instead of the ray with an angle of deflection y reflection and 51 Some of the downstream wavefronts are also drawn in at of the calculated wavefronts necting the points choose points the diffraction patterns with interference fringes observed at former by the refractive index of water, 21.6 Diffracting Objects with Both for the case oftreated mechanical here, waves we (Vol. 1) began as the wella discussion limiting as of case: in diffraction The optics phenomena diffracting as with objectstransparent consisted sections partly and of partly completely of completelyticularly clear opaque example sections. can be A found par- in Vol. 1 (Sects. 12.15 and 12.20, The first diffraction patternslit which of we observed width was that of a simple 21.5 Diffraction by a Step water and elements to its long axis by a scope cover glass (of thickness are obtained in a correspondingreflected manner with waves (rays) which are Fig. see Table covered and the free halves together form a is an even multiple of tern with monochromatic light isperiodically in when the general wavelength asymmetric. ispersion It varied continuously, changes in due to the dis- cases: glass of Fig. the step. We find two symmetric limiting also Vol. 1, Sect. 12.13). ima approach the centralwhen axis they more move closely, they awaytion become from patterns stronger; it, are explained they on become the weaker right-hand side of Fig. odd multiple of same pattern as without a step; and Fig. Diffraction 21 C21.2. Larger numbers of reflections can also occur, but the resulting “rainbows” are increasingly faint or located towards the direction of the sun, making them mostly unobservable in the sky. The concentration of scattering paths near a certain angle (“rainbow scattering”) is also of great importance in modern nuclear, atomic and molecular physics; see the first reference in Com- ment C21.1. 422 Part II 21.6 Diffracting Objects with an Amplitude Structure 423

–3 –2 –1 0 1 2 3 4 λ Order-one position 2 Amplitude Part II

–3 –2 –1 0 1 2 3 (B/λ) α

5 λ Order-two position 2

Figure 21.12 Left: The two symmetrical diffraction patterns from a step; that is, a slit, half of which is covered by a transparent sheet. Comparison with Fig. 12.33 in Vol. 1 shows that the diffraction pattern for D N (N D ; ; ;::: D . C 1 / 0 1 2 ) is unchanged by adding the step to the slit. For N 2 ,in contrast, there is a minimum at the center of the pattern instead of the primary maximum, and maxima appear near where the ﬁrst minima were seen before. (B is the slit width, ˛ the angular spacing from the center of the slit; see Vol. 1, Sect. 12.13). Right: These results of model experiments are a continuation of Fig. 21.1. The wave centers of the glass images were moved along the step as sketched (Exercises 21.1 and 21.2).

Figure 21.13 A line grating magniﬁed 20-fold. The grating lines are furrows in the surface of a glass plate, ﬁlled with an opaque material

point 7): A line grating. In such a grating, transparent “slits” alternate with opaque “beams” or “rods”. Figure 21.13 shows an example of a line grating. Immediately behind the grating, the amplitudes of the transmitted waves are modulated along a direction transverse to their direction of propagation in a “square-wave” form: The amplitudes alternate sharply between a maximum and a minimum value (Fig. 21.14a). The latter is equal to zero for the special case of completely opaque beams. More important, however, is a different limiting case: An amplitude modulation as shown in Fig. 21.14b. Immediately behind the grating, the amplitude varies sinusoidally around an average value along a direction transverse to the propagation direction of the waves. Gratings . c. 0), Z D 21.14 21.16 m 1); beams a b combination D when the curve litude transverse m sine-wave grat- A Photographic plate is equal to m is positioned perpendic- C – can be experimentally varied in D D C C have an important property which can readily be x A

A Modulated amplitudes Modulated is the grating constant (length period of the modulation). Top: A sine-wave Two examples of modulation of the wave amp ). A photographic plate or fil D 21.15 . They can be readily prepared by a photographic method: We represents a line grating with completely opaqueparent beams and slits. completely trans- line grating (enlarged). Bottom: The procedure for photographically producing the grating shown above, using the interference field of two plane waves which intersect atangle an to the propagation direction ofthe the upper waves, curve, directly the behind a constant line average grating. value In Figure 21.15 ings generate an interference fieldlight with beams two which monochromatic are plane-wave (Fig. tilted by a small angle relative to each other Figure 21.14 which produce this kind of modulation are called ular to their midlinebeing and developed, it exposed shows to the the image in interference the pattern. upper part After of Fig. they exhibit only two secondary beams of first order ( Sine-wave gratings shown in demonstrationwhich experiments: is Symmetrically not to deflected the (the beam zeroth diffraction maximum, This curve can beplitude represented curves as sketched the below superpositionnot it simply of in formal: the the The three grating figure. actsual am- physically This sine-wave just gratings like superposition three as is individ- shownthese in gratings the produces curves only d its through two f. secondary beams Each of of first order of higher order do not occur. The method used for fabricatingof sine-wave interference gratings and – photography a many ways. It can alsoulate be used the to amplitudes fabricate of line gratings transmitted which waves mod- as shown in Fig. Diffraction 21 424 Part II 21.7 Gratings with a Phase Structure 425

Modulated amplitudes f Part II x

Figure 21.16 The continuation of Fig. 21.14. Curve c shows an additional example of amplitudes modulated along a direction transverse to the direction of propagation of the waves, immediately behind a line grating. Curves d, e, and f are the three FOURIER components of curve c.

(m D 1) in addition to the incident, practically monochromatic, collimated light beam (m D 0) which passes through without deﬂection (cf. FOURIER analysis). This has a number of applications.

An example: The grating shown in Fig. 21.13 (a raster grating) modulates the amplitudes of the transmitted waves in a square-wave pattern (Fig. 21.14 a). It acts like a very large number of sine-wave gratings. The grating constants of these 1 1 ::: “sine-wave gratings” are in the ratios 1: 3 : 5 (see Vol. 1, Sect. 11.3 and Fig. 11.13). As a result, this grating structure produces a long series of light-wave beams with the orders m D 1, 3, 5 . . . Each of these beams belongs with its order m D 1 to one of the individual “sine-wave gratings” (Exercise 21.3).

21.7 Gratings with a Phase Structure

We can carry out a continuous transition from grating beams which attenuate the light (Fig. 21.13) to completely transparent beams. They need only be distinguished from the “gaps” by their different refractive index (G. QUINCKE, 1867). Such transparent structures change only the phase of the light which passes through them. In the regions of larger refractive index, the phase changes more than in the regions of smaller refractive index. Thus, we call them for short phase gratings or in general phase structures. The diffraction pattern of a phase structure is geometrically no different from that of an amplitude structure of the same form. Differences are seen only in the ratios of the amplitudes and phases between the higher-order and the zeroth-order maxima; the zeroth order may even be completely lacking. An example is given in Fig. 21.17. . illa- ound α β α RAUNHOFER phase grating ). Observations . ˛ , 21.18 ˇ 21.18 illimeter in liquids, and are propagating can serve as , practically mode. With it, the diffracting is the order- , 1932, Fig. ˛ ), the case at ˇ ≈ 1 MHz Quartz oscillator v EARS P 21.12 -S ition of the diffraction pattern. As a result, sound waves experiment. Top: High-frequency s RAUNHOFER EBYE (D EARS -S photographed in red-filter light. EBYE C21.3 D The diffraction spectra from a line grating C21.4 optical phase grating structure can be placed in frontimage of without the changing aperture the of pos the lens used to form an it plays no role that thethe acoustically-produced lens aperture phase grating at is thein moving velocity this past of manner sound. can A be diffraction seen spectrum in the obtained lower part of Fig. are carried out using the F Differences in the refractive index canSound result waves from consist changes of in adecreased the periodic density. density. sequence of Using regions electrical ofsound components, increased waves one and with of can wavelengths readily ofa order generate narrow of basin 1/10 in m which such an is the order-one position, and the case at two position. To fabricate thissilver grating, film a is wedge-shaped deposited ontothe glass “gaps” have in been high inscribed, vacuum. e.g.silver 5 After film per is millimeter, the converted bytransparent exposure AgI. to iodine vapor into only the first odd orderdiffraction pattern to from the a right step and (Fig. the left. In the only the central zeroth order maximum is present; at with phase structure, in which theincreases thickness in of the the direction “beams” of the arrow. At waves in a flat liquid-containing basin are used as an optical The sound waves produce alight 3-dimensional layered passes grating parallel through which to the the layers. It is observed using the F Figure 21.17 tions by an electrical oscillator circuit.phase Below: grating, A diffraction spectrum of this Figure 21.18 observation mode: The propagatinginstantaneous image. sound They waves are are generated by representedin a the here quartz direction as crystal of which an oscillates the double arrow; it is excited piezoelectrically to osc , 21.17 /2 are B ”, VDI- (that is in the ,belowleft. ). ı Exercises 21.4, ( 21.12 Diffraction 21.17 https://en.wikipedia. ). 21 Der Ultraschall Exercise 21.6 C21.4. In this layered grating, the density, and with it the refractive index, is sinusoidally varied. The result is thus a sine-wave phase grating, whose grating constant is equal to the wavelengththe of sound waves. This produces a diffraction pattern which is similar to that from a line grating with amplitude modulation, if the latter has the same grating constant and narrow gaps ingrating. the See L. Bergmann, “ C21.3. When several emitting areas of width Verlag Berlin (1942), 3rd edition, Chap. 2, and especially p. 118. English: see e.g. org/wiki/Ultrasonic_grating ( and they emit alternately in phase and with a phaseference dif- of 180 arranged side by side toa form grating, as for Fig. 21.5 Additional maxima are much fainter, and are not shown in Fig. order-two position), then the maxima appear unchanged at the same angles and strengths as in Fig. 426 Part II 21.8 The Pinhole Camera and the Ring Grating 427 21.8 The Pinhole Camera and the Ring Grating

We can now continue the discussion begun in Sect. 17.2. The central bright spot seen in Fig. 21.3 d as a small white disk and represented graphically in Fig. 21.3 g can serve as the image point of a pinhole camera. It uses a circular opening as aperture, which admits only waves from the ﬁrst FRESNEL zone, the central zone.

The image point of a pinhole camera is however sharpest when the pinhole Part II allows a region with only 4=5 of the diameter of the central FRESNELzone to pass through (Fig. 21.19).

The bright spot within the shadow of all circular disks (e.g. in Fig. 21.3 a – c) can be used for image formation. Here, we are not limited in our choice of diameter for the disk (POISSON’s spot). Fig- ure 21.20 shows an example. There, the disk was replaced by a steel sphere of 4 cm(!) diameter. More intense images than with only one disk or sphere can be obtained by using a large number of concentric, narrow, circularly- symmetric transparent rings which surround a central opaque disk. When the radii of the rings are chosen randomly, only the radiant power of the diffracted waves adds at every point along the symmetry axis. The phase relations which would give a large resulting amplitude are lacking, and thus the radiant power, which is proportional to the square of the total amplitude, is lower. Fixed image positions are also lacking. Both can be obtained only if the radii of the rings are chosen to be proportional to the square roots of whole numbers, and only light beams with a small opening angle are used. Then the path lengths which lead through two neighboring rings from an object point to the corresponding image point differ only by integral multiples m of the wavelength . The only path differences are D m,

Figure 21.19 The distribution of the radiant r intensity in the shadow of a circular hole that allows only 4=5 of the central FRESNEL zone to pass through (the diameter of the hole and the distances are the same as in Fig. 17.9) Radiant intensity across a diamete 42024mm

Figure 21.20 An image (actual size) formed using a steel sphere as the imaging system (the setup was as shown in Fig. 21.2; the diameter of the sphere is 4 cm, a D 12 m, b D 18 m). The object is a metal stencil about 7 mm high, in place of the aperture L in the figure. t They 90 cm Real ) yields focal poin ::: max f !1 ,3 a 2 D max f (Vol. 1, Sect. 12.14) with , the longest focal length ·const. D ∆ = λ = √m m zone plate . This type of ring grating produces r m for the formation of a focal point. Here, = grating max Circular ring f 21.21 RESNEL 7m). : ). D 2 was obtained. Virtual images can be observed subjec- m max f f 21.22 max f 21.22 . Shorter focal lengths occur at . With it, for example the image shown as a photograph on the . = 2 1 r 21.22 21.22 D is found. A simple geometric construction (with In order totems demonstrate with fixed ring image gratingsFig. positions, and we can zone use plates the arrangement as sketched imaging in sys- right in Fig. tively (with the eye directly behind theFig. grating); but see also the caption of max max f quite intense images as anthe imaging right in system Fig. (an example is shown at an opaque central disk.with It a transparent acts disk in at theover its many center. same real Both manner and distribute as the virtualthe its radiant images sudden power “negative” and transitions focal between points transparent (thisare is and thus a inferior opaque result to rings). of lensesobtainable They as with imaging a systems. lens Only shapein allows the image minimal radial formation. losses order of radiant power i.e. all the waves thatThis reach is a sketched given in image Fig. point have the same phase. the object point is tofrom the it left at are infinity, incident andarrows. the on plane The the waves waves coming ringvirtual which grating and emerge in real from focal the the points. direction rings give of For rise the to two both f In a ring grating, theat width the of same the time transparent reducing the ringsthe width can of be overall the increased, opaque areas rings, of thusway, keeping one the arrives individual at rings a roughly F the same. In this obey the relation zone plate with an opaque central disk. It forms ca. 5 real and virtual focal points Virtual Top left: A ring grating which is suitable for use as an imaging system ( focal point Fresnel zone plate Ring grating RESNEL size actual 1.85 x for red light). Atleft: the A right: F The simultaneous formation of a real and a virtual focal point. Bottom which can be observed with red light ( Figure 21.21 Magnified Diffraction 21 428 Part II 21.9 A Ring Grating with Only One Focal Length 429

Ring grating Screen Object

A FL K C ab

Figure 21.22 The demonstration of ring gratings and zone plates as imaging systems. At the right is the real image of a small stencil, photographed actual size. It was produced by the ring grating shown at the upper left in Fig. 21.21 (a D b 1:8 or 0.9 or 0.6 or 0.45 m; K is the crater of an arc lamp, C is the condenser lens for stencils up to ca. 10 cm in diameter). For the subjective observation of virtual images or virtual focal points, it is difﬁcult for the Part II eye not to accommodate to the bright light source. Therefore, it is expedient to replace the lens of the eye by a convex lens, for example at A, with its focal point at FL. Then we can observe the virtual image on the screen at the right as a real image, while the corresponding real image can be made visible on a screen to the left of FL. (The justiﬁcation of these statements can be seen in Fig. 21.21: Imagine that there, directly behind the ring grating, we insert a large convex lens, which encompasses the whole surface of the grating).

21.9 A Ring Grating with Only One Focal Length

Besides the linear line grating with straight-line slits and beams, we have also treated ring gratings in Sect. 21.8. The most well-known of these is a limiting case, the FRESNEL zone plate, with its abrupt transitions between completely transparent and completely opaque rings. An especially important form of the ring grating can be fabricated photographically using interference. We again use two coherent, monochromatic wave trains of light which intersect, but this time, one has spherical symmetry and the other is a plane wave. A photographic plate is placed in the resulting interference field2.Fig- ure 21.23 shows an example. Ring gratings which are fabricated in this manner by photographing an interference field are called sine-wave ring gratings, since two of their characteristics are similar to those of sinusoidally-modulating line gratings (sine-wave gratings; see Sect. 21.6): 1. In a sine-wave line grating, aside from the central collimated light beam which is not deflected (zeroth order), only the two sub-maxima of order m D 1 remain. In a sine-wave ring grating, aside from the central maximum, only the two focal or image points belonging to the order m D 1 remain (this can be demonstrated with the setup from Fig. 21.22).

2 If the general clarity of the experimental setup is not important, we can make use of a more modest arrangement: We pass a collimated light beam through a filter practically perpendicular to the planar side of the lens used for the demonstration of NEWTON’s rings (see the end of Sect. 20.9), and then we project an image of this surface onto the photographic plate, using a lens with a long focal length. . 3 Z O 2 suf- b Z as light .Inorderto 21.24 C16.4 Photographic plate incident light . The spherical waves which P -collimated monochromatic light . illuminated with 21.22 Z Mirror Photo- 5mfor plate graphic : 1 ). In both sketches, the four object points lie in a single f P 18.13 Top: A ring grat- b as centers of Object points spherical waves sources. The setup sketched at the upper left in Fig. fices. There,beam (i.e. a a plane-wave beam) nearly is incident perfectly on a photographic plate On the way there, it is scattered by four small particles (of Al red light). Bottom (on ascale): different Its photographic fabrication in an interference field.demonstrate In order this to grating as aning imag- system and to show theof position its focal point, wemental use setup the described experi- in Fig. ing (magnified 3.4 x) withone only focal length ( Figure 21.23 or ‘alumina’). Theseglass particles, plate, carried serve as by four a object points slightly wedge-shaped 2. Several sine-wavecan line be gratings superposed withanalogous without different behavior losing grating is their found constants gratings, on individual using superposing long character. several wave trains sine-wave of ring The monochromatic light. This important fact can bechromatic readily radiation demonstrated today, has since become mono- available using lasers are emitted by thethe scatterers primary interfere collimated both with lighta each beam. section other through and The with the photographic resulting plate interference records field. It can be seen on ≈ 3° At the right: Four sine-wave ring gratings which overlap each other (photographic or as four sources of spherical secondary waves are Telescope P plane waves 3 m). Lower left: A technically-important variant: The small particles which serve as four object : Monochromatic make room for the mirror,using the a cross-section telescopic of optical the system monochromatic (Sect. plane. light This beam is (laser not beam) necessary, is but enlarged it is experimentally convenient. positives, magnified 2x). Upper left: Their photographic production in an interference field ( points 1 Figure 21.24 Diffraction 21 430 Part II 21.10 Holography 431 the right in Fig. 21.24: It shows the superposition of the four sine- wave ring gratings belonging to the four object points. These, as claimed above, should not have lost their individual character if we use long wave trains of monochromatic light for the experiment. The empirical proof: If we remove the four object points and illuminate the developed photographic plate with a monochromatic, collimated light beam, then at a distance b to the right of Z on a screen (not showninFig.21.24), we find real images of the no-longer-present object points! These are the real focal points of the individual sine-wave ring gratings.

The corresponding virtual images of the no-longer-present object Part II points can be observed subjectively: We look through Z as if looking through a window. It is usually more convenient to observe the virtual images also as real images projected onto a screen; for this, we proceed as shown in Fig. 21.22.

21.10 Holography

Compact disks and magnetic tape can record sound waves (as a linear sequence) with the correct amplitudes and phases, and reproduce them on demand in the same form as the original, no longer extant sound waves. The experiment described in Sect. 21.9 does the same for light waves (as a two-dimensional recording). That experiment demonstrated the principle of “holography”. The image on the plate Z shown at the right in Fig. 21.24 was the hologram of an object which consisted of only four object points. Imagine that these four object points are replaced by the set of all points which make up the monochromatically-illuminated surfaces of some arbitrary objects. Then on plate Z, we will see no recognizable interference patterns from individual sinusoidal ring gratings. To the uninformed eye, the hologram appears to consist of a nearly unstruc- tured grey surface. It is all the more surprising that it can produce a virtual image, for example when we look through the hologram, illuminated with monochromatic light, as if through a window. A normal photograph shows an impeccably lifelike perspective when viewed with one eye from the correct distance (Sect. 18.16). A hologram is however superior: It permits us to – see an object as if we were gazing at the original itself, and – see other, previously hidden objects appear when we change our viewing angle! The origin of this superiority is easily understood: Normal photography makes use of only the first of the two characteristics of a light wave in the recording process on a film or plate, namely its amplitude. The square of the amplitude is proportional to the radiant power which arrives at a given “image point”. Hologra- phy uses in addition the second characteristic of the waves, i.e. their phases. It thus increases the amount of “information” stored in the ,in ,left llimated direction 21.25 of the radia- refraction (1938), then by OERSCH ). has undergone rapid 3 is used in Fig. , or at most its 21.9 C21.6 ). phase reduce the intensity , we have already seen the essen- 16.8 ng of Invisible , a light beam which passes through (1962) deserve mention. These authors no longer gas, and on the right side, the internal struc- 2 gas streaming from a nozzle into the room air 2 PANIEKS . On the screen, therefore, at certain positions some We can expect many more applications of holography Objects. The Schlieren Methods and I. U (from 1948). Among the numerous later works, especially those of simple schlieren method , one without deflection and the other deflected in part by C21.5 diffraction EITH Two groups of light beams are distinguished by their different such beams, which are formed by scattering of a monochromatic, co ABOR The principle was discussed nearly a century ago. Its experimental realization of the light is missing,receive and additional these radiation, points and seem darker. therefore Other seem positions brighter. With this part by tials: directions refraction, in part by diffraction. to come.tion: One Holography point does not is requirewavelength for monochromatic particularly the light production important of of the holograms and(analogously and/or same to for deserves the viewing last them men- paragraph of Sect. the jet of gasmogeneities) is or deflected slightly through to the the sides, interior in part of by a sheet of glass (its inho- ture of a sheet ofilluminated glass. uniformly. Explanation: However Normally, the screen would be progress. side, to show a jet of CO D. G used simply a beam ofmany plane waves as reference frame for the phases, but instead beam of light by a frosted-glass disk (cf. is invisible. A smoothly-polishedin sheet its of interior. glass These showssee; things no the are structure not reason too forthe small their visible or invisibility radiation too is which distanteyes quite passes for cannot different: through us register. them to They in They modify ways do which not our tion, but rather they change only its took place, as usual, in a series of steps, first by H. B 3 21.11 Visible Imagi Some objects have neither visibletures. outlines nor A visible jet internal of struc- CO photograph. The frame ofcase reference for is these a phases plane inplate wave the at which simplest the is same incidentobject time on points. as the the photographic spherical film waves or thatThe emerge from technical the development of holography E.N. L very slightly. Such invisible objects canWe be bring made them, visible like by antical object using axis which a can of simple cast a trick. lamp). a nearly This shadow, pointlike into “simple schlieren the light method” op- source (e.g. the crater of an arc https://spie. Physik Journal . Diffraction (2003), No. 3, p. 37. En- 21 C21.6. “Schlieren” refers to the regions which influence incident light due to inhomogeneities in an otherwise homogeneous material. C21.5. Holography is today very widespread and finds a variety of applications, e.g. on bank notes and identification cards, in data storage devices and even in archeology. See for example B.K. Buse and E. Soergel, 2 glish: See e.g. org/Documents/Publications/ 00%20STEP%20Module %2010.pdf 432 Part II 21.11 Visible Imaging of Invisible Objects. The Schlieren Methods 433

Figure 21.25 Two images, photographed with the simple schlieren method: At left, ajetofCO2 gas, moving downwards in a laminar ﬂow. At the right: a section of a sheet of glass (a rinsed negative, 912 cm). (The distance between the light points to the object and from the object to the screen is several meters in each case; the image is about 1=3 actual size). Part II E S L1 L2 + α Image of light source as pupil Aperture E ab c

Figure 21.26 TOEPLER’s schlieren method. The object plane EE corresponds to the slide in a slide projector. Only a single partial beam belonging to the object point ˛ in the object plane is sketched with its two bounding rays (imagine that there is a small opening at ˛). The diameter of the lens L2 used for imaging must be larger than the entrance pupil. The “aperture” can be either an opaque disk (dark field) or an opening in an opaque screen (bright field). The lens can be tinted in zones outside the pupil, e.g. from inside to outside red, green, etc. Then we see weak schlieren, which deflect the light only slightly, as red, while stronger deflections are seen as green, etc. A numerical example for a demonstration experiment:LensL1: f1 D 1m, diameter 12 cm; a D 1:5m,b D c D 2f2 D 4m.

A second step leads to a more refined method, the TOEPLER schlieren method: This method makes use of either only the deflected beam or only the non-deflected beam. The experimental arrangement (Fig. 21.26) is the same as for the usual projection system (Fig. 18.33), but with one of the following small additional modifications: Either we use a disk to block off the image of the light source (i.e. the entrance pupil at the lens used for image formation, an aperture illuminated from behind), or we block the whole surface of the lens with the exception of the entrance pupil. In the first case, the field of view is in general dark. It is illuminated only where light beams that have been deflected to the side can reach the lens outside the entrance pupil. The structures appear bright against a dark background on the screen: this is called dark field illumination (Fig. 21.27). In the second case, no deflected radiation can reach the lens; the structures appear dark against a light background: bright field illumination. ,in ’in .It Z . IV ˇ .Inthe 21.28 ˇ cross-section, , flowing down- 2 ’s schlieren method diffraction pattern 3 actual size) = small ). The light rays which OEPLER of the empty frame 0 of the disk, black on a light 0 ˇ 17.2 , both groups of light rays act structure: It contains a small,

S , but also the radiation from this come exclusively from this image S deﬂection of light beams by diffrac- (Sect. S , they produce an empty, uniformly ’s Description of 1 S treated visible structures in 1873, and L amplitude . The more important dimensions are BBE BBE to plane A A in otherwise open surroundings. In plane 21.26 Z , corresponding to the column labelled ‘

, the object is a large, empty frame B IV RNST RNST Microscopic Image Formation Two images photographed with T . The light source should have a , the object has an (position arrive at the image plane C Z Z 21.28 predominates. This holds both for invisible as well as for visi- illuminated field of view, i.e. the image opaque circular disk In Part addition to the sharp image offrom the the light small source, disk the appearsThis (both shown time, as not photographic negatives). onlyplane light rays from thediffraction image pattern. of In the the image lighttogether plane source and in produce the sharp image using dark-field illumination; at left: a turbulent jet of CO background (shown as a photographic positive). wards. At right: A section of a sheet of glass (around 1 propagate from plane of the light source. At plane his considerations on theproved role to of be diffraction very in fruitful. the microscope have An experimental setupdemonstrations which to we a small have audience found is to illustrated be in Fig. suitable for tion ble structures. E Figure 21.27 21.12 E When the objects are very small, is analogous to Fig. indicated, and experimentalFig. details can be seen in the caption of of the light source (shownthe as whole a photographic open negative), area produced by of lens the tabular lower part of the figure), there is a sharply-focussed image drawn in the sketch as a square. InthefigurePart plane Diffraction 21 434 Part II 21.12 ERNST ABBE’s Description of Microscopic Image Formation 435

Z S L L 1 0.3mm 2 1mm α 40mm 3mm α' A ca. 0.5m ca. 5m I II III IV V

B β β' Part II 1

2 γ C γ' 1

2 δ D δ' 1

2 ε E ε' 1 Transparent, Non- Non-transparent refractive transparent large refractive index = 1 index

Figure 21.28 Imaging of objects which themselves do not emit light, with amplitude structure and with phase structure. The structure consists of many randomly-arranged circular disks (about 2000 in each case). Their production is described in Fig. 21.29. In the ﬁgure, only one of these disks in the object ( ) and one in the image ( 0) are drawn. The diameter of the diffraction patterns in position IV (ABBE’s intermediate-image plane) is in reality smaller than the diameter of the imaging lens L2.

The requirement that both of the groups of light rays from plane Z are necessary to produce the image in the plane S can now be demonstrated by convincing experiments: 1. We place an iris diaphragm in the intermediate-image plane Z,then gradually reduce its diameter, and thus, beginning from the outer rim, we block off the diffraction pattern. The result is that the image of the disk gradually becomes fainter and fuzzier. 2. In the limiting case, the iris diaphragm allows only the direct light from the light source to pass through. The result is that the image 0 . ı 0 of

dark- ,there is thus with the C S can arrive S . ; these give Z and the uni- in the image B , represented V 1 S Z L , represented by . As a result, the ).

and V . (In Part ). These rays alone S IV ). These rays produce III and ), represented here by 21.11 , at the image points III S IV 21.2 . S rk disk on a light background in becomes dark. The image and interfere with the rays from the S with a small disk. The result is that Z at 2 (position , we decompose the rays formally into Z III arrows at positions ’s theorem (Sect. which is essential for image formation. A correct S downwards (see the conclusion of Sect. ABINET , we describe image formation of an amplitude struc- Visible Under the Microscope C . The parallel directions of the vectors represent the fact appears somewhat fuzzy and bright on a dark background; . II V

). S oppositely-directed whole surface of the lens in the image plane an arrow pointing the diffraction pattern in plane possible only when both the groupthe 1 of group rays 2 from the of light rays source from and the diffraction pattern in plane reproduction of the amplitude structure in the image plane without hindrance at the image plane Between these two groups of rays, there is a phase difference of 180 the field of view on the screen 3. We nowof remove the the light iris source diaphragm in and plane block the sharp image the disk 2. Another group of rays emerging from the object by the arrows pointing upwards at 1 (position Every change in the groups of raysin 1 the or image plane 2 will change the interference the rays cancel each other, leavingposition a da 21.13 Making Invisible Structures Most thin sections of organic materials forin microscopic biology examinations and in medicineically are distinct transparent structural and elements colorless; differ their in chem- terms of visible light only according to B can no longer be seen; the field of view on the screen plane is just uniformly illuminated, as in the figure Part i.e. we see thefield amplitude illumination structure of the object imaged with With the aid offigure, these Part and similar experiments, and referring to the an arbitrary reference pointfrom for the the small square phases image ofplane of those the rays light source which to pass an image point in form illumination of the field of view in plane produce the sharp image of the light source in plane same phases. At position two groups: 1. One group of rays from the whole surface of lens ture in the following way: Afterwe they mark have passed the through phases theposition of object, the remaining light rays by vector arrows at that the rays from individual points reach the image plane are also arrows pointing upwards at positions Diffraction 21 436 Part II 21.13 Making Invisible Structures Visible Under the Microscope 437 in their somewhat different refractive indices. Often, important structures are just as invisible as the inhomogeneities in a sheet of glass. Most such sections, stated briefly, have practically only phase structures. In order to make these structures visible, one has to convert them into amplitude structures, i.e. the small differences in their refractive indices must be converted into large differences in their light absorption. To this end, the sections are colored or “stained” with dyes, which are taken up selectively by the different structures. This staining is a chemical intervention and causes considerable deviations from the original state of the living tissues. For this reason, several procedures have been developed for microscopy Part II which can make the phase structures visible without the use of dyes. These procedures are best explained using the terminology of ABBE (Sect. 21.12). We continue the series of images from Fig. 21.28 and show as Part D an object ı with phase structure: The opaque disk in the figure Part C has been replaced by a transparent disk ı. It differs from its surroundings only by having a somewhat larger refractive index. The light which passes through this disk arrives at the image plane S with a phase delay. This is represented in positions II and V by a rotation of the vectors in a counter-clockwise direction. In the intermediate-image plane Z, the image of the light source is surrounded by the diffraction pattern of the disk, just as in the figure Part C. Light from both propagates to the image plane S. Its superposition there leaves the illumination of the image ı0 just as strong as in its surroundings, so that the phase structure remains invisible. In order to make it visible, some sort of intervention in one of the two groups of light waves leaving plane Z is necessary, e.g. a partial blocking of the diffraction pattern, or blocking of the image of the light source. In either case, the disk will become visible at the location of the image ı0. A partial blocking of the diffraction pattern in the intermediate-image plane Z can be obtained in a particularly simple manner by using an oblique illumination. We shift the light source to one side and thus at the same time, the diffraction pattern shifts (in the opposite direction). Then we can use the lens mount of the objective lens L2 to cut off an outer portion of the diffraction pattern. Such a procedure is however a rather rough intervention. A finer method, which produces much better results, is that published by C21.7 F. Z ERNIKE in 1932, the phase-contrast method. We describe C21.7. FRITS ZERNIKE, it in terms of its most important application, to small differences in 1888–1966, Dutch physi- the refractive indices. Parts D and E of Fig. 21.28 serve to illus- cist. For his invention of the trate the method. In these figures, as mentioned above, the phase phase-contrast microscope, vector behind the disk ı (position II)wasrotated slightly in a counter- he was awarded the Nobel clockwise sense. In position III, the phase vectors are again de- prize in 1953. composed formally into two components. The components denoted as 1 produce the image of the light source in the intermediate-image plane Z (position IV, vertical arrow). The component denoted as 2 produces the diffraction pattern in plane Z (position IV, nearly horizontal arrow). At position IV, when there is a phase structure, the mto 13 , while for the .Todothis,in ı ı ), which delays the , i.e. they are directed IV was obtained; it contains ı m), mechanically-ruled 9 21.7 ). 10 in 1912. Such gratings consist 22.6 2 . Starting from the condition for AUE 000 w a continuous transition to a nor- D ). It can be improved still further by < to an angle of around 180 ,and 21.29 , the two vectors 1 and 2 act in the image ı 00 ı , and this produces a good contrast relative D ; 0 D , the image of the light source is blocked by a small E difference and, in particular, the recording of diffraction spectra A section of a phase structure (magnified about 3x) which is (a “quarter-wave plate”). With their path difference now to each other. We can, however, retroactively increase the ı structure, their mutual angle is 180 m. The fundamental experiments on diffraction and interfer- C21.8 8 m), 10 5 the figure Part transparent disk (the whitelight by circle 90 in position the lattice planes), of a three-dimensional series of latticeture points or with lattice planes, three a struc- (in general different) grating constants (spacings of amplitude invisible to thecircular eye disks without are making madeevaporated use in of high of LiF, vacuum. embedded some As stencilused in special for with the Canada which trick. evaporation, the balsam. the diffraction same pattern mask The The in was Fig. small LiF was gratings play only3-dimensional a gratings provided limited by role. nature intices, as the first form Instead, suggested of by one crystal M. v. lat- makes L use of the oppositely phase angle of around 90 plane as their to the surroundings (Fig. making the quarter-wave plate weakly absorbing, anding thereby the adjust- lengths ofabsorption vectors is 1 increased, and we 2mal follo to microscope be with dark-field more illumination. nearly equal. As the increased to around 180 using typical optical reflection gratings made ofare metal or employed glass. with These nearly grazinggratings incidence, is since sufficiently fine the only rulinga when on very it the flat is incidence strongly foreshortened angle by (see Sect. For short-wavelength X-rays ( 21.14 X-ray Diffraction The wavelengths of X-rays lie in the range from about 10 Figure 21.29 two arrows 1 and 2 form an angle of only about 90 ence can be carried outWe just mention as for well example with10 X-rays diffraction as by with a visible single light. slit (slit width 5 to around 2000 randomly-arranged holes of 0.3 mm diameter. ”, Vieweg and Die Physik der Rönt- Diffraction 21 Son, Braunschweig (1912), Chaps. 2 and 9. tion by the author: Robert Pohl, “ genstrahlen C21.8. As an historical note, we mention here the following relevant publica- 438 Part II 21.14 X-ray Diffraction 439

Figure 21.30 ALAUE dia- 031 131 221 311 301 gram obtained from an NaCl crystal. The number triplets refer to the orders m0; m00 and m000 in Eq. (21.2), corresponding to the three spatial directions. Part II

C21.9. A further histori- constructive interference by waves reflected from a simple layered cal note: Up to the 10th structure (see Sect. 22.7, and also Vol. 1, Sect. 12.15 and Fig. 12.41c): edition of “Optik und Atom- physik” (1958), a discussion m of diffraction from layers sin m D (21.2) and from three-dimensional 2D point gratings was given. For .m is the order, and is the “grazing angle”/; a detailed treatment in En- glish, see e.g. A. Khavasi, it can be seen that this condition must be met in a three-dimensional K. Mehrany, and B. Rashid- grating for all three spatial directions in order to observe interference ian, “Three-dimensional maxima.C21.9 One obtains a pattern of spots, as shown for example diffraction analysis of grat- in the LAUE diagram in Fig. 21.30. Figure 21.31 shows an historical ings based on Legendre experimental setup. expansion of electromagnetic fields”, Journal of the X-ray diffraction has attained its primary importance today in the Optical Society of America field of crystallography. It is the most important method for investi- B24 (2007), pp. 2676–2685 gating crystal structures, currently in particular those of large molec- (online at ular crystals such as proteins. One uses X-rays of known wavelengths https://www.osapublishing. and determines not only the positions of the interference maxima, but org/josab/abstract.cfm? also the distribution of the radiant power over spectra of different or- uri=josab-24-10-2676 )

C21.10. Experimenting in this manner with X-rays in R S B the lecture room is certainly no longer permitted today! The observer at the right in the picture is H.U. HARTEN (Dr. rer. nat. 1949). A detailed description of this experiment by W. MAR- Figure 21.31 A convenient setup for the individual observation of LAUE diagrams.C21.10 R is the X-ray source, with a tungsten anticathode at a volt- TIENSSEN can be found age of 6 104 V; B is a lead shield with the crystal under investigation at its in the video “Simplicity center, in front of an aperture of 2.5 mm diameter; S is a fluorescent screen is the mark of truth” (at with a metal disk at its center to block the direct X-ray beam. ca. 16 min.). , 1916). (220) (200) (111) Opening Diffraction lines CHERRER CHERRER Film and P. S radi- of the ˛ r and P. S K Film 1539 nm) EBYE : 0 :The D EBYE . (Grating constant m r rn is captured by photographic film 10 21.32 powder 10 ) is passed through the powder sample elementary lattice components. 2 Crystalline 539 : , a narrow, collimated beam of X-rays (cross- 1 D 21.32 A supplement to Fig. The experimental setup of P. D 3518 nm; the numbers in parentheses are the in- : 0 X-rays D As shown in Fig. is reflected by three different familiesin of a lattice microcrystalline, planes well-tempered nickel wirestituted (sub- for Ni powder). The radius of curvature film was 121 mm and its length was dices of the reflecting lattice planes.the A primary circular beam hole is for at the center of the film.) D sectional area around 1 mm ation from copper ( and the resulting diffraction patte bent into a circular form (ordetector). by a The digital-electronic diffraction position-sensitive pattern consists of a system of concentric Figure 21.33 Figure 21.32 ders. From thisthe distribution, detailed one structures can of compute the backwards toThis obtain important methodmeans of limited crystallographic to large investigationtalline single is powder crystal can by also samples. be no used Arbitrarily (P. D fine crys- Diffraction 21 440 Part II Exercises 441 rings (Fig. 21.33). Their interpretation is simple: In a powder sample, the orientation of the crystallites is random. All of the lattice planes on which the X-rays are incident at a grazing angle (Eq. (21.2)) reflect this incident radiation. With large-grained powder, one can clearly see that the rings are composed of a series of individual spots.

Exercises Part II

21.1 How does the diffraction pattern of a slit change a) if it is completely covered by a transparent glass plate (a microscope cover glass d D 175 m thick, with a refractive index of n D 1:50); b) if only one half of the slit is covered with the glass, parallel to its long axis? c) By what angle ˛ must the glass be tipped to pass from an order- one position to an order-two position? (The wavelength of the light is D 600 nm and the inﬂuence of refraction in the glass on the phase difference is negligible.) (Sect. 21.5)

21.2 Derive the diffraction pattern in the order-two position in Fig. 21.12, below left, making use of the graphic construction in Fig. 12.34 of Vol 1. (Sect. 21.5)

21.3 Why must the slit width B of a high-resolution grating spectrometer be small in comparison to the grating constant D?Toanswer this question, consider up to which order m one can observe the interference maxima in the intense primary maximum of the diffraction pattern from the entrance slit, a) for D D 2B (Fig. 21.13), and b) for D D 20 B,i.e.(B D). (Sect. 21.6, 22.2)

21.4 From the line grating with an amplitude structure as shown in Fig. 21.13 (grating constant D D 2B,whereB is the slit width), we can construct a phase grating if the opaque lines ruled on the grating are made of transparent material. At which angles do the principal interference maxima lie (cf. Fig. 22.6), if the transparent lines delay the phase of the light waves by a) 360° (order-one position) and b) 180° (order-two position)? (Sect. 21.7)

21.5 From the diffraction pattern shown in Fig. 21.17, determine the ratio of the width L of the ruled lines on the grating to the grating constant D. (Sect. 21.7) -th order at the ), with a sound m 21.18 ) 21.7 experiment (Fig. Hz in xylol, and employing red-ﬁlter light 6 EARS . The grating constant of the sinusoidal phase 10 4 -S D in the liquid. (Sect. 10

c 8 EBYE m D m In the D ˛ 700 nm), one observes diffraction maxima of D angles ( grating is equal to the wavelengthvelocity of of the sound sound waves. Determine the 21.6 wave of frequency Diffraction 21 442 Part II opr twt h htgahcsetu erdcdi Fig. in reproduced spectrum photographic the with it Compare n ls lc ihpaeprle ufcs tsostesetu from spectrum the pos shows at It photographed surfaces. lamp, mercury-arc plane-parallel low-pressure with a block glass a ing pia Spectrometers Optical 1 22.1 Figure ranges wavelength convenient of very range wide as a mainly for instruments available, recording commercially de- highly are technically and currently veloped are apparatus spectroscopy Optical Their and Spectrometers Prism 22.1 length lmnaytetet ei with begin treatments design Elementary the of questions apparatus. basic spectroscopy the of operation only and consider to needs physics ern o hsrao,w rdcdteﬁs eosrto facontinu- a of demonstration Sect. ﬁrst in the spectrum produced light ous we reason, this For ta the stead h hre ilb t mg.Btteei ii oti;i ti ex- is it if this; to limit a is there an obtain But longer no image. we its ceeded, be will sharper the ilbe will ih rqec reeg.Ti uniyi fe ernosy aldthe called (erroneously) often is quantity This number energy. or frequency light .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © sufruae h rs hp sntesnil u ahrthe rather but essential, not is sion shape prism the unfortunate; is t nrneslit entrance its aegasbokt htgahteln pcrmsoni Fig. in shown spectrum line the photograph to block glass with same block glass thick a o oohoai (si monochromatic for rs pcrmtrbsdo iprini kthdi Fig. in sketched is dispersion on based spectrometer lamp. Hg-arc prism low-pressure A a from lines spectral the of consists which eodn pcrmtr fe eitrsetanta ucino h wave- the of function a as not spectra register often spectrometers Recording ftemtra,ta stedpnec fterfatv index refractive the of dependence the is that material, the of h eirclo eghi o number! a not is length a of reciprocal The . u ahra ucino t eircl1 reciprocal its of function a as rather but , imaged ifato pattern diffraction eovn Power Resolving iesetu,otie o ihapim u nta us- instead but prism, a with not obtained spectrum, line A nteosrainsre.Tenroe h slit the narrower The screen. observation the on S 0 silmntdwt monochr with illuminated is λ =35454656578 546 436 405 =365 ngle-fre olsItouto oPhysics to Introduction Pohl’s plane-parallel 16.10 falgtba hc a enrestricted been has which beam light a of uny ae nterwavelengths their on waves quency) image ·10 o ihapim u nta using instead but prism, a with not rs spectrometers prism –9 m ftesi ntesre,btin- but screen, the on slit the of ufcs eas sdthis used also We surfaces. = D mtclgt h slit the light, omatic = ition , c https://doi.org/10.1007/978-3-319-50269-4_22 rprinlt the to proportional , b .Thisname nFig. in 27.3 . 1 22.2 disper- Mod- . 16.30 22.1 wave .If S 0 n . , . , 22 443

Part II , , I n 0, II S ,it d 0is and .We λ / .For D 0 S need to d B S < 0 + d <

D S λ C λ d 16.29 = Screen . n KY . (This condition for a limiting case: KX . Light of wavelength ) with d n . We observe d 22.3 mode. The formation of resolved denote points along the path (that is, the sine of the open- 16.10 Y II 22.3 II and is illuminated with two wave- ; γ (Sect. X d is the length of the line segment B ; γ d on the wavelength scale, we see two K 22.2 ./ andiscloseddowntosuchanarrow XY KY n RAUNHOFER )), the only decisive factor is the numerical . For light of wavelength between these two angles is d S criterion). Y has a refractive index K

C22.1 . For the solid-line diffraction pattern, the images 19.4 , in Fig. 0 and / ,where have a short focal length, nor does the slit S I , the optical path passes through the base of the . To a good approximation, d n I / K . .ThisissketchedinFig. ,Eq.( n C 2 AYLEIGH of the two diffraction patterns just touch each other, d S H . C . This pattern lies in the focal plane of the lens is thus negative). (For individual observations with the eye, D 0. A dispersion B n must be illuminated. ( n . , both light beams, the one drawn with solid lines and Schematic of a prism spectrometer which makes use of disper- < K and (d collimator KY S n n d : 22.2 ), which is refracted by a smaller angle, has a smaller refractive in- normal dispersion S D 0 C . On the screen, two diffraction patterns appear, belonging to d S n B C = KX n is known as the R so that the central maximummum of of the the other. one With pattern this spacing,the falls the two in eye patterns the is as just ﬁrst separate; able mini- they to are distinguish barely diffraction patterns The halfwidths light of wavelength prism such diffraction patterns was describedand quantitatively in 12.13 Sects. of 12.12 Vol.undamped, 1, i.e. and monochromatic experimentally soundca. demonstrated waves 1 cm. there with For using a light, wavelength a of similar experiment is shown in Fig. imagine that the slit be set at an inconveniently narrow width. When combined in a tube, and is thus observed in the F width that instead of two are called a ing angle of the beam).necessary Therefore, that for the the lens observation of line spectra, it is not aperture of the light beam to the right of lens between the points is the one with dashed lines, have practically the same widths In Fig. to the width Figure 22.2 of the upper light beamon in the the ﬁgure). screen In (W/m determining the irradiation intensity dex a transparent wavelengthserved scale from is the right used usingthe a instead prism magnifying at of lens (ocular).) the The screen, whole surface and of it is ob- ( sion. Light of wavelength so that the wavefrontsangles. of the The two light difference beams d are tilted by different lengths, d the two light beams as sketched in Fig. that is, d called . tity, will be in- Optical Spectrometers 22 C22.1. The quantities pre- ceded by the letter d inchapter this are not differentials, but rather small differences in the corresponding quan in this case the wavelength troduced later for the “usable wavelength range”. The notation 444 Part II 22.2 Grating Spectrometers and Their Resolving Power 445

Figure 22.3 The resolving power of λ

a prism spectrometer. The observer looks I from the right, that is into the direction of the oncoming light, towards the screen sketched in Fig. 22.2. The width H of a diffraction pattern line at the points Radiant intensity where its ordinate values on both sides Wavelength interval d =

have decreased to half of their maximum λ value is called its halfwidth H. I

Angular spacing α Part II from the center line 0

ﬁrst minima on both sides of the center line are marked. As seen from the midpoint of lens II, they are separated from the center line by the small angle ˛ D =B (according to Eq. (16.23)). The dashed diffraction pattern, which belongs to . C d/, has practically the same shape, and it is clearly separated from the solid-line pattern when d D˛,orS dn=B D=B (RAYLEIGH criterion). This leads us to the resolving power of the prism: dn DS : (22.1) d d In words: The resolving power of a completely illuminated prism is determined solely by its base length S and by the dispersion dn=d of the prism material (as shown in Fig. 22.2). (In the case of anomalous dispersion, that is when dn=d>0, the minus sign should be removed).

Numerical example: Assume that a prism with a base length of S D 1cm is made of ﬂint glass, with a dispersion in the “yellow” spectral range of jdn=djD103/cm. Its resolving power is then =d D S jdn=djD 1cm 103=cm D 103. The prism can just separate the two sodium D D : D : lines. Their wavelengths are D1 589 0nmand D2 589 6nm.Their spectral separation thus requires 589 nm D 103 : d 0:6nm Using three prisms one after the other, made of the same material, each with a base length of 10 cm, we could attain a resolving power of =d D 3 104.

22.2 Grating Spectrometers and Their Resolving Power

Grating spectrometers can be used in all wavelength ranges, at least from the region of X-rays out to electromagnetic waves of several centimeter wavelength. For visible light, the prism in Fig. 22.2 is replaced by a line grating (Fig. 22.4). Line gratings (Fig. 21.13)were 1). D .For (22.2) m 250, the length . is the order light-wave = 1 = 0 th order, by m m m N N m and wavelength ). Now, we want N th order occurs for m orders m , 1821). th order to the sym- must fall at least in m = m ' 2 f / C and wavelength , this path difference has d / ). m th order is separated from 1 m C by the objective and the screen 2) submaxima, and thus by ’s interference experiment, m C II D . to ( m m RAUNHOFER each two neighboring maxima, . N . It follows that it has increased ) adjacent to the spectral line of OUNG D m III

m Grating become sharper while retaining their by ( is the path difference of the wave trains according to their ˛ 0) and a spectral line of ﬁrst order ( / Exercise 22.1 1 m D sin ofaspectrallineof 1 ). The boundaries of the light beams are drawn ). The spectral line of C f ), which follows the line of m light beams, to a grating with ˛

m between two neighboring wave trains. For the . wavelength 22.6 16.26 0 two S Slit ’s interference experiment. Gratings sort spectral m , namely from wavelength is usually left off and the screen is placed at a distance of whole of II OUNG C for light waves. In the bottom image, with A grating spectrometer (J. F same from the spectral line of order Red filter or “grating constant”; . In this process, between grating lines, a spectral line of K / 22.5 fraction is sufﬁciently small). d is the spacing of two neighboring openings or wave sources, the N 1) minima (Fig. 2) submaxima appear (Vol. 1, Fig. 12.40). This is demonstrated 0 D S from two neighboring openings). ( period C N N to be able to distinguish a spectral line of order only a that purpose, the line of wavelength by the focal plane of a telescope ( For subjective observation, we replace lens in for the central maximum ( of the diffraction maximum or minimum.right-hand lens For demonstration experiments, the several meters (cf. Fig. ( for the next minimum ( increased by a a path difference of next following spectral line of order . (or outside) the ﬁrst minimum ( lines of the metry plane perpendicular to the plane of the grating, we have found: For the angular spacing which makes use of positions beams, the interference maxima ( in Fig. submaxima have practically disappearedhave become and very the narrow diffraction principalslit patterns maxima (when the width of the In making the transition from Y Figure 22.4 treated in detail in Vol.tensions 1 of (Sect. Y 12.15 and Fig. 12.65), based on ex- With the neighboring order Optical Spectrometers 22 446 Part II 22.2 Grating Spectrometers and Their Resolving Power 447

Figure 22.5 The inter- m = –3 –2 –1 0 1 2 3 N = ference pattern of a line grating, showing its depen- a 3 dence on the number of interfering light beams (the number N of grating lines). m is the order. For the fig- b 4 ure Parts a-e, red filter light suffices; for f , the light from a sodium-vapor lamp was c 5 used (photographic nega-

tive). Part II d 6

e 10

f ~250

⎡ + λ ⎤ + ∆ = mλ ∆ = ⎣ mλ N ⎦ ∆ = (m 1)λ ⎡ − λ⎤ Produced by wave trains ∆ = ⎣(m + 1)λ N ⎦ from neighboring grating H H lines with a path difference ∆ (N–1) minima γ β ∆ = m(λ + dλ) ∆ = m(λ + dλ)

H H

Figure 22.6 The resolving power and the usable wavelength range (see Sect. 22.5) of a grating spectrometer. In the interest of clarity, the spectral lines (solid and dashed lines) are not drawn beside each other, as in Fig. 22.3, but rather one above the other. The observer is looking at the image plane of the lens II in Fig. 22.4 along the direction of the light propagation. If N, i.e. the number of grating lines, is increased in this ﬁgure, then the submaxima already present move together on both sides of the neighboring principal maxima; the ratio of their heights to that of the principal maximum remains unchanged. At the center between two principal maxima, new submaxima are Video 20.1: formed, and they become smaller and smaller in height. Compare Sect. 12.15 “Interference” in Vol. 1 (Video 20.1). http://tiny.cc/9eggoy. The second part of the video shows interference patterns wavelength (RAYLEIGH criterion). With this condition, we obtain from different gratings (Grating constants: 20 and 10 m) using light from m. C d/ D m C or D Nm : (22.3) N d a laser ( D 633 nm, beam diameter 3 mm) and the In words: The resolving power =d D =d ofagratingintheﬁrst- lecture-room wall as observa- order spectrum is equal to the number N of grating lines. For spectral tion screen. shape m . Numerical 436 nm) has –9 m . D

577

, sometimes the corre- 546 halfwidths is deﬁned as the halfwidth . The latter is the difference H H λ D

436 ).

60. 405 Wavelength shows the line spectrum of a high-pressure

27.3 =

, it increases proportionally to . 365 m D 22.7

22.6

H 334

313

302

, and second, its halfwidth 297 289 The spectral distribution of the radiant intensity from a high- 0 300 400 500 600∙10

of Spectral Lines 6000, and thus has no measurable inﬂuence on the shapes of the

wavelength difference

Wavelength interval d interval Wavelength D

ϑ I Radiant intensity intensity Radiant d between the two frequencies at which the ordinate (line strength = a large halfwidth, mercury-vapor lamp, registered withspectrum, a for example, prism the spectrometer. “blue” spectral In line this (at of the line. As an example, Fig. sponding or amplitude) has decreasedspectrum to is half plotted its against maximum the value. wavelength When the A spectral line thus hasfrequency two characteristic parameters; ﬁrst, its center examples for the resolving power ofgiven commonly-used later gratings will in be Sect. Figure 22.7 An almost uncountable numberexperimental of investigation articles of have line been spectra.obtained written In with on spectrometers many are the cases, presented the graphicallysities results and of the the inten- radiationFor are some indicated aspects of by the theis investigations, this width not method adequate. of of The the representation frequency and linesﬁcient. the drawn. wavelength An alone additional are not quantityof suf- that the must spectral be lines, characterized determined by is their the lines of higher orders 22.3 The Line Shapes and Halfwidths spectral lines (see also Fig. pressure mercury-vapor lamp (line spectrum),registered using with a very prism broad spectrometer. This spectralof instrument lines, has a resolving power Optical Spectrometers 22 448 Part II 22.4 Spectrometers and Incandescent Light 449 22.4 Spectrometers and Incandescent Light

Incandescent or “natural” light does not consist of monochromatic waves from a wide frequency region, but by using spectrometers2, we can produce quasi-monochromatic waves from incandescent light. (For this application, the apparatus is often referred to as a “monochromator”). Its function can be seen qualitatively directly by considering prism spectrometers which employ dispersion: Every dispersion converts non-periodic processes into periodic processes. Part II Such a reforming of wave groups can be seen on the water surface of a pond into which a large stone has been dropped: At first, we see a non-periodic disturbance of the water surface, then groups of surface gravity waves emerge and become longer and longer (see Vol. 1, Fig. 12.79). Those groups consisting of long waves reach the bank sooner than those consisting of shorter waves, thanks to their higher group velocities. Quantitatively, we can understand most simply the modulation of weakly or not-at-all periodic, transient processes in quasi-monochromatic wave groups, that is wave trains of limited lengths and finite spectral widths, by considering a spectrometer which is based on a line grating. We thus want to treat the observation of a continuous spectrum of first order in such a spectrometer. Figure 22.8 shows a grating with N lines or openings. A collimated light beam from an incandescent light source is incident on this grating, perpendicular to its surface. The line A in this case does not indicate a wave crest, but rather the limiting case of a non-periodic, transient pulse with the profile a as shown in Fig. 22.9. This type of process is shown especially clearly in Fig. 22.9; for that reason, we have chosen it as an example. A second pulse, which precedes the first, has already passed through the grating and has been split there into N pulses, each with the same profile. These pulses propagate in Fig. 22.8 in the form of eccentric circles to the lower right; however, only a segment of the circles is drawn in the figure. Along the direction of an arrow r (or v), the series of these pulses forms a wave group whose form is not sinusoidal, or more concisely, a non- sinusoidal wave group. It is sketched as curve b in Fig. 22.9, but only for N D 6. The spacing of the two pulses is large in the r direction andsmallerinthev direction. Along the r direction, it could for example be 0.75 m, and along the v direction only 0.4 m. Every such non-sinusoidal wave group b can be represented as a superposition of groups of sinusoidal waves of equal lengths (FOURIER decomposition), with the wavelengths , =2, =3, etc. This is showninPartsc, d, e, f of the figure (compare Vol. 1, Sect. 11.3). Now we come to an essential point: We want to observe visually, by eye; but the eye acts selectively, i.e. it chooses what it sees. It re-

2 They can employ dispersion, absorption, or reflection in “filters”. ) c , indi- of A 3, etc. OUNG chirp = .Y H 2, , and not as = can be thought of 400 nm (violet). b chirp D v ,butalsoto each step a whistling r 1 5 is decomposed into four sinusoidal, 10 components). b direction, at emitted by 6 openings or lines of a grat- v a , at the upper left. OURIER From no. 22.9 f of Fig. c d e , 1855) on the street. If we walk on a hard stone surface a , the periodic group f 1 5 PPEL to 10 No. The production of wave groups by a grating. The line The periodic but not sinusoidal wave group c A direction, we see only one sinusoidal wave group (curve 750 nm (red), and in the r ab D a tone or aSect. note, 12.28). which we would hear with a sinusoidal profile (Vol. 1, alongside a garden fence, we can hear with noticeable length. The fence actsthe as a air reflection pulses grating. Each fromperiodic slat our reflects impulses footsteps, into and aless thus non-sinusoidal selective than the wave our grating group. eyes;responds converts the not ear Our the responds only ears non- to to are roughly the 10 much longest octaves. wavelength It thus We therefore hear the non-sinusoidal wave groups as a 1801; J.J. O We often experience the corresponding acoustic experiment (T Thus, we can say conciselytra but consist unmistakeably of that groups continuous of spec- sinusoidal waves produced by the grating. as a series of 6 pulse-shaped groups ing. In Parts cates a non-periodicthe process left, (pulse) perpendicular which toa is its ground surface, incident swell in as on shallow canis water the be or sketched grating demonstrated by in an for Part from ultrasonic example boom by in the air. Its profile sponds only to wavelengths between 750 nmin and 400 the nm. As a result, Figure 22.9 Figure 22.8 at monochromatic wave groups (F Optical Spectrometers 22 450 Part II 22.5 Comparison of Prisms and Gratings 451

The number of individual wavelets (i.e. a wave crest C trough) in the group produced by the grating is, in the first-order spectrum, equal to the number of openings or lines N in the grating. N however, from Eq. (22.3), is equal to the resolving power =d in the first-order spectrum. Then the resolving power acquires a simple meaning: It is the number of individual wavelets which are produced by the grating from a non-periodic process and can be combined into a group.This holds not only for gratings, but also for spectrometers of all kinds. This statement can be verified with the help of all those interference experiments in which light from a narrow region out of a continuous

spectrum is employed. Part II

22.5 Comparison of Prisms and Gratings

According to the numerical example given at the conclusion of Sect. 22.1, with one prism of 10 cm base length, we can obtain a resolving power of =d 104. Connecting several prisms in series, e.g. 3, we can obtain three times that resolving power, that is around 3 104. With gratings, or in general with interference spectrometers, around ten times higher resolving powers are attainable, that is a resolving power of several 105. In a comparison between gratings and prisms, however, we must not consider the resolving power alone. Another very important characteristic is the usable wavelength range . A prism always produces only a single spectrum. In it, each direction corresponds to only one wavelength. A grating, in contrast, produces a whole series of spectra of different orders m, and all of these spectra overlap. Each direction corresponds to several wavelengths, namely for m D 1, =2for m D 2, =3form D 3, etc. A unique correspondence between wavelength and direction is obtained only within a certain limited range . Let us look at Fig. 22.6. A spectral line of wavelength . C / and order m must be spaced at least in the minimum ˇ just adjacent to the spectral line of order .m C 1/ and wavelength ;otherwise,the unique assignment of the spectral lines to deflection angles would be lost. We thus find m. C / D .m C 1/ ; N or, when =N can be neglected relative to , for the usable wavelength range, we obtain: D : (22.4) m The most favorable case is found for m D 1; then, we find D . This means that a spectrum of first order has a unique assignment - to ). RAUN into the 22.4 on a highly- by some sort of is equal to only grooves RAUNHOFER achieved 800 grooves , usually between 1 and 5, m preselection gratings (see Fig. of their average wavelength. .Often,itisalsousedasama- g accomplishment for grooves 3, it must transmit for example OWLAND D remains rather large; in the second m ). gratings in plastic. Gratings are often RAUNHOFER 20.13 . This allows us to achieve a resolving power (in contrast to photographic registration or an 5 )) even in the second order. This means that reflection grating and 10 eye . We can for example observe a spectrum in the 5 : transparent 22.3 1 5 grating must be placed very close together, and all : 20.12 0 D (Eq. ( N 3: Here, the usable wavelength range openings must be next to each other on a surface area of 5 5 grating makes use of small orders D 10 10 m 3. Therefore, even the eye requires a , that is a full octave. If there are wavelengths outside this octave 5 RAUNHOFER : F HOFER trix for pressing per millimeter in 1882,10 an cm long! astonishin The ruled gratingis (e.g. best with employed up as to a 1200 grooves per mm) indispensable scientific instrument that it remains today. A F polished metal surface. Thisa diamond is stylus. achieved In this with way, H.A. a R ruling engine using of 3 1 ca. 15 cm diameter. Thistion cannot of be a accomplished as garden inthe fence the grating using construc- is slats carried and out spaces. in the Instead, form the of parallel ruling of The usable wavelength range the grating is abledifference of to only separate about two 3 light millionths signals with a wavelength order, and has a large number ofhave grating up openings to or lines. Modern gratings The line grating was developed in 1821 by J. F 22.6 Various Forms of Line Gratings between wavelengths and angular deflection over a range from All of the lines orthe surface openings of of a the lensare grating or must only concave be seldom mirror. encompassed available Lenses by than in and concave the 15 cm. mirrors laboratory with For diameters of this more (financial) reason alone, the openings of the visible light range from 750 tostructure 400 of nm complex in line a spectra, singleinvestigated such measurement. using as The large those F from atoms, are best additional apparatus, which must sort out thea undesired waves. filter Often, suffices for this task. For 2 only waves between 450etc. and (cf. 600 nm, Sects. or between 600 and 800 nm, = electronic detector), we needsorting no process. special auxiliary Our apparatusonly eyes to for themselves waves this act in selectively; theAs they range a of result, respond the about eye onehindrance. can octave capture (ca. a 400 whole to first-order spectrum 750 nm). without The situation is however quitee.g. different for spectra of higher order, range, they must be sorted out individually in someFor observations manner. by Optical Spectrometers 22 452 Part II 22.6 Various Forms of Line Gratings 453

Figure 22.10 A piece of this rough millimeter scale, etched into glass, about 15 cm long is sufﬁcient at grazing incidence of the light beam to cleanly separate the lines in a mercury spectrum (actual size)

ruled onto a concave metallic mirror; with such a “concave grating”, one avoids the need for a lens in front of the grating. Part II For X-rays, the usual optical gratings made of glass or metal can be employed, as long as the wavelength is greater than 2 nm (DO 620 eV).C22.2 C22.2. The unit eV (electron They are used as reﬂection gratings at grazing incidence, since the volt) is a unit of energy.Itis ruling of the grating is ﬁne enough for X-rays only if it is strongly equal to the product of the foreshortened as seen in perspective at grazing incidence. magnitude of the elementary electric charge e0 and the unit Perspective foreshortening of the ruling of a grating can be demon- of potential difference, volt: strated by a simple experiment. We use the millimeter divisions on 1eVD 1:6 1019 As V a common ruler as a diffraction grating for visible light (Fig. 22.10). D 1:6 1019 W s. Its relation At grazing incidence, the lines of a mercury spectrum can be cleanly to the wavelength is found separated. from

In using mirrors and gratings for X-rays, another point must be kept in E D h D hc= mind: The refractive indices of all materials are close to 1 for X-rays, (h is PLANCK’s constant, c is and therefore their reflectivities are vanishingly small. But a favorable circumstance comes to our aid here: The refractive index of all materials the vacuum velocity of light). for X-rays is somewhat smaller than 1 (Sect. 27.9). As a result, at nearly grazing incidence, we obtain total reflection of the X-rays. We mention only one of the many possible variations on line gratings: The mirror surface grating (blazed grating): In a highly reflective metal surface, grooves are cut with a one-sided triangular profile (“blazing”), for example as shown in Fig. 22.11. A collimated light beam 1 is incident along the normal to the grating. The greater portion of its energy is reflected along direction 2, according to the law of specular reflection. With a suitable choice of the grating constant d,the first-order spectrum can be adjusted to fall around this direction. It then contains a much larger portion of the radiant power than the spectra of all the other orders on both sides of the normal to the grating; this grating has practically only one spectrum. Such mirror-surface or blazed gratings can be fabricated especially successfully for the long waves of infrared light ( D ca. 10 to 300 m), but they can also be made for the visible spectral range.

Figure 22.11 A mirror surface grating (also d called a “blazed grating” or an “echelette”) λ

2 1 . - re- . 22.3 and ABRY aces 4 .There- (F 5 . By multi- ). The path (sources) are UMMER , it is not nec- f m and 10 20.15 ÉROT of a large grating line structures air plate 4 d is less than 10 = m 0) with transmitted light, and Fig. and A. P = D between 10 ˇ 20.10 m ABRY interference plate as a spectrometer .F H of interfering wave trains very large. are sufficient; i.e. a large path differ- N m EHRCKE -G . This means that we separate out the spectral )). Therefore, at low orders 22.4 of interfering light beams. The observations are car- UMMER N and a higher AL ). . The light leaves a plane-parallel glass plate under a large Extended preselection light source N )), but on the other hand, a small usable wavelength range between two neighboring wave trains. This is experimen- étalon), as described in Sect. 22.12 22.3 m EHRCKE suffices (Eq. ( G A variation on this spectrometer design is named after L angle, i.e. at a grazingcritical angle. angle Within for the total plate, reflection.ple it is reflections This reflected without makes at it nearly applying(Fig. the possible a to reflective obtain coating multi- to the glass surf ÉROT P tally much simpler: We firsttigation limit by the wavelength range under inves- fore, the usable wavelength range difference of neighboring wave trains is,of depending the on air the thickness plate(plate used, spacing generally several some ten-thousands cm).formed of via wavelengths interferences This of means orders that the spectral lines are A smaller region to be investigated froma the rest prism of instrument; theing spectrum, sometimes usually light with is a passed filter through a is thick, sufficient. plane-parallel The remain- ence essary to make the number (schematic). The rays of a collimated light beam are sketched. and yield bright spectral linesof on this a design dark are background named (spectrometers for C (Eq. ( Instead, they consist of alap number with of each components, other. which Frequently,by often a weaker over- strong “satellites”. spectral Briefly lineplex put: is structure. also Many spectral flanked The lines experimental have investigation of a com- quires on the one hand the resolving power Mirror images as grating arrangements ofof wave eminent centers importance for X-ray spectroscopythan at ca. wavelengths of 2 less nm. Inimages Vol. could 1, we be showedconsist produced how of a by grating lattice plane-parallel pattern planes. plates of mirror whose We surfaces refer to Fig. 12.66 there, and repeat Figure 22.12 22.7 Interference Spectrometers Many spectral lines do not have the simple shape treated in Sect. ple reflections between two plates (witha mirror large surfaces), number we generate ried out at nearly normal incidence ( Optical Spectrometers 22 454 Part II 22.7 Interference Spectrometers 455

Sl B S r H β To the γ A detector D 4 lattice planes as mirrors S' S'' S''' S'''' 4 mirror images as wave centers (virtual sources)

Figure 22.13 ABRAGG spectrograph for X-rays. S is the line focus of an X-ray source which is perpendicular to the plane of the page (Sect. 19.5 and Part II Fig. 19.10). H is a narrow beam of radiation, usually selected by a series of slits Sl, and not just its central ray. The receiver is a photographic plate or one of the radiation detectors mentioned in Sect. 15.4. The spacing D of two lattice mirror planes (only 4 are drawn here) is of the order of 3 107 mm. As a result, the beams reflected from many lattice planes overlap. All together, they have an overall diameter Br which is hardly greater than that of the incident beam. Therefore, even without imaging, we obtain sharp but very weak spectral lines. For a demonstration with visible light,theseries of reflecting surfaces shown in Fig. 20.22 are suitable, or also a sequence of reflecting layers produced within photographic plates by standing waves (see Fig. 20.25). the schematic here in Fig. 22.13 with a larger angle of incidence ˇ. Lattice planes are provided by nature in crystals with a high degree of perfection. Their mirror planes occur with a large number N of layers, one above the other. This series of mirror planes yields a large number of mirror images which are arranged as a grating and can serve as wave centers (virtual sources). For X-rays, there are no lenses, and we therefore cannot use the FRAUNHOFER observation mode. This means that we cannot separate broad light beams which have only small angles of inclination relative to each other in the focal plane of a lens (Sect. 16.8). As a result, we cannot readily use extended sources of X-rays together with plane- parallel plates. A narrow, line-shaped source must be employed, and the angles ˇ, under which the individual wave trains are reflected from the same position on the crystal, must be scanned successively by rocking the crystal plate (Fig. 22.13). Generally, one quotes the grazing angle D 90ı ˇ instead of ˇ itself. We find3: m cos ˇ D sin D : (21.2) 2D

3 D is the spacing between two neighboring lattice planes (the optical grating constant, here also a lattice constant of the crystal); e.g. D D 0:28 nm in a NaCl crystal. The crystallographic lattice constant a, by contrast, is the spacing between two similar lattice units in homologous positions. In a NaCl crystal lattice, this is the spacing of two neighboring NaC ions or Cl ions. a D 2D is 0.56 nm in the NaCl lattice. A cube of edge length a forms the unit cell of the NaCl lattice. This means that we can construct the entire crystal lattice simply by translating the unit cell parallel to the edges of the crystal. 1484 4

). Fig-

α 1473 1 α 15.4

). 1299 4

1279 1

21.14 β 1260 group 3

β L 1242 2

β 1096 1 γ method, Sect. is assumed to have a grating constant of the light used? in XU λ 22.4 CHERRER -S D EBYE m. On a screen 2 m away, the first-order maximum m) The line -shell 13 3029 nm; L : ) shows an example of a line spectrum in the X-ray region 0 40 10 The grating in Fig. 1) appears at a spacing of 3 cm from the central maximum D 22.2 D 0). D 22.14 D D D 00302 : m m (See also the D a) What is the wavelength spectrum of radiation from tungsten, photographed with a vacuum spectrograph (using a calcite crystal with photographic negative, 1 XU (X-ray unit) 1 Figure 22.14 ure of ( ( Exercises 22.1 For demonstration experiments, it isas best detector, to e.g. use Tl-doped NaI. a Its fluorescentwith fluorescence crystal a radiation sensitive is photocell measured or a photomultiplier tube (Sect. b) After the whole apparatus isthe immersed first-order into a maximum liquid, is the only spacing 2.25of of cm. the What liquid? is the refractive(Sect. index Optical Spectrometers 22 456 Part II time a 10 fReference of Frames Moving in Light and Light, of Velocity The n17,O 1676, In Remark Preliminary 23.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © tedaee ferhsobti 3 orbit is the orbit along of earth’s points diameter of the opposite diameter by at (the separated sun, is the that around the away, orbit earth’s to furthest the approach was closest it its when at and carried was earth He the Jupiter moon. to when Jupiter measurements equal the the i.e. of out passage delay h, orbital 42.5 The one be for planet. to required the time found of was shadow signals the successive from between emerged it when moment fteerh asn ln ra iceo length of circle great a along passing circumference earth, the the around of waves electromagnetic communications sends Modern technology wave. electromagnetic an is light that know hs ewsal ocluaeteaoevleo h eoiyo light. of velocity the From of value s. above 1320 the of calculate delay to additional able an was showed he signal this, the orbit, the on rqec ag rmaon 10 around from range frequency encompass a spectrum electromagnetic the within today Measurements au hc so h orc re fmgiue namely magnitude, of order obtained correct he the observations, of is astronomical propagation which finite on value the Based a discovered Paris, light. in of XIV velocity Louis of court the at o h eoiyo ih nvacuum, also in thus light and of waves, velocity electromagnetic the best of for current velocity the phase the measurements, for precision value of Ac- types various technology). communications to cording by employed waves, radio (long R ÖMER 8 /.H sdlgtsgasfo n ftemoso uie tthe at Jupiter of moons the of one from signals light used He m/s. t saheeeti otyo diainee oa.W now We today. even admiration of worthy is achievement ’s D LAF 0 : 3 .I olw that follows It s. 133 R ÖMER ae tta iettrt h oa princes royal the to tutor time that at Dane, a , olsItouto oPhysics to Introduction Pohl’s 22 c 10 C23.1 z( Hz D 11 l = ) ttems itn point distant most the At m).

is t as ont bu 10 about to down rays) D c D 3 2 10 : 998 , 8 https://doi.org/10.1007/978-3-319-50269-4_23 l m/s. D 10 000k in km 000 40 8 m/s. c D 2 5 : Hz 3 ei edconstant field magnetic the of fixing the from seE.( Eq. (see is value precise cepted ac- currently The C23.1. 2 c.Sects. (cf. constant field electric the of value termining de- for methods that mean not does however value this Fixing 1). Vol. in C1.5 ment Com- (see value this at fixed was vacuum in light of velocity the meter, the length, of unit the of definition new of the framework the in 1983, In u oterelation the to due ment. measure- of methods other thr available definition precise more no currently is there meter, the for that only means value fixed The measurements. precision ing correspond- the from was found value quoted the all, ter af- textbooks; in superfluous nte osqec sthat is consequence Another : 99792458 23 " 8.7 0 c 5.3 sas fixed also is aebecome have ) n resulting and )), 10 and 8 m/s. 2.13 c 2 0 D ,the ough ). c " D 0 457 0

Part II , , 0 - is 0 H in of ˛ H S the L OU P 00 S' , 0 S of the Planar mirror mirror ˛ 32 m, di- β' D b OUCAULT 5m, is up to 400/s. The : R (1878). The rotating 10 lies within the slit D into the opening of the 00 of the slit on the planar f t S 0 S 2m, : . With (initially slow) rotation, ICHELSON 5 L moves across the whole diameter D 0 S R of the light beam is shifted parallel to the β' β ˇ in the direction of the straight double arrow. . This second image P 0 L light signals in a time S N K α' is 30 cm each, diameter of the rotating double-sided mirror α P of the image of the slit is up to ca. 4 mm). of the circulating light beam of radius s t 0 shows the experimental setup. It makes use of a telecen- Measurement of the velocity of light by the method of F S = of the Velocity of Light fb and N L . Following reﬂection there, the light beam takes the same H

P (1850), as simpliﬁed by A.A. M 23.1 S''

. The first image of the slit Screen . Each signal produces an image R 0 L phase velocity c of light in vacuum has in the course of time ac- ˇ is the light source, an illuminated slit. The axle of a small rotatable 0 mirror CAULT 1850. In thislength method, in a both directions light andsured beam directly the using passes corresponding a along uniform transit rotation time aFigure of is known path rotational mea- frequency. of known tric light path with a clearly-defined position ofS the pupil. and is thus not(a visible. thin But plane-parallel with glass the plate),the the aid side second of and image a projected can half-silvered be onto mirror shifted a to screen. Using this auxiliary mirror we can see the principlemirroristurnedslowlybyhandbackandforth.Thesegment of the setup; for this purpose, the rotatable the first slit image mirror telecentric. (A numericalameter example: of path in the reverse direction and at its end, it projects an image mirror serves as the entrance pupil, and the light path to the right of of the planar mirror Rotating optical axis. Both movements are indicated for the beam segments light beam moves in the directionsame of time, the the curved segment double arrow. At the Figure 23.1 and be lacking in ana introductory method course which on was physics. introduced by We describe the here physician L. F 23.2 Example of a Measurement The quired a fundamental importanceFor for this all reason of alone, the a physical description constants. of its measurement should not 5 cm, rotational frequencyfrequency of the mirror up to about 200/s. The rotational displacement mirror is at the focal point of the lens In spite of these motions of the light beam and the first image S the mirror emits lens The Velocity of Light, and Light in Moving Frames of Reference 23 458 Part II 23.3 The Group Velocity of Light 459 second image S00 remains unchanged and at rest. This the decisive point. The reason is not difficult to understand: At low rotational frequencies, each light signal strikes the rotatable mirror on its return path at practically the same position as on its outward path. This changes at high rotational frequencies. The returning signal finds that the mirror has rotated through a small angle during its transit time. Therefore, the image S00 of the slit is displaced by a small distance s to one side. If we now remove the auxiliary mirror H,we 00 find S no longer directly on the slit S0, but rather displaced to one side by s.Wehave: Part II Path traversed Transit time D Velocity or t s 2.f C b/ D : (23.1) N 2R c The data that apply to the lecture hall in Göttingen can be found in the caption of Fig. 23.1.

23.3 The Group Velocity of Light

In Fig. 23.1, we can place part of the light path in a strongly dispersive liquid, e.g. carbon disulfide. Then for the nearly non-periodic wave groups of incandescent light, the same thing holds as for the similar groups of gravity waves on a water surface: They are ‘stretched out’ into long wave groups. Their front flank consists of longer, nearly sinusoidal waves, while their rear flank consists of shorter waves. The “red” light (refractive index n 1:6) arrives first, while the “violet” light (n 1:7) arrives last. As a result, in Fig. 23.1,the second image S00 of the slit appears as a short spectrum. Only the vacuum is strictly dispersion-free. An experimental proof of this: When a Jupiter moon emerges from behind the planet, we see it immediately as colorless, not first red, then yellow, green, and so forth. When dispersion occurs, we can no longer measure the phase velocity of waves simply in terms of path length and transit time. Instead of the phase velocity c, we obtain only their group velocity c .This concept, which is important for both physics and technology, was explained in detail in Sect. 12.22 of Vol. 1. A group velocity c can be defined for waves only within a limited spectral range. Within that range, we find:

dc0 c D c0 I (23.2) d J δ D D 41 arc (23.3) (astro- .One 0 D m/s. .We c s 8 0 D km D c 10 J ı 30 72 = : s

S km m/s. u J ). The velocity D 8 D 4 m/s. With these ı 30 8 D 10 shows the earth at = δ

10 with a diameter of D 10 orbital plane u is the phase velocity 72 n :

84 = 1 : 23.2 c 1 c D = D D : D u 0 c c n n = D c d d 2

D c n ), yellow light of wavelength 0 D circular orbit 2 c as- is observed between a star near the the orbital axis and the orbital plane ). D n ı 0 of light in this wavelength range. In order to c /m and d d (the ratio 5 D ), we must know the experimentally-determined 10 ) we obtain the result between 23.3 88 63. The phase velocity of this light is therefore : C23.2 and a star near the earth’s : 60 km/s is known. ) 1 m/s. However, the measured value is only 1 1 ) an angular difference of 2 23.2 for sodium D-light as well as the phase velocity 8 2 D 10 D u D aberration 23.3 n , and (after differentiation) D 84 : n group velocity c / Variation of the velocity The apparent angular 1 = of Reference d appear to move on a c = light source outside the moving frame of reference D between two stars changes n d orbital axis D ı . 63 0 : axis is here the phase velocity in vacuum, and 1 c = c in a material of refractive index Example: In carbon disulfide (CS values,usingEq.( find dispersion of CS This is the calculate it from Eq. ( c of the sun relative to the system of fixed stars is unknown; only ( 589 nm, the well-known “sodiumtive light” index or of “sodium D-light”, has a refrac- S u second, i.e. an of the earth along itssun, orbit relative around to the a distantseen star in (the perspective orbit from is the side) spacing tronomical aberration according to the season of the year ( nomical aberration) The earth’s orbit aroundpointlike the when sun observed is from a a large star. carousel, Figure which appears two arbitrary points along itsFrom orbit, these separated points, by an one-half of angle a year. Figure 23.2 Figure 23.3 1. For a 23.4 Light in Moving Frames with earth’s observes (Fig. 41 arc second. Stars As a resultbital of these variations, all fixed stars near the earth’s or- the difference 2 The Velocity of Light, and Light in Moving Frames of Reference 23 C23.2. Pointing the telescope: Think of raindrops which are hitting the window of a moving train. If weto want look at them directly, we have to fix our gazethe along direction of travel and upwards. 460 Part II 23.4 Light in Moving Frames of Reference 461

Figure 23.4 The measurement of the velocity Light from of light using the aberration a star 2 c / 2 u γγc c 1 – c Part II

= S c

uu Orbital velocity of the earth follow elliptical orbits over the course of a year, with a major axis of 41 arc second. This phenomenon was discovered by BRADLEY and explained in 1728 as in Fig. 23.4, left side.

According to EINSTEIN’s theory of relativity (1905; see Com- ment C5.1 in Chap. 5 of this volume), the velocity of light c is a limiting value which cannot be exceeded when adding velocities. Therefore, the left-hand diagram in Fig. 23.4 should be replaced by the diagram on the right. From it, we can read off p p 2 2 2 2 cs D c u D c 1 u =c (23.4) andC23.3 C23.3. Historical note: A derivation using the u u 1 tan D D p : (23.5) LORENTZ transforma- cs c 1 u2=c2 tion (time dilation), was given in the 13th edition of Now, however, u c. Therefore, we can approximate the tangent by POHL’s “Optik und Atom- the sine and set the square root in the denominator equal to 1. Then physik”, Chap. 9, Sect. 4. D = = the aberration becomes sin u c or u c. This result agrees English: see e.g. https:// with the observations of BRADLEY. www.lsw.uni-heidelberg. 2. For a light source within the moving frame of reference de/users/mcamenzi/ DopplerAberration.pdf Figure 23.5 shows a carousel from above; at first, it is at rest. Two co- or https://en.wikipedia.org/ herent light beams, 1 and 2, are emitted simultaneously from point A. wiki/Stellar_aberration_ They are reflected by mirrors at the vertices of the polygon to point B. (derivation_from_Lorentz_ There, they are superposed in a suitable manner, so that they produce transformation) . interference fringes, for example curves of constant inclination. The position of the fringes is recorded (e.g. photographically). Then the carousel is set in motion, rotating in a counter-clockwise sense as seen from above, and the interference fringes are again photographed. Now, the fringes have shifted by some fraction Z of their original . B c D = r m .This and (23.6) (23.8) (23.7) s must be 6 A : D 1 t 0 t = 2 r D B A N 3 is desired; with : is replaced by the 2 . This doubles the 2 equal to = B r B c to D ! . Thenwecansaythat to : 8 A r has moved forward at the 2 2 A B r r D c c ! ! 2 are D D B or rt s ! 2 and 2 m/s. Then the product r D D 8 A c s 10 ! 8 3 /s. This can be accomplished experimentally in 2 D 3 of their spacing. For yellow light, D is the area enclosed by the two light paths 1 and 2). = 2 full circumference of the carousel , and has covered the distance c r is the angular velocity of the carousel at its rotational fre- r t ! = . t The measurement of the ve- N = D N 2 u m, and 7 D ! 10 quency ( , while light beam 2 travels along a path which is shorter by not shown. Numerical example: A path difference of a reversal of thefringes rotation, equal to this 1 will give a shift in the position of the shift of the fringes.during the Secondly, experiment, the so that sense the shift ofThen of rotation the we fringes can find is again be for doubled. the reversed overall path difference can be placed adjacent totravel each along other the so that both light beams have to This path difference in turnIt causes can a easily shift in be the increased interference by fringes. a factor of 4. Firstly, points made equal to 1.2 m Figure 23.5 locity of light by meansexperiments of on interference a carousel. Inplified this schematic, sim- the components ofinterferometer the setup at points various ways. Examples: Therefore, light beam 1 musts travel along a path which is longer by spacing. From the magnitudelocity of of light. this shift, we canExplanation: calculate We the choose ve- our pointframe”, outside of the observation carousel. in the Furthermore,path we “laboratory along imagine the that the edges light of the polygon from gives rise to a path differencerotation: between the two light beams due to the 6 circumference of the semicircle, that is each light beam has a transit time from During this transit time, the target point velocity The Velocity of Light, and Light in Moving Frames of Reference 23 462 Part II 23.5 The DOPPLER Effect with Light 463

1. A carousel with an area of 1.2 m2 and N=t D 1/s, that is one rotation per second. 2. The interference apparatus is put on board a ship. Its light path encloses an area of r2 D 120 m2, and the ship sails around a full circle every 100 s, so that N=t D 102/s (for rotational motion, the angular velocity is independent of the location of the axis of rotation; the latter is in the center of the apparatus in example 1., but off to the side for a circling ship). 3. The light path (protected by an evacuated pipeline) encloses an area of the order of r2 D 105 m2. Then the angular velocity of the Part II earth’s rotation, ! D 2N=t, is sufﬁcient; or, more strictly speaking, the angular velocity of its vertical component at the location of the observations. We thus have constructed an optical analog of FOU- CAULT’s pendulum demonstration (Vol. 1, Sect. 7.6).

Of course, we can neither bring the earth’s rotation to a stop nor can we change its direction. Therefore, the determination of the original position of the interference fringes requires a trick: We ﬁrst let the light beams follow a path enclosing a very small area, and then later use the large area. In this way, we lose a factor of 2 in Eq. (23.8), but nevertheless, the comparable experiment carried out by A.A. MICHELSON in 1925 yielded an impeccable result.

None of these methods is suitable as a demonstration experiment. Protecting the apparatus from perturbations by centrifugal forces and temperature variations requires a considerable effort. That is why we left it at the simple scheme described above, without considering the details of the light path.C23.4 C23.4. The interferometer described here, named for G. SAGNAC, has considerable practical importance today 23.5 The DOPPLER Effect with Light as an “optical gyroscope” or “ring-laser gyroscope”. For mechanical waves, for example sound waves, either the source or the receiver or both may be moving relative to the medium of the waves, e.g. the air. Their velocity u can be clearly deﬁned and measured. With all such motions, the frequency 0 measured at the receiver is different from the frequency emitted by the source. This is called the DOPPLER effect. When the receiver is moving and the source is stationary, we ﬁnd (Vol. 1, Sect. 12.2): Á u 0 D 1 ˙ ; (23.9) c where c is here the velocity of sound in the particular medium. In contrast, when the source is moving and the receiver is stationary (the upper signs hold when the receiver is moving towards the source), we have: Á u u2 0 D Á D 1 ˙ C ˙ ::: : (23.10) u 2 1 c c c , ) S 2 c = 2 23.9 u (23.12) (23.11) cceler- ,there A ; Á between the . and therefore c u (of the ions in ::: transformations c u = C ˙ u , that is when our c 2 2 hydrogen molecules = c u u effect using mechani- 1 2 velocity and the anode ORENTZ C + : C c u Á A c u ) which hold for mechanical ˙ OPPLER relative . The velocity 1 ˙ llide with the 1

23.10 23.6

D C23.5 2 2 D c u 0

)and( effect C23.5 ) must be replaced by a single equation; it . It shows a shift of the spectral lines towards and all the higher terms. Then Eqns. ( 1 2 23.9 c s 23.7 = 2 23.10 OPPLER u Á. c u be applied in optics. In optics, i.e. when dealing with ˙ A simple ion-beam tube )and( ) are the same. The observed change in the frequency, 1 ) was successfully carried out only in 1938, using the light , and the references quoted there). An experimental test of is now the velocity of light.

), then depends only on the cannot c 23.9

7.4 23.10 D 23.12 0 ated, positively-charged hydrogencathode ions (canal emerge rays). from the When channel these in co the is hydrogen gasaround at 30 a kV produces low a pressure self-sustaining of discharge ca. in 0.1 Pa. the gas. An A anode voltage of in the right-hand parthydrogen of atoms the which tube, thendirection of they emit motion produce light. of rapidly-moving, thereproduced ions The excited in with light a Fig. is spectrometer, leading observed to along the image the shorter wavelengths, that is to higher frequencies. Within the tube, and between the cathode

electromagnetic waves, which are completelydistinguish relativistic, between we motion cannot of theand source and there motion of is thetions receiver, no ( ‘medium’ that carries the waves. The two equa- (see Chaps. 7 and 8Sect. in this volume, in particular Comment C7.1 and Eq. ( then the equations ( waves is cally-moving light sources does not justify the required efforts. It and ( where emitted from ion beams. Figure 23.6 for observing the D source and the receiver. This gives with light If we relax our restriction to small values of This demonstration of the optical D Qualitative demonstration experiments canion-beam be tube sketched performed in using Fig. the We ﬁrst consider only small values of the ratio the beam) must be a few tenths of the velocity of light ( This equation can be derived from the L measurement precision also includes the second-order term neglect the term . lon- ”, dis- ,or effect, OPPLER OHL D Journal of the OPPLER D mathpages.com/ http://spiff.rit.edu/ (1938), p. 215. This transverse The Velocity of Light, and Light in Moving Frames of Reference ,or 28 23 Optik und Atomphysik cussed the transverse Doppler effect in the 13th ed.“ of classes/phys314/lectures/ doppler/doppler.html https://en.wikipedia.org/wiki/ Relativistic_Doppler_effect# Transverse_Doppler_effect home/kmath587/kmath587. htm Chap. 9, Sect. 6. Forern mod- treatments in English, see e.g. i.e. parallel to the direction of wave propagation. There is also a experiment observes the gitudinal effect, but it is muchHistorical smaller. note: P C23.5. Herbert E. Ives and G.R. Stilwell, Optical Society of Amer- ica 464 Part II 23.5 The DOPPLER Effect with Light 465

Hδ Hγ = 0.410 μm = 0.434 μm

Figure 23.7 The DOPPLER effect in the spectrum of moving hydrogen atoms. The sharp lines Hı and H are emitted by atoms at rest (the BALMER Part II series, see for example M. BORN’s “Atomic Physics”, 8th ed. (1969) (available for download – see https://archive.org/details/AtomicPhysics8th.ed). The broad lines which are seen to the left of the sharp lines are DOPPLER shifted from moving atoms and exhibit the distribution of their velocities. C23.6. Today, redshifts corresponding to much higher velocities have been shows no more than some arbitrary interference experiment in which observed. These are “cos- a mirror is being moved at a velocity u along the direction of a light mological redshifts” which beam or opposite to it. As an example, we use an air wedge formed are due to the expansion by two silvered plates (Fig. 20.11); we move one plate slowly and of space and not to mutual thereby continuously vary the path difference of the two wave trains motion at a velocity u within that overlap between the plates. The interference fringes slide across space. They are a measure the field of view, and the irradiation intensity at a particular location of the age of the emitting in the field of view varies periodically N times during the time t. object, since the expansion Due to the DOPPLER effect, the frequency of the reflected wave train has led with time to greater is shifted relative to the frequency of the incident wave train by distances and greater ve- D 2u=. The superposition of the two wave trains thus produces locities for distant objects. See e.g. http://curious.astro. beats of frequency S D N=t D (compare Vol. 1, Sect. 12.18). cornell.edu/physics/104-the- The optical DOPPLER effect has acquired considerable importance universe/cosmology-andfor astronomy. In the spectra of distant stars and galaxies, the line the-big-bang/expansion- spectra of known elements are often shifted towards longer or shorter of-the-universe/610-what- wavelengths. This shift can be interpreted in many cases unambigu- is-the-difference-between- ously as a DOPPLER effect. From its magnitude, the radial velocity the-doppler-redshift- ur between the source and the earth can be computed. Generally large and-the-gravitational-or- shifts, always towards longer wavelengths (“redshift”), are observed cosmological-redshift- in the spectra of extragalactic objects (distant galaxies). They lead to advanced or https://en. surprisingly large radial velocities, up to several tenths of the velocity wikipedia.org/wiki/Redshift# C23.6 of light. Highest-redshifts .

All of these observed velocities are directed away from the earth, and their C23.7. The current record magnitudes increase proportionally to the distances of the galaxies. This distance (2016) is about is illustrated by Part a) of Fig. 23.8 (E is the earth). The lengths of the 3:3 1010 light years, corre- lines correspond to the velocities. Distances observable today are up to sponding to a relative redshift 5 108 light years.C23.7 D 0 D : of z 11 1. Here, This relation between distance and (expansion) velocity, discovered by 0 is the observed wavelength E. HUBBLE, appears at first view to attribute an improbable special vantage point to our earth. But this is not the case; the graph a) could represent and is the emitted wave- a race in which many students participate. Initially, they were all clustered length (in the rest frame of around their teacher at position E. Then, at a given time, they all began the source). See the refer- to run away in all possible directions. Their goal is a distant circle with E ences in Comment C23.6. E velocity of the teacher, E in graph a) to all N EN EN is the apparent center point. N unners. The fastest runners have a b ities of the r E in N is no longer at the center point of the general radial E . The graph b) can be constructed very simply from graph a); The radial escape N at its center. At the momentpoint of in observation graph represented a) by shows the theis graph, position given each of by one the student, length and ofare his the proportional or line. her The to distances the covered since veloc starting at moved furthest away. Part b)at of the the same moment figure in shows time, the but now same not race, from the observed location of the other velocitydashed vectors lines). in Now, a) (shown at top left for one example as escape velocities, but instead the location but rather from that oflocation some randomly-chosen participant in the race at the we need only add the velocity vector of the runner at Figure 23.8 in the upper graph, and at motion of distant galaxies, obtained from the “redshifts” of their spectral lines (the location of the observer is at the lower graph) The Velocity of Light, and Light in Moving Frames of Reference 23 466 Part II .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer the © sine represent crest. air can a wave it the a e.g. or represent at compression; then line, maximal of may is regions wavy to ordinate which a density, refer the crests wave, by wave sound is represented a that of be pressure, drawing also the can In wave function. longitudinal any ing, oaie Light Polarized 1 24.1 Figure between distinguish to learned we Figure waves. mechanics, longitudinal and on transverse sections the In Transverse Distinguishing 24.1 ..asudwv ntearisd ie esecompressions see We pipe. a inside air the and rarifications in crests wave and the wave, sound see longitudinal a we a shows snapshot cord; e.g. lower The elastic waves. an the of along troughs represents e.g. one wave, upper transverse The a images. instantaneous or “snapshots” two ( wave transverse A a: sisl nti ln foscillation. of plane observer this the of in eye itself the is when invisible becomes wave transverse the of hshsaoesddeso lnrt hc scle t “polariza- its Fig. called single in a is wave by which characterized transverse planarity is The It or tion”. line. one-sidedness straight a a has visible; as longer thus no cord now the is wave see transverse we the but same, the looks still (fro page the of eaiet htdrcin,wieatases ae ncnrs,is contrast, in Fig. wave, in transverse shown a As while “one-sided”. symmetric direction), decidedly cylindrically is that (it to propagation relative of direction its around all lrt.Te,iaieta orveigdrcinis full direction in viewing appear your phenomena that wave imagine Both Then, perpen- page. clarity. the also of is plane direction the viewing to dicular the con- Initially, and experiments. propagation the of direction sider detail. its some to perpendicular in wave means the View this what see to want We polarized”. “linearly uia wave tudinal longi- A b: Snapshot A Figure sisdeflection). its is 24.1 n ogtdnlWaves Longitudinal and hudb huh fa saso”of “snapshot” a as of thought be should Snapshot bv rblwtewaves) the below or above m 1 ogtdnlwv xiistesm behavior same the exhibits wave longitudinal A . b a olsItouto oPhysics to Introduction Pohl’s 0 A ln foscillation of plane 24.1 24.1 h ogtdnlwave longitudinal The . w experiments two hw sa example an as shows tppr) tcnbe can it part), (top , https://doi.org/10.1007/978-3-319-50269-4_24 within h motion The . .Inadraw- h plane the 24.1 x 24 467

Part II , P no ,we (also P in terms fall perpen- light n , a hand produces birefringence 24.2 P lt, the plane of oscillation of the observation, i.e. the occurrence of , this selected plane lies parallel to ropagation; we initially observe 24.2 , a Dane, discovered positive , the transverse waves may also have no polariza- ARTHOLINUS A slit B as a polarizer RASMUS E a polarization, could eliminateuniquely longitudinal that waves light and consists demonstrate ofpositive evidence transverse in waves. the We following can way: obtain this polarization. But only a Figure 24.2 called double refraction) in 1669. He let a light beam P for mechanical transverse waves a transverse wave on a long elasticfrequency cord. and The hand amplitude, moves with a butcontinually fixed and it randomly. changes As its a resu direction of oscillation waves also changes continually and randomly;fill the a waves completely cylindrical region withaxis. the direction The of propagation intersection asindicated of its by this center two cylinder dashed lines. with Now the comes plane the of essential point: the At page is the cord passes through a narrowselects slit. one This single slit acts fixed as planeplanes a out “polarizer”. of of It oscillation. the mixture In of Fig. rapidly changing the plane of the page. Therefore, to the right of the polarizer can observe a linearly-polarized wave.the Its character polarization of clearly the shows wavesthe which polarizer: are They travelling are from transverse the waves. left towards 24.2 Light as aThe Transverse Wave knowledge of wavesapplied that analogously to we optics. But obtained – from should we mechanics describe can be In mechanics, a polarization can thusBut occur only beware for of transverse the waves. conversedoes of not this rule out sentence: transverse the waves.lation lack The of position of transverse of polarization waves the can planeaveraged of change over oscil- time rapidly and randomly.tion (we Then, then call them “unpolarized”). Nevertheless, even in thisbetween case, longitudinal and we transverse waves. can Weusing make distinguish this a experimentally clear mechanical again demonstration. In Fig. of longitudinal waves or of transverse waves? We employ one of the fundamental observationstrace of of optics, a the light visible beam insuspended a particles cloudy as medium. such We a can usesame medium. water all The with light around fine beam its looks direction just of the p Polarized Light 24 468 Part II 24.2 Light as a Transverse Wave 469

B S eo n o

R Part II Figure 24.3 A demonstration of birefringence. A thick plate of calcite crystal (a natural rhombohedral cleaved crystal) is attached to a disk SS (seen here in cross-section). This disk can be rotated within the ring RR around the n-o direction as axis. When we add a circular aperture B,wehavemadeasimple polarizer. (The direction of the optical axis of the calcite crystal is indicated by shading, as in Sect. 24.4)

dicularly onto a platelet of Icelandic calcite (CaCO3 –Fig.24.3). He observed that the beam was split into two sub-beams. One of the two, denoted by o, passes through the crystal platelet in its original direction without any refraction. It thus shows the same behavior as seen for any glass plate with a beam of light at perpendicular incidence. This sub-beam o is therefore called the “ordinary” ray. The other sub-beam eo experiences a refraction on entering the crystal, in spite of its perpendicular incidence, and it leaves the calcite crystal with a parallel shift of its beam axis. This second sub-beam is called the “extraordinary” ray. There are several possibilities for eliminating one of the sub-beams. In the simplest case, the aperture B in Fig. 24.3 is sufficient. It allows only the ordinary ray (beam) to pass through. By eliminating one of the sub-beams, we have converted the doubly-refracting crystal into a polarizer. It serves the same purpose for light as the slit in Fig. 24.2 for the mechanical waves on the cord. We will see this in our next experiment. We allow the light to pass through such a polarizer and then follow its trace in a container filled with cloudy water (Fig. 24.4). Now, the light beam shows a clear-cut polarization: We can observe the light beam from a direction perpendicular to its propagation direction and look at it from all sides. Along two particular directions, separated by 180ı, the beam becomes invisible;inthese directions, the eye is in the plane of oscillation of the polarized beam. We mark the position of this plane of oscillation on our polarizer with a pointer E. Now, we can make the observations more straightforward. We maintain our viewpoint and make use of the pointer to position the polarizer around the axis of the light beam as optical axis. This allows us to demonstrate the transitions between good visibility and complete invisibility of the light beam to a large audience in an impressive manner. P reflec- light beam are directed 2 .The T and denote it by the (Video 16.1) produces light beams of only E rger beams, we would need thick C24.1 light vector P 24.3 F fixed plane of oscillation . Its “deflection” can be oriented parallel to C , conclusions. Concerning the physical nature of the light vector, 16.9 A demonstration of the plane of oscillation of a light beam. L C24.2 . . This allows us to fix the position of the plane of oscillation E transverse wave , making use of total reflection. For this purpose, we cut a piece Compare Sect. symbol quantities, vectors oriented transverse toof the direction the of wave. propagation a We light will wave correspondingly from refer now to on the as “deflection” the of we need make noto statements limit for our the descriptions of timeessential. optical being. phenomena to We what will is continue absolutely is the polarizer. (Theexpediently by water using is Styrofan (BASF), clouded i.e.is by plastic less spherules adding than whose the suspended diameter wavelength of particles, the light) most tion a few millimeter in diameter;and for expensive plates la of calcite ortal. some To other overcome double-refracting this crys- disadvantage,ers a have series been of developed. other types of polariz- In the first group, one or the other sub-beam is eliminated by 2 24.3 Polarizers of Various Types The polarizer sketched in Fig. Figure 24.4 We summarize: Usingtransverse a waves polarizer, with we a can prepare light beams as becomes invisible when the eyecillation of the observer lies inin its the plane polarizer of and os- to mark itThe with a discovery pointer. of polarizationtion considerably of enriched light thescheme as interpreta- which a we form haveplest of often case waves. utilized, a i.e.of sinusoidal We a curve, can a wavy corresponds now line, in say in optics that the to the sim- the wave picture a plane, i.e. the light“deflection” wave and can its be maximum linearly polarized. value, Therefore, the the “amplitude” Pro- - YN stands T E scattering . OHN effect); see ). J 12.6 26 , 1820–1893, an Irish Polarized Light YNDALL 24 DALL Video 16.1: “Polarized Light” http://tiny.cc/5dggoy (Chap. C24.1. The light beam is made visible by ceedings of the Royal Society (London), Vol. 17 (1868/69), p. 223, an article whichstill is quite readable today. physicist, observed that small suspended particles in liquids can produce light scattering (T at the same time for thetor vec- of the electric fieldwhich with we can describe light as an electromagnetic wave. See Sect. C24.2. The letter 470 Part II 24.3 Polarizers of Various Types 471

68° eo E 90°

Figure 24.5 ANICOL prism, that is a polarizer after WILLIAM NICOL (1828), shown as a longitudinal section and a cross-section. A prism of this form is suitable for modest requirements. The extraordinary light beam is transmitted. Its plane of oscillation (the light vector E) lies parallel to the shorter diagonal of the diamond-shaped cross-section. (The optical axis of the crystal is indicated by the shading) Part II

Figure 24.6 A polarizer with its end surfaces perpendicular to its long axis. In a superior construction as described by GLAN-THOMPSON, the quite uniformly-polarized field of view encompasses around 30ı. Various but superficially similar shapes differ in the orientation of the crystal axes of the calcite. Therefore, one must determine the position of the plane of oscillation experimentally, e.g. as shown in Fig. 24.4, if the particular construction type is unknown. of calcite crystal3 in an oblique direction (Fig. 24.5) and separate the two halves by a transparent intermediate layer of suitable refractive index, i.e. one which leads to a total reflection of the ordinary rays (e.g. Canada balsam or linseed oil). For optimal performance, the two end surfaces should be perpendicular to the long axis (Fig. 24.6). With this shape, the transmitted sub-beam experiences no transverse displacement, so that when the polarizer is rotated, it does not “wobble”. In the second group of polarizers, one of the two sub-beams is eliminated by absorption. For this type, one makes use of “dichroic” materials. These are double-refracting and also absorb the two polarized sub-beams differently. In the most favorable case, one of the oscillating components is transmitted practically without attenuation over the entire visible spectral range, while the other, perpendicular to the first, is completely absorbed (see also Sect. 12.8). The most serviceable types of fabrication in use today are “polarizing foils” or “Polaroid sheets”.C24.3 One type of these foils contains many C24.3. These polarizing foils parallel-oriented tiny dichroitic crystallites. Another type consists are commercially available of films of a transparent plastic with rod-shaped structural elements with dimensions of up to (miscellanea). These rods are oriented parallel to a particular axis 1/2 m and more. They are during the manufacture of the foils by mechanical deformation and quite suitable for demonstra- are treated with dyes which are absorbed onto their surfaces. With tion experiments, e.g. using an overhead projector. 3 All polarizers made from calcite are useless in the ultraviolet spectral range. Calcite, and especially the glue for attaching the parts, absorb strongly in the short- wavelength region. An alternative is shown in Fig. 24.9. In the infrared, calcite can be used out to D 2:5 m. . 4 .Its P optical )isless .Inthe (polariza- E eo C24.4 ). It is thus ex- 25.6 )ofthecrystal ). The long axis of 24.3 section r prism, it propagates parallel is perpendicular to the plane of double refraction , beginning). deflection, the splitting into two E . Important accessories for produc- 16.6 (or principal . Within the upper prism, its optical ), one can also fabricate polarization (so that 24.7 ı is referred to – not very adroitly – as the ÄSEMANN principal plane in crystals (also called direction in Calcite and Quartz . (“Axis” here means a direction, not a line, deviating somewhat ) takes a similar path as in the upper prism, without birefringence. In contrast to the principal plane of a prism, which is a plane perpendicular to o We will often make use of this concept.For our next experiment, weprisms use as two shown geometrically-identical calcite in Fig. axis lies parallel toperpendicular. This the is indicated base by plane the shading of linesThe in the light the prism; figure. is in normallyfirst incident the surface, on lower, from both the it prisms left). is (perpendicular In to the uppe the to the optical axis, and insult, the birefringence lower occurs prism, only it is inwe perpendicular. the see As lower two a prism, separate re- and beams. only The( there beam that do is more stronglyThus, deflected it is the ordinary beam. The extraordinary beam ( strongly deflected. Both beams then pass through a polarizer direction of oscillation is marked by the double pointer position shown, the polarizerpass. allows If only we the rotate extraordinary it beam by to 90 foils for the ultraviolet and the infrared spectral regions. the refractive edge of the prism (Sect. tion by reflection). the hexagonal column is thus opticallythere distinguished; is along this no axis, birefringence. This special 4 24.4 Birefringence, in Particular Polarized light plays a significant role init optics. again We and will again meet in up with latering chapters and investigating polarized light arebirefringence based on the phenomenon of this process (E. K A third group of polarizers will be described in Sect. axis from the usual parlance!).axis is Every called plane a which contains the optical pedient for us to considerbirefringence. some additional facts about the subjectQuartz of crystals arecolumns. Calcite generally is also knownbohedral found cleavage in in fragments the are same the better form,perpendicular although known. to form its We the rhom- put of long twobeam axis surfaces hexagonal to of the enter column thepasses and through crystal allow the parallel a crystal thin tospatially without separated light its sub-beams is long absent axis. (Fig. Then the beam . AND H. L DWIN Polarized Light 24 It is called “H-sheet” and contains a polyvinyl alcohol (PVA) polymer impregnated with iodine. During manufacture, the PVA polymer chains are stretched to form anof array aligned, linear molecules in the material. The iodine dopant attaches to the PVA molecules and makes them conducting along the length of the chains. Light polarized parallel to the chains issorbed, ab- and light polarized perpendicular to the chains is transmitted. C24.4. Another type of Po- laroid foil was invented in 1938 by E 472 Part II 24.4 Birefringence, in Particular in Calcite and Quartz 473

eo E o Part II P eo

Figure 24.7 The birefringence of calcite. The direction of the “optical axis” is indicated by the shading lines, and the plane of oscillation of the extraordinary beam is shown pictorially. The principal plane of the prisms (a plane perpendicular to its refracting edge) is at the same time a principal plane of the crystals, here the plane of the page. the page), then only the ordinary beam can pass through the polarizer. Therefore, the planes of oscillation of the two beams are perpendicular to each other. The plane of oscillation of the extraordinary beam lies along a principal plane of the crystal, while that of the ordinary beam is perpendicular to it. From the angles of deﬂection, the refractive indices can be calculated. For green light, we obtain

neo D 1:49 ;

no D 1:66 :

The extraordinary beam is less strongly refracted (Fig. 24.7, bottom). For this reason, calcite is called negative birefringent. For quartz, the opposite is the case; quartz is positive birefringent. In Fig. 24.7, the beam in the interior of the crystal is either parallel to the optical axis (above), or it is perpendicular (below); i.e. the angle between the beam and the optical axis is either zero or 90ı.The measurements can however also be repeated for intermediate values of , for example as shown in Fig. 24.8, left side. The refractive index no of the ordinary beam is found to be equal to the value quoted above, no D 1:66, for all values of . The refractive index of the extraordinary beam, in contrast, changes with ˛ and . It has its smallest value at D 90ı, and its maximum at D 0ı.For D 0ı, neo D no, i.e. along the optical axis, birefringence is absent. In Fig. 24.8, right side, a prism with a different orientation is sketched. In this case, the optical axis is parallel to the refractive edge of the prism, and thus perpendicular to the plane of the page. This is indicated by the dots. The two beams are perpendicular to the optical axis within the crystal for every angle of incidence ˛,i.e. is , . eo o 66 : NN 1 ,but .We D OCHON ): Both 24.3 o 24.10 n 24.9 prism). When , because the optical ı ca. 1° OLLASTON between the beam and the

o eo is constant at 90 n becomes very complex even for angle and, together with the angle

eo oblique o , it determines a plane of incidence. In the orientation γ ˛ prism). 49. : The birefringence of calcite. At left, the refractive indices can A double prism made of quartz gives two non-achromatized (see . Therefore, for every angle of incidence, we measure the 1 ı ), symmetrically deflected sub-beams (W D eo 18.8 n ENARMONT α S axis lies parallel to the refracting edge of the prism. and joined by a film of water,For it other choices is of suitable the for axis the directionsbeam polarization in of can the halves ultraviolet be of light. made the prism, tobe the pass achromatized. ordinary through However, this the costs prism half without of deflection the and beam can divergence (R thus Sect. same two values of the refractive indices as given above, be measured for different angles of inclination always 90 Figure 24.9 Figure 24.8 optical axis. At the right, in contrast, The examples given thuscases, far which in are this also section important refer for to applications several (Fig. special the first surface onof which the page the were lightWithout either is this parallel condition, incident the or as situatio perpendicular well to as the optical the axis. plane single-axis (“uniaxial”) crystals. The essential point can be demonstrated as shown in Fig. of incidence the light is incident at an make use of the same experimental arrangement as in Fig. as drawn, the plane of incidencecrystal. is Both parallel sub-beams to propagate a in principal the plane plane of ofNow, the the incidence. thick calcite block is rotated slowly around the normal This causes the opticalThis axis is unimportant to for the move ordinary beam; outthe it of plane remains as of the before incidence within plane (planeever, of of the extraordinary the incidence. beam page) continues to over propagate itsplane along entire of a path. the principal crystal. How- Thisoptical principal axis. plane contains It the is normal thus andduring no the the longer in rotation the around planeon the of a incidence ordinary cylindrical and beam surface circles outside. on Apart a from cone the inside special and cases treated Polarized Light 24 474 Part II 24.4 Birefringence, in Particular in Calcite and Quartz 475

Figure 24.10 Refraction outside R the plane of incidence. When the calcite block is rotated around eo the normal NN to the surface of o incidence, the extraordinary beam is refracted out of the N N plane of incidence (the plane α of the page). S

R Part II above, the refraction of the extraordinary beam thus does not take place within the plane of incidence. The elementary law of refraction (Fig. 16.4) fails. The refraction of the extraordinary beam can in general be described only in perspective with a three-dimensional representation. The phenomena become still more complex when the crystals have two axes, i.e. with crystals which have two internal directions that exhibit no birefringence. In such “biaxial” crystals, there is no “ordinary” beam at all. Both beams are “extraordinary”, i.e. for both, the refractive index depends on the direction, and both in general leave the plane of incidence when refracted. The planes of oscillation of both beams remain perpendicular to each other in biaxial crystals. For physical investigations, one often uses cleaved pieces of crystals from the biaxial group of clear mica5. Mica sheets have mechanically distinguished directions. If the sheet is laid onto a blotting pad and a hole is punched through it with a pin, a chatter mark as photographed in Fig. 24.11 results. It shows a six- pointed star with two long arms. The direction of the long arms is called the ˇ direction, and the direction perpendicular to it in the plane is the direction. The two beams resulting from birefringence oscillate parallel to the ˇ direction and to the direction. Red-ﬁlter light which oscillates parallel to the ˇ direction (which propagates faster within the crystal) has a refractive index of

nˇ D 1:5908 : The red light which oscillates parallel to the direction (and propagates more slowly within the crystal) has a refractive index of

n D 1:5950 : Many additional details of birefringence are important for crystallography, but not for its physics applications.

5 The two optical axes in mica form an angle of 45ı within the crystal. The midline of this angle is nearly perpendicular to the cleavage planes (deviations of less than 2ı). The plane deﬁned by the two optical axes intersects the cleavage planes in Fig. 24.11 along the direction . x y B . γ 9.4 per- B and and D A β x A is varied. In this ı /: y A ı/ ; are the semi-axes of the ellipse b C B b / t t and a (Vol. 1, Sect. 4.4; see also Sect. .! .! x . The vertical oscillation along the a ı sin sin 45 A B is the circular frequency D D D ; the phase difference ı y x B 2 D and .! A ), the axes of the ellipse lie at some angle between ). When both oscillations have the same frequency, in shows examples for the special case in which The formation of an elliptical oscillation by superposition of A chatter mark on 24.12 . 9.20 24.13 (Video 9.1) a sheet of mica axis leads the horizontal oscillation. Figure 24.11 Figure Figure 24.12 general the deﬂection followslimiting elliptical cases. orbits; The circlesmention form and two methods: of lines are the ellipses can1. be varied The at two will. mutually perpendicular We sinusoidal oscillations 24.5 Elliptically-Polarized Light In the sections on mechanics, wependicular treated sinusoidal the oscillations superposition of two and Fig. two mutually-perpendicular linear oscillations withand the a amplitudes phase difference of case (Fig. have amplitudes the directions of the two individual oscillations: . Polarized Light 24 Video 9.1: “Circular Vibrations” http://tiny.cc/qcggoy 476 Part II 24.5 Elliptically-Polarized Light 477

x A y B

7 7 5 7 Phase difference δ = 0° 30° = 6 90° = 2 150° = 6 7 180° = 7 210° = 6 7 D δ λ λ 5 λ 7 Path difference = λ· 2 7 = 0 12 4 12 λ 2 12 λ

Figure 24.13 Examples of elliptical oscillations for the special case of A D B. The vertical oscillation leads Part II the horizontal one by a phase difference of ı. That is, the vertical deﬂections begin sooner with positive values than the horizontal ones. If we wish to apply these images to travelling transverse waves (using the values of the path differences shown), they show the sense of rotation for light that is incident on the plane of the page in the positive z direction (that is perpendicular from above). Compare Figs. 24.15dande.

Figure 24.14 The formation of an elliptical oscillation x from two mutually-perpendicular linear oscillations with B the amplitudes A and B and a phase difference of ı D 90ı. The semi-axes of the ellipse, a and b, are equal to the amplitudes of the linear oscillations, A and B. a A b y

2. The phase difference ı of the two individual oscillations is kept constant at 90ı, and the ratio of their amplitudes is varied. Then the axes of the ellipses are parallel to the directions of the two individual oscillations (Fig. 24.14). In a corresponding manner, we can superpose two propagating, linear polarized waves. Their planes of oscillation are positioned perpendicular to each other and their “light vectors” are added at every point along their path. We will illustrate the composition of the waves and the forms of circular and elliptically-polarized waves with two examples using perspective drawings (Fig. 24.15). These represent snapshots – like all pictures of propagating waves. The direction of propagation is the z axis, from the front left to the rear right in the drawings (Fig. 24.15a). In Fig. 24.15 b, the two waves have the same amplitudes and their path difference is zero. When their vectors are added, we again obtain a linearly-polarized wave. Its plane of oscillation is inclined to the vertical by 45ı (Fig. 24.15c). In Fig. 24.15 d, the two waves likewise have equal amplitudes, but now, the horizontally-oscillating wave leads the vertically-oscillating wave by D =4. Adding their vectors yields a circularly-polarized wave. In its snapshot image, the set of all the vectors makes up a helical surface or “spiral stairway” . ı 4. of P d = z is then z (rcp) light area of the he- the propagation (lcp) light wave). , is inclined by 45 E against and onto a polarizer c F , the two light beams have G axis horizontally to the right). In , a nearly perfectly collimated y C right-circularly polarized as its center axis. In every pair of , both inside the crystal and to the z 24.3 left-circularly polarized ). This is a direction. They have equal amplitudes (here “snap- z z screw (a at perpendicular incidence. In the mica plate, the light G . From the condenser left-hand 24.16 . A fixed plane at the right rear which is perpendicular to axis points upwards and the positive x the platelet, in contrast to Fig. to the vertical. The linearly-polarizedmica light platelet then strikes a birefringent with the direction of propagation beam is split through its birefringence intowhich two propagates sub-beams. faster The within beam thevertical, crystal while has the slower its beam plane oscillatestwo of in beams oscillation the overlap horizontal almost plane. completely The within the small thickness a path difference (i.e. a difference in their optical path lengths; see right, after leaving it. After leaving the birefringent platelet points which are spaced atpoint a in distance the of same one direction;one wavelength, one wavelength. the rotation vectors of the helix correspondsThis to general scheme, valid forapplied every type to of the transverse waves, description canconnected of be with some important birefringence. phenomenato which We Fig. are demonstrate this with reference beam of light passes through a red filter Its plane of oscillation, indicated by the pointer towards the source of the waves The superposition of two transverse waves which are oscillating in mutually If the horizontally-oscillating transverse wave were leading with a path difference of (i.e. z y z along https://en. de C24.5 towards 4, the snapshot would show a optical con- x = penetrated in sequence byof the intersection individual rotate vectors in (like a clockwise the sense steps as of seen a by spiral an stairway). observer looking Their lines direction wave. The arrowheads along thelix helix does in Part not e rotatethe simply around entire indicate the area its enclosed z helicalcharacteristic by axis form. of the as The the helix the waves to wave move moves in forward. the z Instead, direction we without should rotation, at imagine the velocity Part d, the horizontally-oscillating transverse wave leads the vertically-oscillating wave by shots”; the positive perpendicular planes and propagating in the Figure 24.15 ab 3 . The “physical , so that the mean- is used. Polarized Light 24 C24.5. The notation described in the figure caption is the so-called “optical convention”; the rotation of the electric field of the lightobserved is looking the source convention” is just the opposite: There, the observer is supposed to be looking away from the source the beam of light rather than towards it ings of ‘rcp’ and ‘lcp’exchanged. are Helpful anima- tions can be seen at wikipedia.org/wiki/Circular_ polarization In this book, the vention 478 Part II 24.5 Elliptically-Polarized Light 479

E x β V z y γ

L CFP G A L M

Figure 24.16 The production of elliptically-polarized light using a mica platelet G. ˇ and Part II are the directions deﬁned in Fig. 24.11. Without the radiometer M, the arrangement is also suitable for demonstrating interference phenomena with collimated beams of light (Sect. 24.6).

Sect. 16.3)of

D d.n nˇ/: (24.1)

We insert the values of the refractive index for red-ﬁlter light as given at the end of the previous section ( D 650 nm D 6:5 104 mm) and obtain

D 42 104d or

42 104 d D d D 6:5 : (24.2) 6:5 104 mm mm As a result of this path difference, the two light beams, which are oscillating perpendicular to each other, superpose to give an elliptically- polarized light beam (this includes of course the limiting cases of circularly- and linearly-polarized beams). To identify the type of polarization, the section of the apparatus to the right of G is employed; its most important component is a second polarizer A, called in this application the “analyzer”. The light which it transmits falls on a lens L, and the lens forms an image of G either on the radiometer M (e.g. a photocell) or on an observation screen. So much for the experimental arrangement, now to its demonstration: We set the analyzer into a slow, uniform rotation. At the same time, we observe the deﬂections of the radiometer for different angles between the plane of oscillation of the analyzer and that of the polarizer. Examples: 1. A ‘dry run’ experiment without the mica platelet G (i.e. for d D 0). Only linearly-polarized light reaches the analyzer. It allows only the component of the light vector E of the incident light which is parallel to its transmission direction to pass through, with a magnitude , / etc. and . 1 12 refers ı I I 1 0 3 is tilted . 4 plate” D P ; D = .Thesame 2 ). The light I

,curve 0° D 24.2 330° 30° 4(a“ 2. We again ob- .At is the angle between 24.17 ı = =

.Thismeansthattwo ı 167 mm. D : D 60° 0 300° ψ . The light is circularly 270 D III D d 154 mm is now inserted. It pro- : 0 and 90° ı , according to Eq. ( to circularly-polarized light 270° D , there are more or less ﬂat minima, ). 90 ı d III ) allow no light to pass through from I D A III D 270 24.17

, however rotated by 90 II I and D 120° P . The light is elliptically polarized, we measure 240°

1 12 and , and we observe curve . This agrees with the measurement; the results are ı D 150°

210° d to elliptically-, and 90 The radiant power (relative values) transmitted by the analyzer 2 , represented as the length of the radius. 180° II . The transmitted radiant power must therefore be propor- , no light is transmitted. The light is thus again linearly po- C24.6 D ı

, and the analyzer allows some light to pass for every angle (notshowninFig.

24.16 II ı 180 cos E D .At

“crossed” polarizers ( duces a path difference of the lamp to the observer. 2. A mica platelet of thickness equivalent to but no longer islinearly-polarized the light. transmitted intensity zero, as it would be with

4. The mica platelet is 0.038 mm thick, or “quarter-wave” plate).dependent of The deﬂection of the radiometer is in- remains linearly polarized, and we again measure curve holds for mica platelets whoseabove thickness thickness, is and an thus integral giving multiple path of differences of the to linearly-, in Fig. the plane of oscillation of the analyzer and that of the polarizer. Curve 5. The mica platelet has a thickness of of Figure 24.17 tional to cos curve The zero values appear at shown graphically using polar coordinates in Fig. larized, but its plane of oscillation relative to the polarizer by 90 polarized. 3. The mica platelet is 0.077mm thick, tain a curve of the form ); they convert 24.3 Polarized Light 24 linearly-polarized light into circularly-polarized light. They can also be obtainedcombinations as of linear and circular polarizing foils (keep in mind the direction of the light!). C24.6. There are also polarization foils available today for producing circularly-polarized light (Sect. 480 Part II 24.6 The Interference of Parallel Beams of Polarized Light 481

6. Thus far, we have kept the amplitudes of the two sub-beams constant and have varied their path differences. Now, we keep the path difference constant at =4, i.e. we use a =4-plate and vary the ratio of the amplitudes. To do this, we change the angle between the plane of oscillation of the polarizer P and the vertical (i.e. the ˇ direction of the mica platelet). In this way, we can produce elliptically-polarized light with any desired oscillation form using only a single mica platelet. We can thus obtain all of the curves as measured in Fig. 24.17 as well as their intermediate forms.

To conclude this section, we replace the red-ﬁlter light which we have Part II used throughout the section by everyday incandescent light. Furthermore, we remove the radiometer and observe the images directly on a screen. The constant in Eq. (24.2) has a different value for every wavelength region; thus for example, with green light of wavelength D 535 nm (from a thallium-vapor lamp), we ﬁnd

d D 7:1 : (24.3) mm The individual wavelength ranges thus exhibit different path differences and polarization states. The analyzer allows some spectral ranges to pass, others less or not at all, i.e. for one range, curve I in Fig. 24.17 applies; for another curve II applies, etc. As a result, the image of the mica platelet appears in a variety of colors, which glow brightly for some thicknesses of the crystal.

24.6 The Interference of Parallel Beams of Polarized Light

In the last experiments discussed above, we superposed two coherent, transverse waves which were however oscillating in perpendicular planes, with arbitrary path differences. This yielded elliptically- polarized waves (including the limiting cases of linear and circular polarization), but no interferences, i.e. no changes in the spatial distribution of the waves, no maxima or minima as for example in Fig. 20.10. In order to produce “interference fringes”, the coherence of the two beams alone is not sufficient; instead, they must also have a common plane of oscillation. A common plane of oscillation can always be obtained by inserting an analyzer (e.g. A in Fig. 24.16). An analyzer permits only those components of the two waves oscillating perpendicular to each other that are parallel to its own plane of oscillation to pass through. In Fig. 24.16, the plane of oscillation of the polarizer and that of the analyzer are perpendicular to each other. They could also be set to be parallel; then all the maxima and minima in the interference patterns would exchange places. Take note of both possibilities. Following these general preliminary remarks, we will present two examples – in this section, only for parallel beams of polarized light. E F G H I .The wedge / .Itex- , right), 2 etc., the F = Thin edge Thick edge its long di- of the wedge 24.8 e f g h 24.18 C and f , across m E . D and e is replaced by a long, flat and is now parallel to that of the se the spacing of neighboring in- etc., the path difference of the two ı g ; 24.16 f ; e pending on the form of the ellipse (compare in Fig. G quartz plate, about 1 mm thick, also cut parallel etc. appear as deep black minima. The maxima ) g ; f Equidistant interference fringes in a quartz ; 24.19 e . Therefore, the light behind the birefringent crystal is po- ). etc. occur for path differences of G cut parallel to the optical axis (collimated beam of m plane-parallel ; 24.17 F D ; light is elliptically polarized.light to The pass analyzer through, de allowsFig. some part of the analyzer. In the transition regions between wedge red-filter light, length of thedecreasing wedge from 38.5 0.79 mm, to thickness 0.48just mm; as photographic in positive, Fig. Figure 24.18 1. The mica platelet Using ordinary incandescent light, thecolor-shaded interference fringes bands. appear as This isterference becau fringes is reduced asincandescent the light, wavelength the decreases. interference Thus, fringes of with regions the overlap various in wavelength different locations.phenomena. This is true of all interference 2. A to its optical axis,whose is planes placed of between oscillation are theThe set analyzer image to and of be the perpendicular thewhole polarizer, to length quartz each the plate other. same in chromaticsame hue thickness. incandescent as We now light the cast the wedge showsbut image where over of it rather the its plate had onto not the onto thespectrum a screen, entrance on the slit screen. of a The spectrometer, spectrum is and striped observe its cut from a birefringentas crystal optical (e.g. axis quartz). is parallel The to the direction edge denoted of the wedge (Fig. and this edgeperfluous is and placed can horizontal. bewe removed. see The the On image the radiometer of screen, M the using wedge is red-filter as photographed light now in su- Fig. sub-beams is equal to an integral multiple of the wavelength, so that hibits interference fringes parallel to the edgetion: of The the wedge. interference fringes Explana- areThe curves crystal of produces constant two path difference. sub-beamspath due difference to depends its on birefringence. thethrough. The thickness Their interference of fringes are the thus a layerthickness. kind that of At curves the they of positions constant pass larized just as before entering it.the It fringes cannot pass through the analyzer; E light is again linearly polarized behindplane the of birefringent oscillation crystal, is but its tilted by 90 Polarized Light 24 482 Part II 24.7 Interference with Diverging Beams of Polarized Light 483

Figure 24.19 Interference fringes in a continuous spectrum, produced by a plane-parallel quartz platelet cut parallel to its optical axis and ca. 1.1 mm thick; here shown as a function of wavelength rection by dark interference fringes (Fig. 24.19). The missing waves in these dark bands remained linearly polarized at the right of the apparatus, behind the birefringent plate, just as they were at the left, before entering the plate. Therefore, they could not pass through the Part II analyzer.

24.7 Interference with Diverging Beams of Polarized Light

Interference with divergent polarized light can be dependably produced in the focal plane Z of a lens. The light source must have a large area. It is expedient to make the optical path on the image side of the lens telecentric (Fig. 24.20, top). Then one requires only small birefringent crystal plates. The light beams belonging to the image points 1 and 4 are dotted in the ﬁgure. They penetrate the crystal plate, just like the light beams of all the other image points, with parallel boundaries. Furthermore, all the light beams pass through the polarizer and the analyzer, in this case two polarization foils (Sect. 24.3). The plane of oscillation of these two foils are perpendic-

E f1 f1

B 1 From a large χ illuminating area X 4

L1 Polarizer Entrance Crystal Analyzer Focal plane Z Exit CYE L Light 2 pupil plate L1 L3 pupil source To the screen

Image of light source Image of light source

f2 f2 f1 f1 f3

Figure 24.20 Top: Using divergent polarized light, interference fringes are produced in the focal plane of a lens. Bottom: The same as a demonstration experiment. The glowing surface X is an illuminated lens L2. The image of the light source (a carbon arc) projected by L2 acts as entrance pupil. In Z, there is not only the image of an infinitely distant plane, but also one of the plane Y determined by f2.Afree-hand experiment: Place the crystal platelet between two crossed polarization foils, hold it close to an arc lamp and observe the light on a screen. one c. , top). is thus Z 24.21 b. It is vis- 24.20 .Thenwe ı ), we observed (Fig. 24.21 to its optical axis, 24.16 parallel b. nd 4, in contrast, there is only to its optical axis (see Fig. 24.8, illation of the light coming from to the optical axis, gives practically in the center of the image was equal 24.21 ı , the dark and the light regions are exchanged. / (top). Furthermore the penetration points of 1 2 perpendicular C m . 24.20 D b, the path difference the image of a plane which is infinitely distant at the left. a. It exhibits circular interference fringes and a dark as seen from in front and enlarged. The numbers 1 and 4 Z 24.21 .For m 24.21 24.22 only the center part of this image, using the same3. plate. A uniaxial crystal plate, cut at 45 to With a parallel beam of light in earlier figures (Fig. In Fig. 4. We put two such plates together and rotate one of them by 90 linear interference fringes. Theythe branches could of be the considered hyperbola to in Fig. be extensions of In incandescent light, one ofdue the to middle rays fringes with appears a colorless; zero it path is difference, thus and is a fringe of zeroth order. Its obtain the complicated interference pattern photographed in Fig. ular to each other (“crossed polarizers”). The image plane ible only with monochromatic light (e.g.With from incandescent a light, sodium-vapor lamp). the orders ofhigh. the The interference curves fringes of are constant too pathThe difference have detailed a explanation hyperbolic shape. of thisafield here. phenomenon would take us too far initially dark. Only afterwe the see in birefringent crystal plate is inserted do shows the interference pattern photographed in Fig. three additional beam axesof are incidence shown. (a principal For planedicular each of to of the it crystal) them, are and indicateddouble the the arrows by plane show plane the the perpen- dashed plane intersecting ofthe lines. osc polarizer. Each The beam thick isan decomposed ordinary at and the an positions extraordinarythin 2 sub-beam. double and arrows. This 3 is At into the indicated positions by 1 the a an extraordinary beam, and at positionbeam 5, alone only an can ordinary never beam. producelight One interference. remains unchanged Therefore, and the thusthe incident cannot corresponding pass areas in through the the image analyzer; remain dark. 2. A thick, uniaxial crystal plate, cut It shows interference fringes. Examples: 1. A calcite plate, cut mark the penetration points of thesketched beam in axes for Fig. the two light beams Therefore, the curvesence of fringes, constant are path circularinclination”). (they difference, are i.e. thus a the sort interfer- of “curvesThe of crosses constant are interference-freepolarized regions. beam. In them, there TheFig. is reason: only We have drawn the crystal plate in cross. Explanation: Thebeams depends path only on difference their of angles of the inclination two polarized sub- left), givesFig. the interference pattern shown as a photograph in Polarized Light 24 484 Part II 24.7 Interference with Diverging Beams of Polarized Light 485 ab Part II

Figure 24.21 Three interference patterns of uniaxial crystals in divergent polarized light, photographed in the image plane Z of the apparatus in Fig. 24.20 (photographic positives). Part a shows a calcite plate (d D 2 mm) cut perpendicular to its optical axis (using circularly- polarized light, the black cross can be eliminated). b: A quartz plate cut parallel to its optical axis (d D 9 mm, Na-D light). c: Two quartz plates, cut about 45ı to their optical axes and put ı one above the other, rotated by 90 (SAVA RT’s double plate). All of the interference patterns obtainable with crystals and polarized light are noticeable for their strong intensity. This is a result of the coherence condition (Eq. (20.1)). Compare e.g. the interference pattern in Fig. 24.21 a with the one in Fig. 20.10. There, the angle 2! was already very small; when using polarized light, it is zero. This means that both of the “rays” in each pair which can interfere with each other have the same direction. They nevertheless have a path difference, since they are polarized perpendicularly to each other and propagate in the material at different velocities. For sin 2! D 0, one can use light sources of arbitrarily large diameter and thus obtain high light intensities.

neighboring fringes on both sides are colored, and the rest of the structure of the interference pattern is not visible with incandescent light.

F. S AVA RT set two such crossed quartz plates, cut at 45° to their optical axes, together with a polarization prism in the same mount, and thus obtained a very sensitive polarimeter. It serves in many kinds of observations to detect small admixtures of polarized light in natural light. If you look at is by ro- ˛ E G o (24.4) 5 ”, Vol. 3, 10th E E /mm, and it in- eo eo 4 1 ı . Such a “sensi- 6 http://www.quartzpage.de/ E E 2 3 o o : eo eo d ough such an instrument and rotate Effect const and replace the mica platelet of the plate, that is D d C24.7 ˛ Lehrbuch der Experimentalphysik 24.16 a ARADAY 24.21 . The explanation of the Rotation of the Oscillation Plane. The F 2004), Sect. 4.9). English: See for example the plane of oscillation of the light. The angle of rotation the sky or some illuminatedit object around thr its long axis,low then order, you the will colorless alwaysfraction center see the fringe of interference with the fringes its of light light colored is neighbors. is an ideal nearly A limiting case. small always polarized; completely unpolarized These are two mirror-image forms of quartz which rotate the light in either the tive double plate” exhibits a uniform purple color only when it is creases strongly with decreasingincandescent wavelength. light Therefore, instead if ofthe we red-filter analyzer use light, for which theretion, no is we light see no is a position bright transmitted; field of instead, of at viewFor which each demonstration has posi- experiments, a a different color. quartzticularly plate suitable. 3.75 mm Two thick of is theseother, par- plates one can of be them setthe made up of other adjacent right-rotating to of (“dextrorotatory”) each quartz, left-rotating (“levorotatory”) quartz dark cross in Fig. Figure 24.22 edition ( one or the otherexample Bergmann-Schaefer, direction “ (“right- and left-rotating”, enantiomorphism; see for 6 24.8 Optically-Active Materials – We now return to Fig. gen_phys.html tates a quartz plate whichthis setup, is a cut new phenomenon perpendicular can to be observed: the The optical quartz axis. plate With proportional to the thickness The value of the constant for red-filter light is 18 Optik and at . ”, Chap. 10. Polarized Light 24 und Atomphysik Some relevant references in English can be foundhttps://www.osapublishing. at org/josa/abstract.cfm? uri=josa-65-3-352 https://www.osapublishing. org/josa/abstract.cfm? uri=josa-50-9-892 C24.7. An historical note: Further details, in particular on the analysis of elliptically- polarized light, were given in the 13th edition of “ 486 Part II 24.8 Optically-Active Materials – Rotation of the Oscillation Plane. The FARADAY Effect 487

Figure 24.23 The superposi- AAδ tion of two oppositely-rotating 2 circular oscillations of the same δ frequency and amplitude. The R R direction r in the left-hand im- llr age is shown as the long dashed 0 0 r line outside the circle in the right-hand image.

A' A' Part II placed between strictly parallel-oriented NICOL prisms. Even with very small angular deﬂections, the color hue on the one side changes towards the red, and on the other side towards the blue. This setup can be used to align the oscillation planes of polarizer and analyzer strictly parallel to each other, which is useful for example in calibrat- ing measurement instruments, e.g. the saccharimeters which will be described below. The ability to produce optical rotation (rotations of the oscillation plane of light), usually termed optical activity, is not speciﬁc to a crystalline structure of the material. It can also be found in molecules in solutions, for example for sugar dissolved in water. The angle of rotation of the plane of oscillation is in this case proportional not only to the layer thickness of the sample, but also to the concentration of the solution. Therefore, we can determine unknown concentrations from the value of the angle of rotation measured (for example in “saccharimeters”). Sugar molecules can also be right-rotating (“right-handed” or dextrorotatory) or left-rotating (“left-handed” or levorotatory). A 50 % mixture of the two is called a “racemic mixture”.

Every linearly-polarized oscillation can be treated as the superposition of two circular oscillations of the same frequency and amplitude, but with opposite senses of rotation. In Fig. 24.23,attheleft,l denotes the vector which is rotating to the left, and r the vector which is rotating to the right, while R is the resultant (sum) vector. Its end point passes along the double arrow AA0. The half-length OA is the amplitude of the linear oscillation (that is, the maximum value of its deﬂection). At the right, the same superposition is drawn, but now the oscillation of the vector rotating to the right leads that of the left-rotating vector by a phase difference ı.Asare- sult, the resultant linear oscillation vector R is rotated by the angle ı=2in a clockwise direction. Applied to the case of light, this means that a dextrorotatory material transmits a right-circularly-polarized light wave (see Fig. 24.15) more quickly than a left-circularly-polarized wave. The right-circular wave propagates more rapidly in the material than the other wave; it has a smaller refractive index. An optically-active material shows a new type of birefringence, called circular birefringence: It splits natural light not into two linearly-polarized beams (as in linear birefringence), but instead into two circularly-polarized beams. This strange form of birefringence can be seen in all spectrometers which use simple quartz prisms. During the fabrication of such prisms, the line of symmetry SS (Fig. 24.24) is chosen to be perpendicular to the long in G 4- = S S axes, we can see that

and the ˇ ence. Conclusions , we find a similar situation with , we distinguish between conductors 436 nm by only 7 units in the fifth place , with the optical axis parallel to the optics D Looking perpendicular to the field direc- , i.e. a rotation of the plane of the linearly- C24.8 birefringence effect the solid materials, there are numerous birefringent electromagnetism A quartz prism that exhibits birefrin- , then we find that the field of view always shows brighter double lines. These two lines are circularly polarized in opposite ARADAY 24.16 one of the two spectral lines vanishes. plate of mica in frontThen, of depending the on ocular, the together positions with a of the polarization analyzer. direction of the quartzNevertheless, we can column, see that that all the spectral isspaced lines are perpendicular split into two to closely- thedirections. optical axis. The magnitude of thisindices circular differ birefringence for is example very at small. The refractive after the decimal point.optical axis to be We the can direction whichlikewise therefore is in free in of calcite, general birefringence, in simply quartzcrystals. and and define in the allBecause the of other the non-optically-active smallsuitable birefringent magnitude for of demonstration this experiments.blue circular spectral For birefringence, line individual it from observations, is a the not mercury-arc lamp is suitable. We place a optical axis field direction. the division intostances. single- Among and double-refracting (birefringent) sub- and insulators.materials There (in are particular numerous thean conductors idealized metals), limiting among but case. a the In perfect solid insulator remains tion, we observe gence in the dashed direction,its which is denoted as substances, including the crystals ofa all the strictly non-regular systems, single-refracting but we material place thick can layers only (ofsingly-refracting several be centimeter materials approximated. thickness) ofplastics) (regular apparently between If crystals, crossed polarizers, glasses, e.g. in transparent place of the plate Fig. and darker regions,used: which The are materials are coloredregions. birefringent when in many incandescent more-or-less light extended is Figure 24.24 In the field of Paramagnetic and especially ferromagnetic materials rotate theof plane oscillation of lightthe when direction of they light are propagationis is placed parallel in the to a the F field magnetic direction; field this and 24.9 Strain Birefring polarized light beam. ) for effect and http://www.osti. ARADAY Polarized Light https://en.wikipedia. 24 a brief history and references. other magneto-optical and electro-optical effects. See also org/wiki/Faraday_effect “Among the solid materials, there are numerous birefringent substances, including the crystals of all the non-regular systems, but a strictly single-refracting material can only be approximated”. for a detailed discussion of the F C24.8. See for example the book by Y.R. Shen, “Prin- ciples of nonlinear optics” (Wiley-Interscience, New York 1984) ( gov/scitech/biblio/6102640 488 Part II 24.9 Strain Birefringence. Conclusions 489

Figure 24.25 Strain birefringence in a model of a hook for a crane (the polarizers are perpendicular to each other and tilted by 45ı to the vertical; photographic positive). (The holder, the lever to provide the load, and the outline of the hook were drawn into the photo to make them clearly visible) Part II

This birefringence comes about through internal strains which vary from place to place within the material. Their practical elimination is tedious and expensive; the material must be heated to almost its melting point and then cooled very slowly. Glass blanks for large astronomical lenses have to be cooled over a period of many months. “Tempered” glass approaches the optical ideal of a solid object without birefringence rather closely. However, it must be carefully protected against mechanical stresses. Even pressing between the fingertips produces a noticeable birefringence. For optical technology, strain birefringence is a source of annoying disturbances. For another technical field, however, namely the science of materials strength, it is extremely useful. With its help, the distribution of compressive and tensile stresses can be investigated in model experiments. For example, Fig. 24.25 shows a model profile of the hook for a crane, made of plastic and placed between two crossed polarizers. A load is applied via a lever. The regions which are stressed due to compressive or tensile strains appear brighter in the image. The dark boundary stripes between them show the strain-free transition regions, the “neutral fibers”. The quantitative evaluation of such images is not simple. It is treated in the voluminous technical literature on this subject. Our treatment of polarization has been limited to experiments using visible light. In the ultraviolet and infrared spectral ranges, one finds nothing new. Polarizers for the ultraviolet are described in Fig. 24.9, and for the infrared, they will be discussed later in Sect. 25.6.Itis also more expedient to treat the polarization of X-rays later; it requires special experimental techniques (Sect. 26.8). riiga h aimtr hs powers These radiometer. the at arriving nFig. measurement In a by index: defined refractive is the like It just procedure, it). from derived quantity equivalent ln t ah n ftoset ftesm aeil(u ihdif- with (but material same the of sheets thicknesses, two ferent of one path, its Along mltd ftelgtoclaigwti htpaei eoe by denoted is plane that within oscillating light the of amplitude n ercino Light of Refraction and Reflection Absorption, Between Relation The enee nyasnl aeil osat aeyterfatv in- refractive the namely case, constant, that materials dex In single matter. a of only layer needed a radiationwe through the passing that on assumed attenuated have not we far, was thus observations our of all In Absorption and Extinction The 25.2 reflec- of plane the experiments Sect. as the known see all (also In tion; incidence lamp. spectral of metal-vapor plane individual a the presented, from use those as we monochromatic such be lines, measurements, to presumed for is radiation (single-frequency); This waves. plane with pure dealing are we that assume beams we chapter, entire this In Remark Preliminary 25.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © aimtrgv eaiemaueo h ain power radiant the of measure relative a give radiometer ifrneo h thicknesses the of difference oprdt h thickness the to compared ufcso h hes hs rcin r h aefrbt sheets. both for same the rear are and fractions front These the at sheets. reflection the to lost of is surfaces radiation the of fraction some with pnigradiant sponding eodmtrascntn,teso-called the constant, materials second a n hto ih siltn epniua otepaeby plane the to perpendicular oscillating light of that and n h hesta ihu hm hsi o w esn:First, reasons: two for is This them. without than sheets the ftemtra.I hr shwvra teuto,te eneed we then attenuation, an however is there If material. the of ie aallbuddbas flgt hti practically is that light, of beams) parallel-bounded (i.e. 25.1 Constants olmtdlgtba sicdn naradiometer. a on incident is beam light collimated a , 16.3 intensities x ftelgtle ntepaeo h ae The page. the of plane the in lies light the of ) 1 or olsItouto oPhysics to Introduction Pohl’s x 2 x r lentl lcdi h em The beam. the in placed alternately are ) e q 1.) r nbt ae smaller cases both in are (19.2)) Eq. see ; 1 h deflections The . x D . x 2 xicincntn K constant extinction W x P 1 1 / and S scoe ob small be to chosen is te’inl)o the of ’signal’) (the , https://doi.org/10.1007/978-3-319-50269-4_25 W P 2 W o h corre- the (or P E ftelight the of collimated ? . C25.1 o an (or E k , 2..Tequantity The C25.1. chapter. previous the in as light, constitute which waves netic electromag- the of field electric the of vector the notes ftelight. the of direction propagation the to perpendicular are both always, as and, incidence, of plane the to perpendicular and allel par- field the of components E 25 k and E ? r the are E de- 491

Part II . D b). x (25.1) (25.2) 1 .b:Its 25.1 x through ;thatis, and then – 2 d and x x x d )isdiffi- 1 ; . The use of ) d er “absorbed” extinction con- : mm is nearly always ; in the absence of x 4 K / 1 penetrates 10 x 1 D absorption constant that is lost by a colli- P > 2 W P 2 P W W K x absorption constant . ; K which ln x 1 D P into the sheet and is thus re- V W is termed the 1 2 1 S P P d radiation is eith W W K is larger for the thicker sheet than Radiometer C25.2 to consist of thin layers d extinction constant K 1 the extinction constant. For its prac- / 1 P const W x penetrates thus ln is also called the 2 ; x define D D K extinction : . x d Kd / P 2 2 1 W or it is “scattered” out of the beam. The fraction e x x S K extinction constant from scattering 1 , the thickness difference d ), P 0 W Z 5 1 )servesto S D D . – and phosphorescence). 2 a: The definition of the P P P W 25.1 W W d Reflection and Refraction of Light The proportionality factor 1 16.10 P 2 W P Z W measurement . If scattering plays no role in comparison to absorption, then The measurement of large extinction constants ( cult. It requires extremely thin layers. In such layers, interference occurs, ab C25.3 . These energy forms can later be converted back into radiation (fluorescence – mated beam of light withinscattering a is layer proportional of to material the due power to absorption and In words, this means that the radiant power We imagine the segment chosen to be of the same magnitude as the layer thickness not small compared to the thickness, as assumed above (Fig. d tical the concepts ‘extinction’, ‘extinctionleaves constant’, it open etc. asabsorption by to and themselves which to scattering. relative importance isEquation to ( be attributed to If, in contrast, wewe can will neglect absorption call relative it to the scattering, then of the radiant power which moved from the beam by scattering out of the beam, measurement with thick absorber layers sum over the absorption of all of these layers, obtaining see Sect. (i.e. taken up by theelectrical material, energy that is converted into heat, chemical or 5 Figure 25.1 Second, a fraction of the transmitte the layer of material, and also to the thickness of the layer, stant we will call the extinction constant simply the for the thinner one. The measurements yield - ’s )), are lity ’s law of Optik 25.2 AMBERT ). AMBERT . ”, Chap. 15. 19.2 gaveadetailed was treated previ- AMBERT OHL ’s law”). (L The Relation Between Absorption, EER 25 B absorption” (or “L cosine law ously in Sect. between the absorbed radiant power and the layer thickness, and the exponential law which results (Eq. ( C25.3. The proportiona C25.2. P often referred to in the literature as “L treatment of fluorescence and phosphorescence in the 13th edition of “ und Atomphysik For a modern discussion in English, see for example https://en.wikipedia.org/wiki/ Phosphorescence 492 Part II 25.3 The Mean Penetration Depth w of the Radiation. The Extinction and Absorption Coefficient k 493

and furthermore, the reflectivity depends on the layer thickness. These difficulties can be avoided by using the following procedure: We first measure P P the ratio of incident to transmitted radiant power .Wi=Wt/ as a function of P P the layer thickness d.Thenweplotln.Wi=Wt/ graphically against d.The P P larger values of .Wi=Wt/ will lie on a straight line. Its slope is the extinction constant that we are seeking.

25.3 The Mean Penetration Depth w of the Radiation. The Extinction and Absorption Coefﬁcient k Part II

In this section, we first list some values of the absorption constants of various materials for light waves in the visible spectrum. They are set out in the third column of Table 25.1. The reciprocal of the absorption constant K (or, more generally, the extinction constant), has an intuitively clear meaning: Along the path w D 1=K, the radiant power of a collimated light beam decreases to 1=e D 1=2:718 37 % of its initial value. This distance w will C25.4. In carefully purified be called the mean penetration depth of the light. Examples of this silicate glasses, for infrared D : useful quantity can be found in the fourth column of Table 25.1. light ( 1 55 m), absorption constants of : 8 1 The transparency (everyday language) of a layer of material of thickness 3 6 10 mm can be d depends on the ratio d=w. The smaller this ratio, the more transpar- attained, and thus mean ent is the layer. Thus, at a thickness of a few m, even pitch becomes penetration depths of transparent(w 7 m); and at a thickness around a hundred times smaller, around 28 km. With the aid metals are also transparent (w 10 nm). of intermediate amplifiers, this makes it possible to man- For wave phenomena, the wavelength is always the appropriate ufacture and use optical-fiber length scale, in particular the wavelength in vacuum (or air). We thus cables for transmitting data use the ratio of the wavelength in vacuum (air) to the mean pene- around the globe (see Com- tration depth w of the radiation in the particular material, i.e. =w. ment C16.10).

Table 25.1 Absorption constants, mean penetration depths, and absorption coefficients of some materials Material Wavelength Absorption Mean pene- Penetration depth w Absorption in nm constant K tration depth of Wavelength coefficient in mm1 light w D 1=K 1 k D 4 w 7 Water 770 0.0024 42 cm 550 000 1:4 10 7 Heavy flint 450 0.0046 22 cm 500 000 1:6 10 glassC25.4 “Black” neutral 546 10 0.1 mm 180 4:4 104 glass Pitch 546 140 7 m 13 6 103 Brilliant green 436 7000 0.14 m 0.32 0.25 Graphite 436 20 000 0.05 m 0.11 0.72

Gold 546 80 000 0.01 m 0.022 3.6 –3 4 m = 26 liter mol mol (or in (25.3) (25.4) (25.5) C25.5 .Whether ): c 25.1 ). In both cases, 1 3 0.5 : : O 2 2 25.2 mol cm , for an aqueous copper . in H n 4 : 1.5 4.5 10 Concentration Concentration of the solution). 0.25 0.75 600 nm 25.2 K extinction coefﬁcient 1710 = in a particular case depends V λ 1 D k I 4 CuSO Number density the of molecules % absorption coefﬁcient 1 K c K or 1

D –1

liter

1.5 0.5 D = K K constant Extinction D cm w % c 71 cm , or that of a solution to be proportional ’s law”; see Fig. K : K 1 1 % 1mol 4 is the concentration, i.e. the amount of sub- c EER D D c K k (“B specific extinction constant D c c extinction constant in chemistry, as suggested by the units ’s law. The Interaction K are shown in the last column of Table ’s law k molar EER EER B Cross-Section of a Single Molecule is the density and of the dissolved molecules/volume Reflection and Refraction of Light n % Example: From the slope of the line in Fig. sulfate solution we obtain the specific extinction constant stance (here, c = It is called the one uses the extinction quantity only upon which ofmulation the of two a makes statement. possible a more convenient for- Some values of one can then define a and and the measurement of the specific extinction constant K employed; this is however nototherwise referred completely only consistent, to since the molar amount quantities of are substance Figure 25.2 1 25.4 B Sometimes, one finds the extinction constant ofbe proportional a to uniform its material density to To simplify later trigonometric calculations, we multiply by 1 to its concentration the special case of no scattering, the and define the resulting quantity as the K coeffi- )isalso 25.1 The Relation Between Absorption, . 25 called the absorption cient C25.5. In some textbooks, the absorption constant defined by Eq. ( 494 Part II 25.4 BEER’s Law. The Interaction Cross-Section of a Single Molecule 495

Figure 25.3 A model experiment demonstrating the interaction cross-section of individual molecules

If the proportionality mentioned above is experimentally obeyed, i.e. when K=% or K=c can be treated as constants, then the extinction takes place without interactions between the individual molecules. Then it is expedient in Eqns. (25.4)and(25.5), instead of the density Part II % or the concentration c,tousethe

N Number density D Number of active molecules of the molecules NV Vo lu m e V of the body or the solution

(as in Fig. 25.2, upper abscissa scale), that is:

K%Mn Kc K D K% % D NV and K D Kc c D NV NA NA

(Mn is the molar mass D M=n, n is the amount of substance, and NA is the AVOGADRO constant D 6:022 1023 mol 1).

The extinction constant K is the reciprocal of a length. Therefore,

K KV D (25.6) NV N has the dimensions of an area. We call it the interaction cross-section of a molecule. In the limiting cases discussed in Sect. 25.2, is an “absorbing” or a “scattering” cross-section (i.e. the area of the “target” for absorption or scattering).

Example: From Fig. 25.2, we ﬁnd the following value for the interaction cross-section:

D 2:82 1025 m2 :

The physical significance of the interaction cross-section can be explained in an intuitively apparent manner. Figure 25.3 shows a “snapshot” of a model gas composed of steel balls, with a layer thickness of 1 cm. We C25.6. Here, we thus have see the projection of the cross-sectional areas of the individual molecules. D 2 When brought into a collimated beam of radiation, each of these areas acts r . In general, we find as if it were completely opaque; the radiation can continue in its original N l D 1 ; direction only in the gaps between the molecules. If we stack such lay- V ers of a model gas with a statistically-disordered distribution of molecules, where l is the ‘mean free then the overall area of the remaining free gaps decreases according to an path’. For radiation, l D w, exponential function, and this yields Eq. (25.2).C25.6 the mean penetration depth; for gases, it is the mean free path between collisions (compare Fig. 2.15). 0 is O : γ β α 1 : (25.7) O 0 > k 2 of the radiation, or w . of the incident radiant < 1 K torsional-wave machine D 0 w e brushes scrape over rough paper : ), apart from the minor losses due d small fraction K 1 : 0 < in the figure); it consists of small brushes at k O 2 , a short portion of a or O 25.4 60 cm), starting from below and travelling upwards. We Some of the arms of > and Strongly-Absorbing Materials ) 1 Reflection and Refraction of Light K have physical terms been chosen in such a misleading way as D Weak absorption means: Strong absorption means: w If we round 0.08 up to 0.1. Video 25.1 or else the absorption constant Seldom a torsional-wave machine. The upper portion is fitted withjustable an frictional ad- damping mechanism ( 2 25.5 Distinguishing Between Weakly- Distinguishing between weakly- and strongly-absorbingof materials is great importance for whatis follows made in by this employing the chapter. mean This penetration depth distinction “Weakly” absorbing materials, for exampletheless diluted absorb the inks, entire can incident never- radiantficiently power or large intensity layer (at a thickness suf- the words “weakly” and “strongly” absorbing to reflection. “Strongly”in absorbing contrast, materials, can for absorb example only metals, a power. The greater partreflected cannot by its penetrate surface. the material atThis all, and is is quitewaves. general, In Fig. as can be demonstrated using mechanical surfaces when the machinecan is be set raised or into lowered together, motion. thusof varying the These the frictional waves. paper damping Along surfaces thegroup axis ( of this machine, we excite a short wave the ends of the oscillating arms. Th with no effect. sketched. This machine isthe fitted dashed with line a damping mechanism (above Figure 25.4 can make three observations: 1. Without damping, the wave group passes over the boundary The Relation Between Absorption, 25 Video 25.1: “Absorption” http://tiny.cc/sfggoy The absorption process is demonstrated using a torsional-wave machine. “Seldom have physical terms been chosen in such a misleading way as the words ‘weakly’ and ‘strongly’ absorbing.” 496 Part II 25.6 Specular Light Reflection by Planar Surfaces 497

2. With strong damping: The dumbbell arms ˇ are strongly impeded by the damping. They can take up only a small fraction of the oscillation energy from the arms ˛. The major part is reﬂected, so that the amplitude of the wave group which travels back down is barely smaller than that of the original group which moved upwards. 3. The energy which is transferred to ˇ in spite of the damping is mostly converted into frictional heat. A small remainder is passed on to , etc. Thus, the wave motion dies out over a short distance within the “absorbing material”. Its mean penetration depth w in our example is only a small fraction of the wavelength . In the case of “strong” absorption, i.e. w<, the waves cannot penetrate into the Part II material. Only a small amount of energy is absorbed, and this occurs over a short distance (cf. Sect. 25.8).

25.6 Specular Light Reﬂection by Planar Surfaces

Following our detailed treatment of the second optical materials constant, the extinction constant K or the extinction coefficient k, we will now discuss on an experimental basis the specular reflection of light from the planar surfaces of homogeneous materials. In Fig. 25.5, a collimated light beam passes through a polarizer P, and the now linearly-polarized beam is incident on a radiometer, ei- P ther directly (lower radiometer deflection S1 / W1), or else after P reflection by the surface M (upper radiometer deflection S2 / W2). The plane of oscillation of the light is alternately chosen to be parallel (Ek) or perpendicular (E?) to the plane of incidence; furthermore, the angle of incidence ˛ is varied (the limiting case of ˛ D 0, i.e. perpendicular incidence, can be only approximately achieved with this simple setup). The analyzer A is initially not present in the optical path. Each time, we measure the P D Reflected radiant power D W2 : Reflectivity R P (25.8) Incident radiant power W1 The amplitude of a light wave is proportional to the square root of its radiant power (or of the deflection of the radiometer).C25.7 Then, C25.7. Light waves are elec- for the ratio of the amplitude Er of the reflected light vector to the tromagnetic waves (see Com- amplitude Ei of the incident light vector, we can write: ment C24.2). The statement s that their amplitudes are P proportional to the square Er W2 D : (25.9) roots of their radiant powers E WP i 1 was dealt with in Com- The results of several measurements can be seen in Fig. 25.6 a–c. ment C12.5. In Fig. 25.6 a and b, the mirror is made of materials with “weak” absorption (typically dielectric materials), while in Fig. 25.6 c, it is made of a material which exhibits “strong” absorption (typically . - n : REW is ob- (25.10) V ˚ V a). Fur- P slaw ˛ ’ Radiometer 25.6 cos n . It is called the A P D / ˛ P REWSTER ˛ m, one can utilize sub- //: ı α α 90 16.4 . α n sin 90° n M Pr D was thus able to discover the linear D P ecessary, cooled) glass plate. ˛ ˇ , the reflected beam is perpendicular P principal angle of incidence P sin ALUS ˛ tan n D in the range of small and medium angles of is defined as in Fig. P ˛ .ˇ can be utilized to measure the refractive index is much larger for strongly-absorbing materials or Brewster’s angle, and it occurs for the follow- P i ˛ E = ), r . Instead, the P E : , only that fraction is reflected whose vector is perpen- ˛ a–c is the best way to illustrate their common character- P P . ( ˛ :ana- ˛ 25.10 Derivation: sin ˛ A Reflection and Refraction of Light . 25.6 angle polarization of light in 1808this by method studying we its lose reflection. 84 Unfortunately, % with of the incident radiant power (Fig. thermore, the kink in the opticalIn path is the inconvenient. infrared, thisFor example, method at of wavelengths greater polarization than is about 3 indispensable even today. stances with very high refractive indices, e.g.thus selenium avoid or the lead large sulfide, and lossesror which surfaces made occur from in such the materials canto visible be most spectral prepared metallic in region. surfaces: a Mir- similar Thecondensed manner material onto is a evaporated in polished high (and vacuum if and n The French researcher E.L. M STER polarization angle ing reason: When theincidence incident light is unpolarized, at the angle of polarizer, lyzer). The plane of incidence is the plane of the page to the refracted beam. This is consistent with B than for weak absorbers incidence 2. If the lightweak vector absorption, lies there parallel is to a the plane characteristic of angle incidence, then for dicular to the plane of incidence.linearly Thus, the polarized. reflected light has become istics and their differences.aspects: We want to point out in1. particular four The ratio 3. At the polarization angle With Eq. ( a metal). Placingin the Fig. results for these various cases side by side as Measuring the reflectivity at various angles of incidence 4. With strong absorption, there is no polarization angle or B Figure 25.5 The Relation Between Absorption, 25 498 Part II 25.6 Specular Light Reflection by Planar Surfaces 499 as Pr 45° ║ Φ Φ Φ E α → 60° ┴ E 2.5 40° 80° 40° 80° 40° 80° =

k Part II 0.436 μm 0.436 0.9; = Angle of incidence

20° 60° 20° 60° 20° λ n Air → Stainless Steel = = = 135° α α α relative to the plane of incidence, as shown ı 0°

90° ⎫ ⎬ ⎭ 1.0

0.6 0.4 0.8 0.2

(not only for the incident beam, but also along 180°

i E Incident due to reflection to due

Amplitude

┴

r ║ Reflected E E between E and Plane of difference Phase incidence ), which are derived from them). was inclined by 45 ı 25.40 0 ˛ )and( α → Total ref lection 25.22 ║ T Crown glass → Air T T E α α α P P P in the direction of light propagation α α α ┴ 1 E 1.5

Angle of incidence 20° 60° 80° = 20° 60° 80° 20° 60° 80°

n = = = α α α 0°

1.0 ⎫ ⎬ ⎭

0.6 0.4 0.8 0.2

90°

i E 180° Incident due to reflection to due

Amplitude

┴ r ║ Reflected E E and E between

’s formulas (and Eqns. ( Plane of difference Phase incidence RESNEL ║ E α → P P P ┴ α α α E 1.5 ). In graph c, the light is reﬂected from the surface of a metal. The graphs in the center row, d–f, show the paths that would be traced out =

n Angle of incidence 20° 40° 80° 20° 40°20° 80° 40° 80° 25.5 Air → Crown glass = = = α α α 135° 130° 113° 80° 45° 135° 113° 45° 0° ⎫ ⎬ ⎭

90° 1.0 0.6 0.4 0.8 0.2

180° due to reflection to due

The influence of the angle of incidence on the reflection of linearly-polarized light. In the upper row of graphs, the light is reflected in graph a at the

Incident

┴

║

E and E between

Amplitude . The graphs in the bottom row, g–i, show the phase differences which are required to describe the experimentally observed orbits in graphs d–f. These =

r Reflected E Plane of difference Phase incidence abc de f gh i 25.7 phase differences agree with those calculated from F in Fig. the reﬂected beam). For these observations, the plane of oscillation of the incident light at sketched below in Fig. by the tip of the light vector for an observer looking at all angles of incidence Figure 25.6 boundary between air and crown glass, and in graph b at the boundary between crown glass and air (to carry out these measurements, we require a prism , z ı z be- ,and axes of or 180 x ? .Thisis ║ z ║ ı ı x r E E

E , Reflected beam Reflected 45 k E D ,and αα y 135° axis).

║ = point upwards, per- upwards, point i ,

┴ axis in a clockwise x y E Ψ r, x phase differences E y x ┴

z Incident beam E and ┴ z Mirror i, endicular to the plane the of page d–i : Reﬂection produces not E p ,butalso is now complemented by adding ? 25.6 E 0. The ﬁgure depicts a special case, 25.5 and . When these are not equal to 0 k Plane of incidence ? . It can be rotated around the beam axis of E z ˛ A E y for ┴ and E k E c). It can be employed when the two optical con- direction, we must rotate the 25.7 45° are to be measured in strongly-absorbing materials = z

To the mirrorTo From the mirror Ψ k 25.6 ). without a perspective view, again making the plane of the The orientation of the light vectors in the special case of nearly The orientation of the of Light .TheyareplottedinFig. ║ ˛ Reflection and Refraction of Light and x E 25.8 n 25.12 24.17 sense to bring it into the original direction of the sketched in Fig. follow each other in a similar sequence as the the reflected light is elliptically polarized. With weak absorption, this a right-hand coordinate system.look (In along such the a coordinate system, if we the polarization analyzer The experimental setup in Fig. perpendicular light reflection andthe direction an of light observer propagation who is always looking along drawn in perspective, of our general arrangement.in The Fig. latter is shown page coincide with the plane ofment incidence. states: Our agreed-upon In arrange- every case, the positive directions of tween the vectors light vectors for an arbitraryincidence angle of the reflected beam. We thenFig. obtain measurement results as shown in only different amplitudes Figure 25.7 Figure 25.8 We now no longer alternateoscillating between incident perpendicular light and beams which parallelstead, are to we fix the the plane anglevector (called of and the incidence; the azimuthal in- angle) plane between of the incidence light at a value of of served (Fig. 25.7 Phase Changes on Reflection stants (Sect. The Relation Between Absorption, 25 500 Part II 25.8 FRESNEL’s Formulas for Weakly-Absorbing Materials. Applications 501 occurs only in the region of total reflection (indicated in Fig. 25.6 b). For strongly-absorbing materials, however, it occurs at all angles of incidence. At the principal angle of incidence ˚, the phase difference between ı Ek and E? becomes 90 . After two reflections at the principal angle of incidence ˚, the light is thus again linearly polarized. This is the basis for a convenient measurement method for ˚,usefulalsofor demonstration experiments (J. JAMIN, 1849). Part II 25.8 FRESNEL’s Formulas for Weakly-Absorbing Materials. Applications

The entire empirical content of the left-hand and the center columns in Fig. 25.6 (graphs a and b, d and e, and g and h) was summarized by A. FRESNEL (1788–1827) in simple formulas. If we write the law of refraction as sin ˛= sin ˇ D n, then for the reﬂected radiation, we have:C25.8 C25.8. POHL gave a derivation of FRESNEL’s formu- E ? sin.˛ ˇ/ r D (25.11) las in the 13th edition of Ei? sin.˛ C ˇ/ “Optik und Atomphysik”, Chap. 11. For a derivation and in English, see for example http://physics.gmu.edu/ E k n cos ˛ cos ˇ tan.˛ ˇ/ r D D : (25.12) ~ellswort/p263/feqn.pdf Eik n cos ˛ C cos ˇ tan.˛ C ˇ/

For the radiation that penetrates into the material, we ﬁnd

E ? 2sinˇ cos ˛ p D (25.13) Ei? sin.˛ C ˇ/ and

E k 2sinˇ cos ˛ p D : (25.14) Eik sin.˛ C ˇ/cos.˛ ˇ/

In the special case of perpendicular incidence, it follows from Eq. (25.11)for˛ ! 0that:

E n 1 r D : (25.15) Ei n C 1

By squaring this equation, we obtain for one boundary surface the Â Ã Reflected radiant power n 1 2 Reflectivity R D D (25.16) Incident radiant power n C 1 1. 1. D ) P > 2 < ˛ n C25.9 n (25.17) for direction co- , 1934). ough a glass ). In the first z C20.3 are directed op- 2 ˇ ˇ ˇ ˇ ˇ i AUER ? ? E 20.12 = E E llimated light beam (1802): In a demon- P P W W /: or ). ; however, making use and C ı r k k . For the perpendicular E E 4 %; for germanium, with E OUNG P P W W Y ) hold for the light that pen- 25.10 ˇ ˇ ˇ ˇ ˇ D 25.9 ), he bounded the air layer by lline layers (e.g. of KBr or CaF D R Q 25.14 20.9 5, : HOMAS ; this is an important and often-used 1, and they are parallel for 1 , 208 (1961). > is the radiant power 17 D ) means that 25.6 P )and( n W n for an oblique passage through the bound- . ? ection-free glass surfaces 25.15 E 25.13 = k ’s rings (Sect. 36 %. The penetration of radiation can thus by . 3 E D ), it seemed for a long time that it would not be pos- Degree of polarization R EWTON 25.16 Physikalische Blätter 1, in contrast, the phase remains unchanged. ’s formulas ( Reflection and Refraction of Light < n , but rather it continues to increase with increasing angle of 0 4, we find 19 a piece of camberedplanar glass polished glass with with a aof small large the refractive refractive air index. index, gap Then with and hetwo a a pieces filled liquid of a whose piece glass. section refractive of In indexexchanged this lay region, positions between the those bright of and the dark interference fringes practically successful method, thin crysta were evaporated onto quartz glass in high vacuum (G. B When a collimated light beamplate, passes one at obtains an oblique partially-polarizedand angle linearly-polarized light, thr light. i.e. Quantitatively, a thisits mixture light is of characterized natural by light The demonstration experiment of T From Eq. ( sible to manufacture refl of interference in thin“anti-reflection” evaporated or crystal “dressing” layers, can a be considerable achieved (Sect. degree of If we produce the partially-polarized light from a co which passes at an oblique angle through a glass plate, then the degree of stration of N ı D RESNEL R. W. Pohl, F When positely to one another for ary does not reach its maximum value at the polarization angle 56 The reflection produces a phase jump of 180 incidence, as will be shown in the following. incides with the direction of incidence of the light. 3 With our knowledge of thisular phase reflection jump, at we the illustrategraphically the planar for perpendic- surface of two a examples weakly-absorbing in material Fig. The minus sign in Eq. ( which we introduced in Sect. relation. Examples: For glass, with n etrates beyond the boundary surface anddient into to the illustrate material. them It graphically is (Fig. expe- The amplitude ratio reflection, we use a single coordinate system whose no means be prevented by strong absorption alone. , , 434 AUER 39 B ERHARD The Relation Between Absorption, 25 Dr. rer. nat. Göttingen 1931; Annalen der Physik C25.9. G (1934). 502 Part II 25.8 FRESNEL’s Formulas for Weakly-Absorbing Materials. Applications 503

S is the resultant of the incident and the reflected waves 0 Continuous transition = = δr 180º δp 0º Ei S Ep

Er z

0 = = Air A; nA 1 Solid object B; nB 2 = = In the direction air → object, n nB/nA 2 0 Continuous transition

= = Part II S δr 0 δp 0º Ep Ei Er z 0 = = Solid object B; nB 2Air A; nA 1 = = In the direction object → air, n nA/nB 0.5

Figure 25.9 Two examples of a perpendicular passage of travelling waves through the boundary O O between two materials of different refractive indices. ‘Snapshot’ images of this type continually change their form over time, but they repeat themselves periodically. In each snapshot, at each moment in time, the sum of the incident and reﬂected light vectors at the boundary is thus equal to the light vector of the light that penetrates the boundary.

Figure 25.10 The 1.0 penetration of light with the polarizations E? 0.8 and Ek into an optically-denser material which

Amplitude E absorbs weakly, 0.6 ║ corresponding to Air → Crown glass = Eqns. (25.13) and n 1.5 E┴

(25.14) Incident 0.4 Penetrating = p i E E 0.2

0 020°40°60°80° Angle of incidence α

polarization will be

1 cos4.˛ ˇ/ Q D (25.18) 1 C cos4.˛ ˇ/ 1 .˛ is the angle of incidence, and sin ˇ D sin ˛/: n ), : 2 a 5 : 1 16.7 tunnel D D k ? p (25.19) (25.21) (25.20) litudes, surpasses p n E E ˛ α of the layer is sandwiched d A aces ), we ﬁnd for the n is thus determined , it follows that the n ˛ : 25.14 1 1 1.5 =

C n 4 ). 4 )and( a a . boundary surf Angle of incidence shows an example for ˛ ; by a glass plate, D 4 25.18 Partial polarization ˇ ˇ ˇ ˇ ˇ two a 25.13 ? ? with planar surfaces and a larger E D E 25.11 P P B W ace: W k ? 020°40°60°80° E E 0

C P P

W 60 40 20

W k k Q Degree of polarization polarization of Degree with a refractive index E E yields Eq. ( P P . Figure W W A and through ˇ ˇ ˇ ˇ ˇ ˛ ˇ/ ): From Eqns. ( are proportional to the squares of the amp D a also increases with for total reflection, as defined by Eq. ( P ? Q boundary surf ) D W .˛ . They undergo total reflection when . Waves are incident from the lower left at the T E . 25.18 ˛ B = ˛ B with increasing angle of incidence k cos ˇ/ n one n E Q = 25.17 = is at least of the order of the wavelength (Vol. 1, 1 A 1 n .˛ , a thin layer Influence D A or “frustrated total internal reflection”. a of Total Reflection , D cos Reflection and Refraction of Light T D 25.12 ˛ k C25.10 ? p p E E amplitude ratio which is practically important. Fromof the polarization continuous increase of the degree that is The radiant powers passage through Derivation of Eq. ( by the angle of incidence The degree of polarization for a given refractive index Inserting and, from Eq. ( i.e. sin between two sheets of material of the angle of incidence on the degree of polarization of the light transmitted by a glass plate of material Sect. 12.9). Thinnerthe layers waves; are they not can anthough pass they insurmountable through were obstacle the passing for through layer,effect a although tunnel: attenuated, This as is called the Total reflection can occur only when the thickness In Fig. 25.9 Detailed Description the critical angle angle of incidence refractive index, Figure 25.11 The Relation Between Absorption, 25 C25.10. Total reflection and the tunnel effect are discussed in detail in Vol. 1 using water waves as anample ex- system (Sect. 12.9). In earlier editions, the series of images in Fig. 12.23, Vol. 1, could be found here inchapter this on optics. 504 Part II 25.9 Detailed Description of Total Reflection 505

Figure 25.12 Avoidance of total reflection. B The tunnel effect (Video 12.2 from Vol. 1) Video 12.2 from Vol. 1: A d “Water-wave experiments” http://tiny.cc/tfgvjy. B The experiments on total reflection and the tunnel effect α>αT are shown after the time mark at 5:30 min. For light, the demonstration is carried out using waves from the infrared spectral region. In Fig. 25.13, the crater of an arc lamp is imaged onto a radiometer M by two lenses made of rock salt. The colli- Part II mated beam between the lenses is split into two beams by a mask B1. An aperture B2 which can be shifted in the vertical direction allows one or the other of the two sub-beams to pass. The two sub-beams are then incident on three 90ı prisms made of rock salt. The bases of the small prisms are separated from the base of the large prism by small strips of metal foil, above with a thickness of 15 m, below with 5 m. The visible part of the two sub-beams undergoes total reflection; it exits to the sides in the direction of the arrows. Likewise, the infrared radiation of the upper sub-beam undergoes total reflection. For the lower beam, in contrast, the radiometer indicates a large deflection. Radiation thus passes through the lower pair of prisms. This means that a 5 m thick layer of air behind the base of the large prism avoids total reflection. But a 15 m thick air layer allows the total reflection to occur without hindrance. As a result, the infrared radiation in the two beams contains waves of up to about 15 m wavelength. (Waves of wavelengths longer than 15 m were already absorbed by the first rock-salt lens. Details are given in Sect. 27.2).

This experiment using two prisms is also technically important. We can make the spacing of their basal areas variable; we then would have the possibility of changing the transmitted radiant power through tiny shifts in the spacing, i.e. of continuously varying the intensity. Furthermore, the two prisms can be used in the infrared spectral range as a ﬁlter. They stop shorter wavelengths and allow the longer ones to pass through.

15 μm

M B1

5 μm V

Figure 25.13 Demonstration of total reflection of infrared light and its avoidance by the “tunnel effect” D its ˇ> ˛ (25.25) (25.26) (25.24) (25.22) is the (25.23) 2 n RESNEL , the phase n D ! ;thewaveis t ; : 2 n 1 : 2 ˛ yields values of sin p n Ã 2 ˛ n ; D and at time sin z = . Thus, linearly-polarized / i c c z sin ˛ ? p is its amplitude, = ˛ 2 1 n n 1 E zn A 2 ): t T i t sin D '; component of the light vector (the Â sin !. for two angles of incidence, at ˇ D and i p x ! ı sin ˛ k ˇ . In optics, the phase velocity is the 1). This is inserted into the F i 2 Ae c 45 E ˛>˛ sin C cos sin p D D 1, A its phase velocity in a material, and ' . x . In optics, an undamped travelling wave ı direction, and D ı < D 1 n D E D 5 h, in the region of total reflection, there is z C25.11 : 5, = . i n ı 2 : x q cos c 1 54 E n between = ), we can write D = D 1 c tan D ı 25.6 25.11 ˇ ' i . Within a material of refractive index ˛ D e c cos 1. Then we find n 25.24 quantity ( < Representation of Damped Travelling Waves n Reflection and Refraction of Light and also at ’s relation: ı is the momentary value of the imaginary 5 : x E refractive index of that material). circular frequency, 1 only for travelling in the positive vector of the electric field) at the location Derivation: The law of refraction sin ( Example: For formulas, and then thescheme computation as is in carried Sect. out according to the same an 48 ULER E Instead of Eq. ( light which has a component withinperpendicular the to plane it of will incidence as bereflection. well converted We as to find elliptically-polarized (for light by velocity is reduced to We can carry outtions computations than more with easily trigonometric withtrigonometric functions. exponential functions func- by Therefore, an we exponential replace function, the making use of 25.10 The Mathematical Travelling waves were treatedvelocity in was Sect. denoted 12.1 there by of Vol. 1. Their phase AccordingtoFig. velocity of light can be represented by the equation a phase difference ). ) can be 25.22 https://archive.org/ The Relation Between Absorption, 25 details/PrinciplesOfOptics found in Max Born andWolf, Emil “Principles of Optics” (4th ed.), Pergamon Press (1970), Sect. 13 (available online at C25.11. A detailed derivation of Eq. ( 506 Part II 25.10 The Mathematical Representation of Damped Travelling Waves 507

+ 3i Complex number ζ = a + ib = Aeiφ + = A (cos φ + i sinφ)

2i part Imaginary φ

+ sin 1i A A

= φ b

– 1 + 1 + 2 + 3 Real part Part II a = A cosφ – 1i

Figure 25.14 Graphical display of a complex number and then we can use complex numbers for calculations and employ separately either the imaginary or the real part.

Complex numbers are pairs of numbers with particular rules of calculation, developed especially for such pairs. The words “imaginary” and “complex” are of only historical importance. For the following sections, we need to note only a few items: A complex number

D Aei' D A.cos ' C i sin '/ D a C ib (25.27)

(A is the “magnitude” and ' the phase angle of the complex number) can be represented graphically (Fig. 25.14): To calculate the angle ', we make use of the equation sin ' Imaginary part tan ' D D of the complex number : (25.28) cos ' Real part

The “magnitude” A of a complex number .a ˙ ib/ is found by multiplying it by its “complex conjugate” .a ib/, so that for example

A2 D .a C ib/.a ib/ D a2 C b2 : (25.29)

In these two equations, we ﬁnd pure real numbers as the results. In other cases, one ﬁnds complex numbers on both sides of the equals sign, e.g.

a C ib D C C iB : (25.30)

This then means that a D C and also b D B is a physical result, i.e. a relation between similar and comparable quantities. Example: Consider a sinusoidal oscillation which begins at time t D 0 with a phase ı (positive or negative). Then instead of A sin.!t C ı/,we could write (in the complex representation):

D Aeiı ei!t : (25.31)

The product Aeiı D A0 is called the complex amplitude. It contains two parameters of the oscillation, namely both the real amplitude and also the ) in and n of its 25.3 .Our n ( ’s for- (25.33) (25.34) Kz ). (25.32) 25.6 e to the frac- the phase ı 25.33 RESNEL C25.12 complexindexof : / c )toEq.( = ı , we must also take the i zn e n t % 25.26 is replaced by the extinction !. : i D e / is the magnitude and k K 2 ik ı % 1 = 4 .ı of the real amplitudes as well as the i ) (a wave without extinction) to n kz e 2 . D 2 ı A i D 1 2 = e 0 1 K A A e % t n there by the complex index of refraction A 25.26 . Making use of the complex index of k D n A D 0 1 0 2 A A % D x ’s Formula for Perpendicular E between them. Here, , the power has decreased to the fraction ı z and at time numerical quantities, both the refractive index EER z . The ratio of two complex amplitudes ı Reflection by Strongly-Absorbing Materials using the relation . It performs excellent service as a formal computational two ) (a wave with extinction) can be carried out formally ) by a complex quantity, namely the . If the extinction constant : ik 2 Reflection and Refraction of Light = Kz n 25.33 25.26 e angle of the complex number contains both the ratio phase difference phase angle D : wavelength in vacuum), then we obtain for the momentary value 0 mulas. In addition to the refractive index The transition from Eq. ( Eq. ( in a different way: We need only replace the refractive index Eq. ( refraction original value, and the amplitude has thus decreased It contains tion at the location ( quantity and is indispensablewaves. in any treatment of the extinction of 25.11 B We have already presented the experimental facts in Sect. In a material withAfter extinction, a the distance waves are damped exponentially. quantitative treatment is based on an extension of F the extinction coefficient refraction, we can go directly from Eq. ( coefficient k n This result is important.of extinction on It the canstart propagation with be of the used formulas a derived to waveplace for a from the compute wave refractive a the without index simple extinction influence and rule: re- We The Relation Between Absorption, 25 C25.12. The power is proportional to the square of the amplitude (see Com- ment C12.4). 508 Part II 25.11 BEER’s Formula for Perpendicular Reflection by Strongly-Absorbing Materials 509 extinction coefficient k into account. This is accomplished by applying the general rule introduced above: We replace the real refractive index n by the complex refractive index n0 D n ik. In the special case of perpendicular incidence, we found for the reflection E n 1 r D : (25.15) Ei n C 1 Inserting the complex refractive index, we obtain the ratio of two complex amplitudes: Part II 0 E n ik 1 ı r D D % ei r : 0 C (25.35) Ei n ik 1 Here, the magnitude % (see Eq. (25.32) in Sect. 25.10) is the ratio of the real amplitudes, that is % D Er=Ei;andır is the phase angle between Er and Ei, i.e. between the reflected and the incident amplitudes. Both are to be computed according to the rules in Sect. 25.10. We begin with the calculation of the ˇ ˇ ˇ ˇ2 2 ˇEr ˇ Reflectivity R D % D ˇ ˇ : Ei For this calculation, we multiply the complex number in Eq. (25.35) by its complex conjugate, thus .n ik 1/.n C ik 1/ R D (25.36) .n ik C 1/.n C ik C 1/ or ˇ ˇ ˇE ˇ2 .n 1/2 C k2 R D ˇ r ˇ D : (25.37) ˇ ˇ 2 2 Ei .n C 1/ C k

This is the often-used formula of AUGUST BEER (1854). For every value of the reflectivity R, there are many pairs of values of the optical constants n and k that satisfy Eq. (25.37). The set of all these pairs forms circles, as is shown in Fig. 25.15 for values of R between 20 and 80 %. In metals, the summand k2 in the numerator and denominator of BEER’s formula (25.37) is generally predominant. Then R is comparable to 1. A large fraction of the incident radiant power is reflected. 2 2 In the example in Fig. 25.6 c, this fraction was .Er=Ei/ 0:78 60 %. Silver can reflect more than 95 % in the visible range. In the longer-wavelength infrared range, all metals have a reflectivity of R 100 % (cf. Fig. 27.8).

To calculate the phase difference, we put Eq. (25.35) in the form a C ib. To do this, we multiply the numerator and the denominator by the complex conjugate of the denominator, that is

2 2 ı n ik 1 n C ik C 1 1 n k C i2k % ei r D D (25.38) n ik C 1 n C ik C 1 n2 C 2n C 1 C k2 D ) n (top). 25.14 ,andtheir (25.40) (25.41) (25.42) / 25.9 R at perpendic- : R 60% 1 =

formula ( and the incident shows ‘snapshot’ R k /=. 1. 1 plays practically t 2 : : R i E 0 ); at the left, for 0 „ƒ‚… C Imaginary part D D ’s formula shows pairs of 1 25.16 . RESNEL C k k 2 : 2 n 25.42 … k D k EER 2 between the amplitudes. For k : t n C ı ’s formula for perpendicular 1 2 2 40% / 2and ƒ‚ k n 4 2 = k 1

C 2 Real part n R n D C „ 1 n n 2 1 D ) through ( . / D t of the complex quantity , (25.39) RESNEL R r ), ı 20% D ı D i = r

e Refractive index 2 ı tan ˇ ˇ ˇ ˇ R % 1 25.37 t i 25.28 E E =. tan / 2. Analogously, Fig. ˇ ˇ ˇ ˇ 2 80% R k = 4

D R C Real part D n 2 / 2 Imaginary part r 1 which give the same value of the reﬂectivity C D r , and likewise the phase angle (right) is hardly distinguishable from Fig. n k i ı A graphical representation of B , we illustrated F .. E

5 4 3 2 1 0 12345678 4, and at the right, for tan k Extinction coefficient Extinction and Reflection and Refraction of Light n 25.9 D 25.16 are given by k r and calculate the ratio of the transmitted amplitude In a similar manner, we could start from the F and Then we make use of Eq. ( perpendicular incidence, we obtain and obtain for the phaseplitudes: angle between the reflected and the incident am- amplitude or values of ular incidence. The center of the circles is at incidence and weak reflectionmerical with example a of ‘snapshot’ image, for the nu- In Fig. Figure 25.15 radii images to elucidate Eqns. ( 2and Figure This means that an absorption coefficient of The Relation Between Absorption, 25 510 Part II 25.12 Light Absorption by Strongly-Absorbing Materials at Oblique Incidence 511

S is the resultant of the incident and the reflected waves 0 0 Continuous transition = = = δr 157º δt +53º δr 176º = δt +2º Ei E Ei SSr Et Ed E z r z R = 68% = Air A nA 1 Object B Air A n =1 = =4 A Object B nB 2; k = =0.1 0 0 nB 2; k = = = n nA→B nB/nA 2

Figure 25.16 ‘Snapshot’ images of waves, continuing those in Fig. 25.9, to illustrate the appli- Part II cation of Eqns. (25.37) through (25.42). At the boundary between air and the object B, at each moment the magnitude of the light vector of the transmitted light is equal to the sum of the light vectors of the incident and the reflected light. The left-hand image represents for example the reflection of red light from a platinum surface. The right-hand image exaggerates somewhat the situation for dye solutions of very high concentrations. no role in reflection. k D 0:1 (or, more precisely, k D 0:08) means that w D , i.e. the mean penetration depth of the light is equal to its wavelength (in vacuum). w D was introduced in Sect. 25.5 as the boundary between strong and weak absorption. That definition finds its justification here. If, for a strongly-absorbing material, we have measured two of the three quantities R, n and k D K=4, then we can use BEER’s formula (25.37) to calculate the third. We could however also measure R and ır, and then combine Eqns. (25.37)and(25.40) in order to obtain k and n.

25.12 Light Absorption by Strongly-Absorbing Materials at Oblique Incidence

In Sect. 25.11, we have discussed the reflection of light with strong extinction at perpendicular incidence (˛ D 0) rather thoroughly. The significance of the equations derived there extends far beyond the field of optics. These equations also play an important role in acoustics and electrical technology. They indeed contain only two formally-defined materials constants, the refractive index n and the absorption coefficient k, independently of any considerations of the exact nature of the waves. When the light is incident at an oblique angle (˛>0), the situation becomes more complicated. If we insert the complex index of refraction into the law of refraction, we obtain a complex angle of refraction. It contains two pieces of information: First, the positions of surfaces of equal phase, and second, the positions of surfaces of equal amplitude. Figure 25.17 serves to illustrate this. In the figure, ) 0 d d 0 0 ,for ˛ 25.18 wave wave ). Once Reflecte Reflecte ’s formulas is presumed 25.12 O Obliquely dampedObliquely dampedObliquely damped .Now,thewave ı O RESNEL Longitudinally )and( 0.7 0.7 0.35 0.35 0.35 2 1 1 1 ======0 25.11 n n k k n n k n n 0 0 measured with prisms is no is about 33 ˇ ˛ sin , ˛= 25.17 0, so that the light is incident perpendicular to D . The lines of constant phase (the wave crests) and ˛ O O , bottom, the refractive index of the material below the The various forms of spa- , top, Reflection and Refraction of Light 25.17 25.17 for weak absorption, that is with Eqns. ( the lines of constantcoincide: amplitude We (equal have thickness longitudinal damping. of the lines drawn) In the center image of Fig. and can increase by more than a factor of two with increasing example in the case of Cu. In spite ofstrongly-absorbing these materials can complications, be the treatedpendicular just case incidence. like We of the once again case oblique start of with incidence per- F on longer constant; it depends upon the angle of incidence (Fig. crests below the boundary nostant amplitude, longer i.e. coincide in with the figure the withwave lines lines is of of “inhomogeneous” constant and con- thickness. “obliquely The damped”. In Fig. tial damping of travelling waves (the thickness of the lines indicates thetude ampli- of the waves) the boundary to be smaller than that of the materialIn above Fig. the boundary. the wave crests aresupposed marked to by indicate broad the amplitudes.fractive dark index of lines. In the the material Their below first the thickness two boundary images, is the re- boundary is larger thandamping. it is above. Here, again, weExperimentally, this see oblique an damping oblique makespleasant itself manner: known in The an ratio un- sin Figure 25.17 The Relation Between Absorption, 25 512 Part II 25.13 Conclusion: Pictures Used in Physical Descriptions 513

Figure 25.18 In materials with 2.0 strong absorption, the ratio Pt k = 4.4 sin ˛= sin ˇ depends on the angle of incidence ˛ (this was measured 1.5 by D. SHEA usingverythinmetal β 3 1.0 k = prisms) /sin Cu α

sin 0.5 λ = 0.64 μm

020°40°60°80° Angle of incidence α Part II again, we replace the real refractive index n by a complex index of refraction, which also takes absorption into account:

n0 D n ik : (25.34)

Unfortunately, the ensuing computations are rather extensive and complicated, if carried out in rigorous form. For this reason, we limit the problem and ask only, “How can we determine the optical constants n and k from reflection measurements with oblique incidence of the light”? For the special case that ˛ is equal to the principal angle of incidence ˚ (Sect. 25.6,Fig.25.6c), we have, according to CAUCHY’s formulasC25.13, C25.13. POHL gave a deriva- k D n tan 2 (25.43) tion of CAUCHY’s formulas in the 13th edition of “Optik D ˚ ˚ ; n sin tan cos 2 (25.44) und Atomphysik”, Chap. 11. where is defined by See also PAUL DRUDE, Â Ã Annalen der Physik 271 E k tan D r : (25.45) (1888), p. 508–523. A mod- Er? ˛D˚ ern derivation in English is We thus have two equations for the determination of the two optical given in Born and Wolf’s constants n and k. The measured quantities are the principal angle of book, Sect. 13.2 (cf. Com- incidence ˚ and tan , that is the ratio of the two reflected amplitudes ment C25.11.). at the principal angle of incidence (Eq. (25.45) and Fig. 25.6c). The two equations (25.43)and(25.44) are rather important in measurement technology. They were published already in 1849 by A.L. CAUCHY. They should thus not be considered to be results of MAXWELL’s theory, contrary to popular belief.

25.13 Conclusion: Pictures Used in Physical Descriptions

The quantitative treatment of “strong” light absorption, when w<, is not a pleasing chapter. One has to carry out numerous calculations and nevertheless, for oblique incidence, only approximate solutions lead to formulas of acceptable simplicity. AYLEIGH mm thick. 4 ics students , we are again using planar 1. Light within this range is trans- Even beginning phys D n nhomogeneous transition layer of fi- m), and the increase in its refractive index liquid, ! 03 : . These filters consist of a layer at least 1 cm thick of fine, cm (0 6 the thickness of the layer that is influenced by handling to within a few percent has to be considered satisfac- , hout attenuation, while everything outside the range is deflected 10 k Reflection and Refraction of Light C25.14 and HRISTIANSEN C mitted wit off to the sides byrealized, because diffuse the reflection. grains of This glassrefractive index is powder near have however their no only surfaces. unified approximately value of their This fact is particularly noticeable in the filters fabricated by the method of carefully purified glass powderfide. in a At the mixture correct of mixingof benzene ratio, the and the liquid carbon dispersion can disul- curvepractically be of made the the to glass same intersect. and refractivethe that transition Then index from the over glass glass a and narrow the frequency liquid range; have for n a simplifying picture. Physically, a fresh liquidleast surface, for irregularities. example of But water, every exhibits the liquid has a vapor pressure, e.g. for These layers, in contrast toagainst the all kinds bulk of of external the influences.over material, time; Their are structures they unprotected depend are not on stable theence history of of impurity the molecules material nearto and the that on of surface. the external pres- The friction situation between is two similar solidNo objects surface in close layer contact. showsterial. the same properties For aspolished example, the surface we bulk into a of can(for liquid the place with the ma- exactly light a the being glassowing same used). to refractive block reflections index But of with we uplight. a to can some always carefully The tenths see of refractive the a indexdifferent surface percent of from of layer the that the of surface incident the layer bulk is glass. always somewhat According to Lord R is about 3 All physical descriptions make usethe of end simplifying are pictures useful which onlymay in as be approximations. encompassed The by same different empiricalbe pictures. facts kept The to simplifications must that minimumpurpose which of is the still pictures. compatibleexplanations: An with example the is intended more instructive than wordy In drawings, for example in aaries sketch of of a lens, a we indicate bodyreality, the we by bound- are a dealing surface. withnite an thickness. A i If a surface surface is is described a as simplified picture: In (1937) But another aspect isassociate still optical worse. measurements with aare high degree aware of that precision; they many refractive significant figures. indices, In wavelengths, thesion case etc. goes of strong out absorption, are the thisof preci- window. known to Being able to reproduce measurements tory. The reason is clear:take With strong place absorption, within all of very thethe thin processes main surface contributions layers are of the made absorbing by material; layers less than 10 can be up to 10 %. , pp. 507 – 160 (London), Proceedings of the The Relation Between Absorption, 25 “Even beginning physics 526 (1937). Series A, Vol. C25.14. Royal Society students associate optical measurements with a high degree of precision; they are aware that refractive indices, wavelengths, etc. are measured to many significant figures. In the case of strong absorption, this precision goes out the window”. 514 Part II 25.13 Conclusion: Pictures Used in Physical Descriptions 515 water at room temperature, it is 24 hPa. Therefore, at the boundary between the liquid and its vapor, there is a statistical equilibrium between evaporating and condensing molecules. Per square centimeter and second, around 1022 molecules make the transition from the liquid into the vapor and vice versa. In a square centimeter of the surface, however, there is room for only 1015 molecules. Each individual “This clamoring, swarming molecule can thus remain at the surface for only about 107 s; then throng is the best approxi- it again flies off the surface with a velocity of around 700 m/s. This mation that physicists can clamoring, swarming throng is the best approximation that physicists use to approach the ideal can use to approach the ideal picture of a surface as formulated by picture of a surface as formulated by mathematics!” mathematics! Part II All pictures and words are contingent on their time. They must be “All pictures and words are adjusted in the course of years to the continual extension of our ex- contingent on their time. perimental knowledge. They must be adjusted in the course of years to the continual extension of our experimental knowledge”. Electronic supplementary material The online version of this chapter (https:// doi.org/10.1007/978-3-319-50269-4_25) contains supplementary material, which is available to authorized users. a brations gto frdainfo t oret eevrusing receiver a to source its from radiation of agation cteig cteiglast hl eiso te important dispersion other and of refraction series with connection whole in (Chap. a of example to significance for leads the insights, exhaust Scattering means no scattering. by however examples These pnte.Satrn losu oietf oaie ih through light polarized identify (Fig. to asymmetry us based its allows is Scattering phenomena interference them. and upon diffraction important its “s some as and of visible beam become lines. not light to do straight light a themselves that emit of as objects allow concept drawn scattering the and rays’ important reflection to Diffuse ‘light an us using play lead representation Both They graphical considered. optics. in also role were scattering and tion Scattering nyaseilcs.I eea,teosal ilb nojc which object an be will obstacle (an however the vibrate is general, itself This In can immobile. case. and special rigid a be only to presumed is new, obstacle a The of source the 12.17). Fig. from becomes away 1, (Vol. directions then all it in obstacle propagates The which train wave “secondary” train. encountered wave is rod, obstacle a a An e.g. wavelength, by waves: the to water compared to small qualitative analogy is which Their the of use chapters. makes illustrated earlier interpretation been in already experiments have demonstration scattering by of aspects fundamental The Underlying Ideas Basic The 26.2 described have we chapters, preceding the In Remark Preliminary 26.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © opeesvl nti chapter. this in comprehensively sal h ercieindex refractive the usually fhroi siltr snsia siltos eetetdi depth in treated were oscillations) (sinusoidal oscillators harmonic of qualitative 27 sa as .Ti swyw att ra h oi fsatrn more scattering of topic the treat to want we why is This ). fScattering of Treatment Quantitative the ecito,tepeoearltdt ifs reflec- diffuse to related phenomena the description, resonator 24.4 oscillator olsItouto oPhysics to Introduction Pohl’s ). yteocmn ae.Fre vibrations Forced waves. oncoming the by n n h xicincoefficient extinction the and ,adi a eectdto excited be can it and ), cnayeitr” h treatment The emitters”. econdary quantitatively , https://doi.org/10.1007/978-3-319-50269-4_26 two quantities, ocdvi- forced k h prop- the Within . 26 517

Part II , : ' m (26.1) (26.3) (26.2) of sec- . S : 2 emission

2 0

2 : Ã '/; 2 , and also on the eigenfre-

, but with a phase shift of

Â m . High-frequency alternating 0 ): t

’s Experiment = S

2 C 0 0

F 2 . Here, we extend that discussion / 2 c components, etc.) are contained ). Its essential component is easily . 2

12.8 cos rating spring pendulum with a mass 26.1 0 D 2 l 0 URCELL m by quantitative data. which acts periodically on a harmonic os- ' . D / Exercise 26.1 . Quantitatively, we found for the amplitude t / s t . Here, we review brieﬂy the most important illation of an RLC circuit (tank circuit) was also de- tan

. 1 2 l 2 1 and on the frequency . 4 0 . F cos D 0 0 l 11.7 F Dipoles. P of the free oscillator, and on its damping, expressed as the D 0 F

The analogous forced osc The oscillator vibrates at the frequency These forced vibrations, for their part, cause the ondary waves. In themechanism case of this of emission light quantitatively. scattering,the We undertake we following this section. must task describe in the logarithmic decrement where aspects and supplement the Aforce quency scribed in Sect. 1 The analogous behavior ofready electrical been waves pointed and out light in waves Sect. has al- 26.3 The Radiation from Oscillating in Vol. 1 (Sect. 11.10) by a comparison which forms the basis forConveniently-operated a treatment of transmitters scattering. formagnetic waves of linearly-polarized short wavelength electro- make are use readily of available one today. of We them (Fig. recognized, namely the short antenna current ﬂows in it.the The technical devices accessories required (electroni toin generate the this shielded current box and K.lies in The planes electric which ﬁeld contain produced the long by axis the of transmitter the antenna (cf. Comment C11.8 and cillator (e.g. a sinusoidally-vib see Vol. 1, Fig. 4.13) producesFig. forced 11.42b vibrations, as for shown torsional intude oscillations. Vol. depends 1, on In a steady state, their ampli- Scattering 26 518 Part II 26.3 The Radiation from Oscillating Dipoles. PURCELL’s Experiment 519

Figure 26.1 Left: A transmitter G or source dipole for emitting undamped waves ( 10 cm). Right: A non-tuned receiver K dipole with a rectiﬁer and a gal- E vanometer as detector S

Radiant Iϑ ϑ intensity Part II

Figure 26.2 The graph shows how the radiant intensity I# of the waves emitted by the source depends on the angle # between the direction of propagation of the waves and a plane perpendicular to the long axis of the source. For the measurement, the receiver (antenna E) is perpendicular to the direction of propagation of the waves, while the angle is varied by tilting the source antenna S.At# D 0, the source and the receiver antennas are parallel.

As receiver, we use a short antenna E (as described already in Sect. 12.6). At its center, it contains a rectifier (diode), which produces a direct current that is measured by the ammeter (galvanometer G). With this setup, we measure the radiant intensity I# (Eq. (19.2)) of the linearly-polarized radiation as a function of the angle #. The result is plotted in Fig. 26.2. This corresponds to Fig. 12.24. Now, we show a corresponding experiment in optics. In Fig. 24.4,we produced linearly-polarized light by scattering. We repeat that experiment in quantitative form here. In Fig. 26.3, the shaded circle P is the cross-section of the primary light beam within a cloudy medium. Its plane of oscillation is marked by a double arrow E. Along the large dashed circle, we can slide a radiometer M around the beam P which forms the center of the circle. We measure the intensity of the scattered radiation (i.e. the deflection of the ammeter) as a function of the angle #. The result can be seen in Fig. 26.4, as the solid curve. The similarity between Figs. 26.4 and 26.2 is evident. In both cases, for the radiant intensity I# in the direction #, we find to a good

Figure 26.3 The measurement of the scattered radiation as a function of the angle. At P, M the primary beam of linearly- E G polarized light is incident ϑ perpendicular to the plane of P the page. u in M (26.4) OUNG Y OPPLER will be ob- y light is pro- . # e HOMAS forced electrical , small print at the , which radiate as 26.9 cos visible ϑ ϑ c u . This corresponds to 22.4

d I = : 1 ). d sources Á U #: D # , as a result of the D = 2 26.9

# T intensity intensity Radiant D cos cos 0

x E c u This demonstration is the electrical P , an electron passes with the velocity const d visible D C26.1 26.5 D # 0 I . In a direction d = u ), the frequency D described an experiment in which

as central axis (see also Fig. indicates its plane of oscillation. The length of the radius cor- 23.5 E The scattering of polarized light by spherical dielectric parti- The production of . E ). The figure can be thought of as rotationally symmetric around the URCELL ,and P 26.3 a frequency of closely above a corrugated metalwith sheet. the Its positive influence negative charge, chargea together in dipole. the The spacing metal of (mirrorare the charge), two varied form charges periodically and with thus the the period dipole moment This analogous behavior leadsoptical to the experiment, following the conclusions:pended incident In particles the in polarized the light cloudy converts liquid into the sus- oscillations E.M. P dipole antennas. Thecause it light consists can itself of excitecan electromagnetic the produce waves. periodically Their suspended electric alternating particles fields suspended electric be- dipole particles; moments or, in put the briefly, excite them to This relation is shown by the dashed curve in Fig. double arrow served, corresponding to the wavelength duced as dipole radiation. analogue of the acoustical experiment with which T dipole radiation effect (Sect. Figure 26.5 in 1801 explained the action of a grating (Sect. Figure 26.4 approximation end). Its principle: In Fig. Fig. cles. The primary lightpoint beam is perpendicular to the plane of the page at the responds to the radiant intensity (or to the deflection of the radiometer Physical Re- , 1069 (1953). 92 Scattering 26 view C26.1. See S.J. Smith and E.M. Purcell, 520 Part II 26.4 Quantitative Treatment of Dipole Radiation 521

Example: An optical grating with d 1:7 m serves as a “corrugated sheet”. Close to its surface and at right angles to its grooves, there is a thin electron beam (diameter 0:15 mm, accelerating voltage U D 3 105 V, I D 5 104 A, u c). One can see its path as a colored streak whose hue changes with #.

26.4 Quantitative Treatment of Dipole Radiation Part II In an electric field, every object becomes an electric dipole: Ev- ery conductor through influence (Fig. 2.25 b), and every insulator through the “polarization of the dielectric”. This can occur in two different ways: First, through the action of influence on the individual molecules (Fig. 2.56); and second, through a parallel alignment of the dipoles of “polar” molecules that are already present without the field, but are randomly oriented due to thermal motions. These are molecules with permanent electric dipole moments, e.g. H2Oand HCl (Sect. 13.10). These polar molecules will initially be left out of our considerations. They will be treated later in Sect. 27.16. An oscillating dipole is the archetype of an electromagnetic wave source (HEINRICH HERTZ, 1887). In the simplest case, its electric dipole moment p varies sinusoidally, so that we have:

p D p0 sin !t : (26.5)

Let the amplitude of the dipole moment be p0 D Ql0.Thenatalarge distance r (i.e. r l , the length of the dipole), the radiant intensity 0 C26.2. A quantitative from the dipole in the direction # is given by derivation can be found 2 2 for example in F. Hund, c p0 2 I# D cos # (26.6) 4 “Theoretische Physik”, 2"0 Vol. 2, Sect. 61 (B.G. Teub-

(Units: watt/steradian (Eq. (19.2)); c is the velocity of light, and "0 is the ner, Stuttgart, 1957) or in electric field constant D 8:86 10 12 As/Vm). P. Lorrain, D.R. Corson, andF.Lorrain,“Funda- C26.2 How Eq. (26.6) comes about is easy to understand qualitatively: As- mentals of Electromagnetic suming that the dipole is undergoing forced oscillations at the circular Phenomena frequency ! D 2 , then the electric field that it emits is produced by ”, Chap. 25 (W.H. Freeman, New York, an induction process, so that its amplitude E0 is proportional to the time derivative of the electric current. Furthermore, the current flowing in the 2000). oscillating dipole is dp=dt. Because of this second-order differentiation, The integration over solid an- !2 the amplitude E0 of the emitted field is p0, and its power is thus gles leading from Eq. (26.6) !4 2 2=4 !2 p0 p0 . (The minus sign before p0 means that there is a phase to Eq. (26.7) can be found difference of 180ı between the field emitted and the dipole moment). in standard textbooks on integral calculus. Note that Integration of the radiant intensity in Eq. (26.6) over the solid angle the angle # as defined here d˝ (that is over # and ', total solid angle ˝ D 4;seeFig.19.1) (Fig. 26.9) is the complement yields the total average power emitted by the dipole at the frequency of the usual polar angle # in : spherical coordinates. ' is 3 2 3 4c p 1 4 the azimuthal angle around WP D 0 D !4 p2 D 4 p2 : (26.7) 4 3 0 3 0 3"0 12"0c 3"0c the z axis. ), )) 25.4 (26.9) 25.1 (26.8) and cross- x due to scattering , i.e. it has parallel , we illustrate these K : 4 2 0 24.4 p (Defining equation ( proportional to the number A 3 0 K collimated " scattering particles. They give c c 3 2 0 x 4 E x ’s method of calculating the wave- 0 2 A " the amplitude of the electric field of the V A Scattering 0 V N D E N p : P x W D For this reason, the average spacing of the AYLEIGH 1 P scattering is characterized by the following as- W p P is composed of the sum of the intensities given W P W C26.3 P W AYLEIGH D , there will be A K is the velocity of light) AYLEIGH ) from all of the scattering particles within the volume c of the scatterers. Thus, at a given wavelength, the quotient indicates the average radiant power (or the radiant inten- of R C26.4 . , yields an extinction constant V P A 26.7 W N is the electric field constant, , called the scattering or interaction cross-section (Sect. 25.2 0 V : " x light, and ( N = rise to an extinction constant of: particles is supposed torangement be is larger statistically than disordered. the In wavelength, Fig. and their ar- conditions by using fine particles of apended weakly-absorbing material in sus- water. Scatteringthe by primary light these beam. particles Its leadsSect. measurement, as to described extinction for of example in density K Here, sumptions: The scattering particles areare spherical and small their compared diameters toThey the are transparent wavelength in of the visiblebegins the only spectral in light range the and that ultraviolet, and their is thus absorptionmore, at their scattered. higher arrangement frequencies. in space Further- allows noestablished fixed among phase the relations rays to of be secondary radiationscattering from the particles. individual sectional area bounds; then in a given section of the beam of length length dependence of the extinction constant alone. The light beam is assumed to be is constant. We begin by following R by Eq. ( is the radiant power in thethe primary beam area of light that passes through 26.5 The Wavelength Dependence Now, we have allscattering. R the prerequisites for a quantitative treatment of sity) of the secondary radiation, and A Scattering 26 C26.4. The derivation is given in Comment C27.9. C26.3. The important role played by the phase relations between the sources of scattered radiation is emphasized by the footnote near the end of this section. These relations lead to a reduction of the overall scattered radiant intensity; that is, they increase the fraction of the light which continues along the path of the incidentbeam. light 522 Part II 26.5 The Wavelength Dependence of RAYLEIGH Scattering 523

In this expression, p0 D Ql0 is the amplitude of the dipole moment of a scatterer which is induced by the ﬁeld E0 of the primary light that excites the scattering. Combining Eqns. (26.8)and(26.9) with the deﬁnition (25.1) yields the extinction constant due to scattering: Â Ã 83 p 2 1 K D N 0 : (26.10) V "2 4 3 0 E0

The relation between p0 D Ql0 and the ﬁeld strength E0 can be calculated quite generally from Eq. (26.1). The amplitude of the Part II force is given by F0 D QE0. The scattering particles are small compared to the wavelength; therefore, considered as antennas, they have very high eigenfrequencies 0. Compared to these eigenfrequencies, we can neglect the frequency of the primary light in Eq. (26.1). Then the amplitude and thus also the polarizability ˛ D Ql0=E0 D p0=E0 are independent of , so that only constant quantities occur in Eq. (26.10)before1=4, and we obtain

1 K D const : (26.11) 4

The extinction constant resulting from this so-called RAYLEIGH scattering is thus proportional to 1=4 (like the power emitted by the dipoles). This important relation (26.11) is realized experimentally only as a limiting case. A good example is the scattering in a NaCl crystal CC C 3 with small additions of SrCl2 (Sr ions/Na ions D 1 W 10 ). These additional ions produce local lattice perturbations in the crystal. In reflected daylight, the crystals appear bluish, but they are reddish- yellow with transmitted light. Figure 26.6 shows measurements of the constant K due to scattering in the wavelength range between D 0:2 mand D 1 m. The coordinate axes are logarithmic. The measured data points lie on the solid straight line, which corresponds to K D const=3:8. The dashed line would correspond to K D const=4; thus, Eq. (26.11) is fulfilled to a good approximation, but not strictly verified. The approximation would in any case be sufficiently good to determine one of the two quantities NV or p0=E0, that is the particle number density and their polarizability, if the other quantity is known. Qualitative examples for preferential scattering at shorter wavelengths are easily found.C26.5 Water containing some drops of milk C26.5. See for example looks bluish. Thin skin over the dark background of veins near the Video 16.1 (“Polarized surface also looks bluish, for example on the inside of the wrist. light”) http://tiny.cc/5dggoy, The most famous example is provided by the earth’s atmosphere. It or, in Vol. 1, Video 10.4 scatters the shorter-wavelength light in the visible spectrum; thus, (“Smoke rings”) the clear sky appears blue. http://tiny.cc/ocgvjy. During the day, even when we are standing in shadow, we cannot see the stars; we are dazzled by the secondary light from scattering in the . ˛ " lity ) 3 ) can (26.12) (26.13) Liquids are (1 : 10 26.10 2 hundred °C to C26.7 0.4 0.6 0.8 μm NaCl crystalNaCl doped SrCl with Wavelength λ : 0.3 )). Then Eq. ( 4 1 . For its polarizability, we 0 /: 1 5 1

–1

2 from the dielectric constant

Ql 13.26 0.1 0.5 K Extinction constant constant Extinction ˛ 0.07 m); but the large local thermal density cm becomes independent of the fre- 9 D ." 3 2 0 13.9 ˛ 0 0 )), it was shown there that " V 0 10 p ,Eq.( 3 " 8 N V N D 13.9 13.27 ˛ bout 3 D ,thatis K 0 0 C26.6 p . Therefore, from the measured extinction con- 2 1(Eq.( (cf. Sect. ˛ " , the polarizability scattering 0 D The wavelength de-

) holds. However, we now denote the dipole moment of 0 E of the atmosphere, we can determine the number density = 0 0 K

p 26.10 AYLEIGH The average spacing of the molecules is in fact small compared to the wavelength With have Figure 26.6 pendence of the extinction constant for R quency and has the same value as in a static field. The polarizability of single molecules of a materialtheory; is well it known was from determined electromagnetic in Sect. of the molecules.Eq. ( This is carried out as follows: Quantitatively, stant (near the earth’s surface, it is a fluctuations in gases actfrom to individual eliminate molecules. phase relationsless This between compressible can than the be gases secondary and shownmuch rays vapors. smaller quantitatively. statistically-distributeddensity Their fluctuations than thermal in motions gases and thereforeAs vapors. a produce result, light scattering by liquidswe is first relatively weak. have To to demonstrate it removevacuum. clearly, all suspended For particles demonstration fromin the experiments, both, liquid benzene light by or scattering distillationfluctuations in can diethyl in be ether solids observed are are using even suitable; smaller red-filter than light. in The liquids. local In density a block of good-qua optical glass with polished faces, theA scattering cone similar can block still be of readilymake crystalline observed. the quartz scattering has visible. to be heated to several 2 upper atmosphere. The longergreater the is path the loss of by lightthe extinction sun’s through disk due near the to the air, scattering. horizon with the yellow-orange As a to bearable a red. brightness result, and we colored see In the clear, dust-free atmosphere,as scattering only centers the individual molecules act be rewritten as: a single molecule as For gases with IE Journal , Sup- , p. 383 33 )) was explained Berkeley Physics , Vol. 3 (McGraw Hill, 26.11 Scattering varies by more than a fac- 26 by the fact that the scattering particles have a diameter of ca. 150 nm (onset of M New York 1968), p. 559. Course C26.7. See for example Max Born and Emil Wolf, “Principles of Optics” (4th ed., Pergamon Press 1970), Sect. 81; available online: See Comment C25.11. A brief introduction can also be found in F.S. Crawford, “Waves”, scattering). (K.G. Bansigir and E.E. Schneider, of Applied Physics plement to Vol. C26.6. In the wavelength range of these measurements, K tor of 200! Light scattering with a similar wavelength dependence was also observed in NaCl:Mn and KCl:Ca. The deviation from the expected wavelength dependence (Eq. ( (1962)). 524 Part II 26.5 The Wavelength Dependence of RAYLEIGH Scattering 525

Combining Eqns. (26.12)and(26.13), we obtain

83 ." 1/2 N D : (26.14) V 3K 4 Observations (e.g. from the Pic de Teneriffe) have given a roughly constant value of the product K 4 D 1:13 1030 m3 at 0 °C and 1013 hPa between D 320 and 480 nm. Thus, for example, at D 375 nm, the extinction constant is K D 5:7 105 m1.This is an extraordinarily small value. It means that a reduction in radiant = D

intensity to 1 e 37 % occurs only after a path length of 18 km! The Part II dielectric constant of the air is " D 1:00063. With these numerical values, Eq. (26.14)gives

25 3 NV D 2:9 10 m :

The number density NV;id of an ideal gas under these conditions of pressure and temperature is known (Vol. 1, Sect. 14.6); it is

p 25 3 N ; D D 2:7 10 m V id kT

(p D 1:013 105 hPa, T D 273 K, k (BOLTZMANN’s constant) D 1:38 1023 Ws/K).

The good agreement of the two number densities veriﬁes RAYLEIGH’s theory, and with it, the perhaps initially unclear footnote on the previous page.

Finally, we investigate the connection between RAYLEIGH scattering and compressibility. The isothermal compressibility (Vol. 1, Sect. 14.9) is:

D dV 1 : dp V

For an ideal gas (pV D NkT), the magnitude of is

D 1 : p

With this, we obtain from Eq. (26.14) for the extinction constant K

83 ." 1/2 K D kT 3 4

In ideal gases, the product T D T=p is independent of T, and therefore, so is K.Inreal gases, the product T in the neighborhood of the critical point becomes very large (Vol. 1, Sect. 15.1), and thus so does K;that is, the light scattering is strong (“critical scattering”). This illustrates the importance of local density ﬂuctuations near the critical point (compare the footnote on the previous page). , n n % = M ;the M m . /g ,there 2 (26.15) n D –10 n M 0.3 0.2 0.1 cm 26.7 M EWLETT is the extinction K ). 27 g/mol = n 26.8 M : 2 kg m he possibility of producing and as abscissa. Here, 02 of nucleons in their atomic nuclei Wavelength λ : 0 A D 12 g/mol = % K n CAl M the amount of substance) of the irradiated mate- n 0.2 0.4 0.6 0.8 1.0 , is independent of the molar mass of the scatterer and of 0 0 0.2 0.4 0.6 ∙ 10 The inﬂuence of the wavelength on the scattering of X-rays by Electrons in Light Atoms by Scattering ). Second, it has provided t

=% /kg

2 0.03 0.02 0.01

K ϱ

m Density

of electrons in atoms with moderate molar masses is close 26.7 K K constant Extinction Z , is plotted against the wavelength is the mass and =% M For this scattering by materials with a small molar mass rial. But here, also, aby special great case simplicity of scattering has which been is found. characterized It is illustrated in Fig. portion of the total extinction constant due to absorption was subtracted. its chemical bonding, where it takes on the practically constant value is a wavelength range indensity, which the extinction constant relative to the ( (Sect. The scattering in thistwo characteristic important physical wavelength perceptions: range First,ber has we led can us see that to the num- to half as large as the number investigating linearly-polarized X-rays (Sect. is the molar mass of the scatterer). Figure 26.7 26.7 The Number of Scattering Scattering of short-wavelength X-rays is independentof of the the bonding atoms in molecules or crystals. This is because for X-rays, 26.6 The Extinction of X-rays The extinction of X-rays byplex scattering manner depends on in their general wavelength and in on a the com- molar mass K light atoms. On the ordinate, the extinction constant relative to the density constant from scattering alone. (After measurements by C.W. H Scattering 26 526 Part II 26.7 The Number of Scattering Electrons in Light Atoms 527 only the electrons in the inner shells of the atoms act as scattering centers. If an atom has Z electrons, then the number density of its electrons is

NA% NV D Z (26.16) Mn

23 1 (NA is the AVOGADRO constant D 6:022 10 mol ,andMn is the molar mass D M=n).

The atomic electrons are somehow able to oscillate around the positive charge which binds them to the nucleus of the atom. The oscil- Part II lating electric ﬁeld of the incident radiation excites the electrons to forced oscillations around their average rest positions. The positive charge remains at rest, due to the large mass of the atomic nucleus. The diameter of the electrons is small compared to the wavelength of the radiation, and their distribution within the atom is on average statistically disordered. Thus the conditions agree with those for RAYLEIGH scattering. For the extinction constant due to scattering, we can again use the equation

83 1 K D N ˛2 : (26.12) V "2 4 3 0 Now, however, there is an essential difference: The eigenfrequencies 0 of the bound electrons in light atoms are small compared to the frequency of X-rays. As a result, their polarizability ˛ is no longer constant, but rather it increases proportionally to 2. Therefore, K becomes independent of in Eq. (26.12). Derivation:

We again put F0 D eE0 into Eq. (26.1)(e is the electronic charge), but this time, we neglect 0 as small compared to . We thus obtain for the amplitude of the oscillations of the electrons

1 e l D E ; 0 42 m 2 0 or, after multiplication by the charge e,

el 1 e2 e2 2 0 D ˛ D D : (26.17) 2 2 2 2 E0 4 m m 4 c

On inserting this expression for ˛ into Eq. (26.12), the wavelength drops out. The remaining expression is

e4 K D N (26.18) V "2 2 4 6 0m c

(K is the extinction constant due to scattering, NV the number density of the electrons, e the electronic charge D1:6 1019 As, m is the elec- 31 tron’s mass D 9:1 10 kg, "0 is the electric field constant D 8:86 1012 As/Vm,andc the velocity of light D 3 108 m/s). . ) ): 2. V = N r 26.15 atomic 26.16 (26.21) (26.19) (26.20) ( ) was found 26.6 ; n A : M N : ). The equation contains 2 2 : m Z kg m , a pure number proportional 2 : electron 29 r 2 26.18 )gives kg m r A m Z A 10 5 one : 02 29 : 0 0 26.15 6 : 04 : D 6 10 D 0 (1906). Z D 6 % K D : )and( In the X-ray range, scattering is the only 6 in the inner shells of an atom with a moder- V scattering not only for the detection of linear % K K N ), but also for producing it. D 26.20 % K HOMSON 24.2 relative atomic mass kg/kmol, due to scattering of X-rays is independent of their r does not occur in Eq. ( A AYLEIGH K This fundamental piece of knowledge of atomic struc- D of Producing and Detecting Polarized X-rays n M C26.8 electrons per atom, we find from this, also using Eq. ( ). Z is the quantity r A ate molar mass, the effective number Z of electrons is equal to A In words, this means that With Evaluating the constants yields for The polarization of radiationdamentally by important means only ofmethods in scattering which the can becomes X-ray be fun- diation, region. used such as for polarization ultraviolet, There, prisms visiblecan and the foils, and no or other infrared reflection longer polarizers, ra- be used. polarization (Sect. or, with to be: ( Experimentally, however, the measured value (Sect. The comparison of ( ture is due to J.J. T to the atomic mass of each element; formerly, it was called the weight 26.8 Scattering as a Means In the visible spectral range,make and use in of R the neighboring regions, we can Once more, in words:tion In constant the spectral range considered, the extinc- wavelength; only constants, apart from the number density of the electrons, ), Z corre- r A , the total A Scattering 26 so this means that in lighter elements, the number of neutrons equals the number of protons in the nucleus. This is well known from nuclear physics. sponds roughly to the nuclear mass number number of nucleons (protons and neutrons) in the nucleus of an atom of ament given ele- (i.e. a given isotope). The statement made here thus means that the number of scattering electrons is half as large at theof number nucleons in the nucleusthe of corresponding atom. In a neutral atom, the numberelectrons of equals the number of protons in the nucleus ( C26.8. The number 528 Part II 26.8 Scattering as a Means of Producing and Detecting Polarized X-rays 529

a β Analyzer Tertiary Tertiary α

Secondary

Primary radiation, unpolarized

Polarizer Part II bc A o A J o L V P P

Figure 26.8 The production and detection of linearly-polarized light by means of scattering. a: Schematic, no tertiary radiation in the direction ˇ. b: A demonstration experiment with visible light. c: A demonstration with X-rays (analyzer A fixed, polarizer P and source can be pivoted together on an arm around the vertical axis. The X-ray source requires AC at 220 V. J is an ionization chamber (Fig. 15.7), V is a static voltmeter with power supply and a light-beam pointer; L is a lens). A and P are cloudy water for visible light (compare Fig. 24.4), and for X-rays, they are made of materials with a small molar mass, for example paraffine. The flat shape is intended to reduce absorption losses. The openings o, which are not visible in the silhouette, are indicated by outlines; likewise, an insulating column is shown by shading. Of course, the eye of the observer at left could be replaced by a suitable radiometer. method of polarization. However, this holds only in the characteristic wavelength range, as we have seen in Sect. 26.6. The scatterers which are used for producing and detecting the polarization of X-rays must contain only atoms of light or moderate molar mass. Figure 26.8 shows the procedure, both for the visible region and for X-rays. The polarization of X-rays was detected for the first time in 1905, thus 10 years after RÖNTGEN’s discovery of X-radiation. It was the first new characteristic of X-rays, one which was not discovered by RÖNTGEN himself and not contained in his original publications. , : 3 O 2 large S 2 Bottom AYLEIGH suspended par- small photographic positive; , top; the light used is 26.9 contains suspended particles of S matte surfacematte Screen with a added. This causes sulfur to pre- S 4 SO 2 Primary light beam light . The glass tube contains a solution of Na : The experimental setup for demonstrating scattering (about 26.9 Instead, we see mainly “forward scattering”, i.e. scat- The lack of symmetry of the scattered radiation from Top by Large, Weakly-Absorbing Particles ). ). C26.9 26.4 26.10 cipitate in theof form the of particles small, increasesthis solid in process, the suspended forward scattering course particles. becomes(Fig. of more several and The minutes. more size prominent During with a small amount of H beam, predominates (the setup is shown in Fig. screen without touching it.sulfur in The water. glass The tube screen isThe illuminated only rough by the symmetry scatteredticles. of light. The the primary light scattered beam (red-filterof radiation light) oscillation is from is linearly polarized. parallel Its tocf. plane Fig. the surface of the screen ( sulfur particles: “forward scattering”, along the direction of the primary light 1/6 actual size). The primary light beam passes above the matte-surface unpolarized, from ansize). incandescent lamp. This image is about 1/10 actual tering along the direction of thestration, incident small beam sulfur of light. particles For inshowninFig. a water demon- are suitable. We use the setup Figure 26.10 Figure 26.9 length of the light.scattering no longer Then apply. thetered For simple radiation example, around characteristics the the of direction symmetrypresent. of R of the the incident scat- light is no longer 26.9 The Scattering of Visible Light Often, the scattering objects are not small compared to the wave- - AYLEIGH were extended 26.4 Scattering 26 the figure caption. scattered radiation, imagine that Fig. to three dimensions, as suggested in the last sentence in C26.9. In order to seesymmetry this in R 530 Part II 26.9 The Scattering of Visible Light by Large, Weakly-Absorbing Particles 531

Figure 26.11 The inﬂu- 4 ence of the wavelength on cm–1 the extinction constant of the suspension of ﬁne sulfur K 3 particles used as scatterer in Absorption due Fig. 26.9.Below D 350 nm, to sulfur the sulfur begins to absorb the 2 light strongly, i.e. the light is no longer scattered, but in- 1 stead is converted to heat (see Extinction constant Sect. 27.13) Part II 0.2 0.4 0.6 0.8 1 μm Wavelength λ

An additional important point is the dependence of the scattering on the wavelength. For large particles, RAYLEIGH’s law no longer holds, that is the wavelength dependence of the extinction constant:

1 K D const : (26.11) 4

The exponent becomes smaller and smaller, the larger the scattering particlesC26.6. In the example shown in Fig. 26.11, K has become practically independent of . The eigenfrequency 0, which depends on the size of the scattering particles (the “antenna length”), is much smaller than the frequency of the incident light. From forward scattering, we continue on to diffraction, when the size of the particles on which the light is incident reaches the same order of magnitude as the wavelength of the light. This case can be readily demonstrated in a model experiment using water waves. The particles are constructed from individual ‘building blocks’, which are small steel balls of about 3 mm diameter, placed below the surface of the water. Each of these invisible “obstacles”, when struck by the primary wave, becomes the source of secondary scattered wavelets. The wavelets interfere with each other, and this produces a diffraction pattern from the round particles. Figure 26.12 shows some snapshot images on the background of the primary waves. When the particles move or rotate, the well-defined preferential scattering directions disappear, and the superposition of different diffraction patterns with different shapes and orientations gives a washed-out diffraction pattern, concentrated along the direction of the incident primary waves. C26.10. An example of We could consider the arrangements of the steel balls in Fig. 26.12 aandb such an investigation is the as models for ring- and rod-shaped molecules; the balls themselves would discovery of the double- be the atoms, and the waves would be X-rays. The directional distribution helix structure of DNA of the scattered waves, which interfere and are combined into a diffrac- (F.H.C. Crick, J.D. Watson, tion pattern, permits conclusions to be drawn about the structure of these and M.H.F. Wilkins, in No- ‘molecules’.C26.10 bel Lectures in Molecular Biology, 1962 (Elsevier, NY 1977), pp. 147–215). ’s formulas e crystallites. 1 μm RESNEL shows an example. 26.13 boundary surfaces of th 10 μm from matte surfaces. Matte surfaces consist ). Only at large angles of incidence do the di- by numerous extremely small and disoriented ’s cosine law up to moderate values of the angle of 19.2 Surfaces Photomicrographs of a matte zinc oxide layer, prepared by con- ), they are more intense than the rays from mirrors struck reflection ab AMBERT 25.8 diffuse reflection ,the at more nearly perpendicular incidence. (Sect. incidence (Sect. rections pointing away fromIn the these light directions, source the becomeflat rays predominant: angle from are mirrors concentrated, that were and, struck according at to a F very densation from the vaporright, (at an left, electron an microscope optical image) image using visible light; at the of fine, usually crystalline ‘dust’weakly-absorbing particles materials. Figure or fibers (paper!), made of Figure 26.13 26.10 Diffuse Reflection from Matte What we have learned thus far aboutstand scattering will allow us to under- We can distinguish three contributions to diffuse reflection: First ‘mirror surfaces’, which are the The radiant intensity ofobeys the L light reflected from these micro-mirrors A model experiment showing the transition from scattering to diffraction by weakly-absorbing Scattering 26 particles whose diameter‘building is blocks’ larger (steel than balls theimage. under wavelength. the In Part water c, In surface) the is Parts balls shown a form at a and the triangular b, upper object, left the and on in arrangement the Part of same d, scale it the is as individual circular the main Figure 26.12 ab c d 532 Part II 26.10 Diffuse Reflection from Matte Surfaces 533

Figure 26.14 Demonstrating P S LAMBERT’s cosine law for scattered radiation from the matte R surface of a piece of chalk

α = 0° Part II

α = 45°

To demonstrate LAMBERT’scosine law in the case of diffuse reflection, we can use the arrangement shown in Fig. 26.14. The primary radiation P is in the form of a collimated beam. It grazes a flat ramp R (at the end of a board, shown as a shaded cross-section) and thereby indicates its direction and diameter. Then it falls on the matte, planar surface of a piece of chalk (S). The radiation scattered there by diffuse reflection illuminates the board and produces a second diffuse reflection. Its rays can be seen directly or photographed by a camera. Both views are perpendicular to the board. Chalk exhibits a nearly “ideally-diffuse” reflection: The radiation is itself still symmetric around the surface normal of the chalk, even when the angle of incidence ˛ of the primary beam is 45ı. Paper and porcelain also produce very diffuse reflections. Their glazing has no effect on this; it only produces additional specular reflections which remain within the plane of incidence.

The second contribution to diffuse reflection is a genuine type of scattering, the secondary radiation coming from tiny powder crystallites. For larger scattering particles, it is confined mainly to the direction of the incident light and a narrow cone which surrounds it. This forward scattering is generally directed into the powder layer and produces multiple scattering by the deeper layers within its interior. This again obeys LAMBERT’s cosine law for the rays which finally emerge from the surface. Only when the angle of incidence ˛ becomes large, that is for grazing incidence, is the direction away from the light source again preferred (Fig. 26.15). A third contribution to diffuse reflection comes about because even matte surfaces act as good mirrors at large angles of incidence. We offer two examples of this:

Figure 26.15 was obtained using the zinc oxide layer already shown in Fig. 26.13. It shows the distribution (as a polar diagram) of the secondary radiation for an angle of incidence of ˛ D 80ı. We can see strong forward scattering, superposed onto a specular reflection (giving the sharp peak Sp). The radiant intensity of the light reflected from the matte surface exceeds , 126 , a reflected –2 10 above Sp –3 10 Zeitschrift für Physik –4 10 Intensity incident of light Intensity scattered of light (H.U. Harten, –5 Sp 10 ) from a matte glass plate. ı 5 : –5 89 ). Instead of the printed block letters, we could also ı , a direct photograph of printed type; 10 5 D : Wavelengths of 400–440 of Wavelengths mμ mμ Wavelengths of 400–440 Length of radius = radius of Length ˛ shows two images of the same printed block letters. The 89 Below The forward scattering from a matte zinc oxide surface, super- D –4 26.16 ˛ 10 lower one was photographed directly, whileing the incidence upper ( was reflected at graz- that of thescale!). scattered light byFigure a factor of around a hundred (logarithmic image a slit ontoat a screen grazing with incidence. abecomes lens With and brighter increasing use due angle a to ofsee matte forward incidence, the glass scattering. reflected plate the image as screen On ofbrighter mirror, at this the and first bright whiter slit, at background, first we weak and reddish, then becoming 27 (1949)) posed onto a specular reflection image, after reflection atincidence a grazing angle from a matte glass plate (angle of The explanation foruppermost this ‘peaks’ mirror-imaging of is thea not crystallites two-dimensional hard on point theing to grating matte constants. find: with surface statistically-distributed act The The these grat- ‘partial like zeroth gratings’, order i.e. hasreflection. the the The direction same flatter corresponding to the(grazing direction incident specular incidence), for light the all beam smaller of to relative the perspective to foreshortening. grating the This constants eliminates surface finally the become, higher the due orders, whole and illuminationintensity diffracted comes by from the grating the points. zeroth-order radiant Figure 26.15 Figure 26.16 Scattering 26 534 Part II Exercises 535

Exercises

26.1 In our quantitative treatment of scattering, we mentioned harmonic oscillators (in Sect. 26.2). Consider a spring pendulum with a mass m (deﬂection l, logarithmic decrement ). It is excited to forced vibrations by a periodic force Fp D F0 cos.2 t/ (see Eq. (26.2)). Their amplitude l0 and their phase shift ' are given

by Eqns. (26.1)and(26.3). These two equations are to be derived Part II from NEWTON’s fundamental equation, F D md2l=dt2 (equation of motion, Vol. 1, Chap. 4). a) In a first step, show that the equation of motion for a freely- vibrating spring pendulum which is damped by a frictional force FR D˛ dl=dt that is proportional to its velocity, is solved by the ıt equation l D l0e , with ı D .1=2/.˛=m/. Find the relation between ˛ and . b) Write the equation of motion for forced vibrations, and solve i.2 t/ it using a complex trial solution, l D l0e . Derive from this Eqns. (26.1)and(26.3). (See the footnote on complex numbers in Sect. 25.10). (For Sect. 26.2) iprinadAbsorption and Dispersion e srcl Sect. recall us Let Dependence Wavelength The 27.2 index refractive The Overview and Remark Preliminary 27.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © t aeegh sw aese nE.( Eq. in seen have we as wavelength, its the coefficient nisrfatv ne.I h -a ein i.e. region, attention X-ray our the focus In first index. We refractive its salt). Fig. on (rock in NaCl begin on we Therefore, measurements the with halides. solids, alkali of the simplest This of the crystals for substances. regular minimal most however is for knowledge curves of extinction lack the dis- and the curves both of persion knowledge fragmentary only have we Unfortunately, constant ino aito,adi,i un eed togyo h wavelength. the Sects. on strongly absorp- following depends the the turn, to In in it, related and closely radiation, is of tion Dispersion “dispersion”. exhibits it opin hswl hwcoete oorqatttv ramn of treatment quantitative our to ties Chap. close in show will scattering ab- will we and This refraction Then of sorption. dependence absorption. wavelength the and quantitatively dispersion treat relating evidence pirical ntrso h xicinconstant extinction Either the ways: two of in terms measurements same in the represent appli- can illustrated the we we on clearly index, depending cation, extinction, refractive most the the For be represent curves”. extinc- “dispersion can To draw measured facts representation. graphical basic the a The on by scattering negligible. of are effects tion the when coefficient h vrg eerto et fterdain(i.e. radiation the of depth penetration average the ntergo flne aeegh,terfatv ne approaches index refractive the wavelengths, figures. longer the of of scales region ordinate the the In on seen be cannot 1 from deviations h ercieidcsaealsihl esta (Sect. 1 than less slightly all are indices refractive the opin ntrso h xicincoefficient extinction the of terms in sorption, extinction K fRfato n Extinction and Refraction of Chapter the of k costespectrum. the across aual a eydfeetfr rmteextinction the from form different very a has naturally coefficient 26 25.2 n . eed ntewavelength the on depends 27.2 erfrt the to refer We : olsItouto oPhysics to Introduction Pohl’s k sthe as through absorption C27.1 K 27.5 r ntecs fsrn ab- strong of case the in or, , 25.3 ewl osdrteem- the consider will we , extinction osatadabsorption and constant .Tu,teextinction the Thus, ). k h atrcompares latter The . < , https://doi.org/10.1007/978-3-319-50269-4_27 w fteradiation; the of constant a 5 ca. 27.9 D .Tetiny The ). 1 = K 10 K with ) 27.1 8 and m, hc stega fthis chapter. of goal the is which dispersion concepts optical the of tions applica- general illustrate to intended are These physics. solid-state from details ular partic- many with deals and optics of topic the beyond 2..Ti hpe oswell goes chapter This C27.1. 27 and absorption , 537

Part II

n index Refractive w depth penetration Mean 27.2 with w ,that mand " 12 8 7 6 5 4 3 2 1 0.1 μm0.1 10 μm 10 mm 10 cm 10 m m. In these 10 eV cm 6 ε n decreases D –2 –2 10 10 95 = –1 reaches significant values only w λ –3 k 10 n m and from about 30 to 90 NaCl NaCl –4 10 ). In most spectral regions, the refractive 11 is largest. This is documented in Fig. ttice planes in the crystal. The occurrence of the = Visible k 12.8 w λ Wavelength λ –5 10 1 10 10 The extinction coefficient II,III L (see Sect. Na are recorded. The smallest penetration depth that occurs, C27.2 2 " –6 is varying most rapidly, and those where it has an ‘anoma- increases with decreasing wavelength; the dispersion is then 10 p II,III 10 n L n =w D Cl n K 3 lous’ wavelength dependence, coincide with thoseabsorption regions where coefficient the decreasing wavelength. Theni.e. the it dispersion deviates from is the called rule. ‘anomalous’, The distinguished regions in the dispersionwhere curves, that is the regions index termed ‘normal’. In some spectral regions, however, the square root of the statically-measured dielectric constant with five additional examples.band, the At rate the of boundary change of of an the absorption refractive index with wavelength, is –7 Absorbed by air Na 10 10 K 7 = kk λ 4 D Cl D K hc etc. in the X-ray region is due to the fact that with increasing wavelength, the energy of n = 4 K D k –8 10

, 10 h 2 X-rays D ’s constant m/s, 1 eV Ws). Ws 8 Refraction and extinction (absorption) of light by an NaCl crystal between E 19 34 5 10 hard soft –9 10 10 10 10 LANCK m, is around 30 times larger than the spacing of the la 0.01 nm 0.1 nm 1 nm 10 nm 0.1 μm 1 μm 10 μm 100 μm 1 mm Dispersion and Absorption 2 6 4 1 998 1 8 7 6 5 4 3 2 1 : – –2 –4

2 is P

10 10 10

602 626 27 (velocity of light) Extinction coefficient coefficient Extinction k Extinction constant constant Extinction K 10 10 : : 01 h : mm c D 1 ( C27.2. The abscissa shows the wavelength, above in cm and below in (nano, micro, milli)-meter. On the center abscissa axis, the associated quantum energy is quoted in electron volt (eV), corresponding to 6 1 mm, thus over a range of about 28 octaves. absorption “edges” Cl in two narrow wavelength regions, namely from about 0.04 to 0.2 0 regions, the highest values of the ratio the light quanta is at some point no longer sufﬁcient to knock the electrons out of their inner shells Figure 27.1 538 Part II 27.2 The Wavelength Dependence of Refraction and Extinction 539

Visible 2 n λ = n w 14 1 k λ = 11 w CaF2 KI n k k 0.11 10 0.1 1 10 μm 3 PbCl2 Cyanine Water n 2 n λ = Refractive index w 20 n Part II Absorption coefficient 1 λ k = 9 k w 10 k 0.11 0.1 1 0.1 1 10 μm Wavelength λ

Figure 27.2 Five additional examples of dispersion and absorption. The birefringence of PbCl2 cannot be seen on the scale of the ﬁgure. The values of k for water are magniﬁed by a factor of 10, as indicated

λ = 405 nm 436 546 578

Figure 27.3 A demonstration experiment illustrating the strong dispersion in the spectral region just before the steep onset of self-absorption: The visible part of the Hg line spectrum, obtained under similar conditions with two different 60ı prisms; below, a prism made from a single crystal of ZnO (embedded in water), and above, a quartz prism (E. Mollwo, Zeitschrift für Angewandte Physik 6, 257 (1954)) i.e. its dispersion dn=d, can become very large. This is shown in Fig. 27.3 with the aid of a prism made of ZnO. The relationship between dispersion and absorption can be demonstrated by an impressive experiment. For this purpose, neither solids nor liquids are suitable1; we must use vapors, gases or very dilute solutions. Sodium vapor is most convenient. Figure 27.4 shows a suitable experimental arrangement. It uses a prism P to project the continuous spectrum of a sodium-arc lamp onto a horizontal screen. An iron tube R ﬁlled with Na vapor is placed directly behind the projection lens L1. It has glass windows at each end and is evacuated. Then the sodium is evaporated in the middle of the tube. A low pressure of H2 gas and air cooling prevent the windows from being clouded by condensing Na vapor. The vapor produces a strong extinction around D 589 nm. The horizontal spectrum is interrupted

1 The justification of this statement will be seen later in Eq. (27.7). n attains high 2 2 values only when the difference of the squared frequencies, that is 0 ,is very small. In solids and liquids, with their broad absorption bands, this leads into regions where the material is opaque. 2 S The lower ( and ). R 27.5 C27.4 1 downwards, P > D n 2 L , 1880, improved by ceable influence on the ). Here, the contribu- UNDT 2 S bands or lines. is a direct-vision prism upper image P , H 27.5 band are due to Na molecules. Because of R absorption D (Fig. ). At the hottest point, i.e. at the bottom of D (photographic positives). The absorption bands which 27.4 a vertical slit, and Cooling water Cooling 27.4 2 S The anomalous dispersion of Na vapor, demonstrated with the R 1 are deflected upwards. In the example shown in Fig. Violet Red < n 1 is imaged upwards onto the screen. A cylindrical lens between L 2 are visible in additiontheir to low the particle-number densities, they have no noti refractive index of the vapor. apparatus in Fig. by an extinction band Figure 27.5 Following this preliminary experiment, not only theupper ends part but of also the the tubein are the cooled. tube This to causes takeupper the on cloud right a of in prismatic Na Fig. shape vapor (schematic in the inset at the tion due to absorption is larger thanwe that nearly due always to speak scattering. of As a result, the center of the tube,towards the both density ends, of it thethe decreases. vapor spectrum is This high; unchanged. ‘vapor upwardsrefractive and prism’ In index leaves these of most unchanged the of spectralsides Na regions, of vapor the is the practically absorption equal band, to however, 1. the light On is both deflected verti- S 1 is a horizontal slit, S 1 S C27.3 C , McMil- A demonstration of anomalous dispersion in Na vapor (A. K , 1904. OOD ), the lower end of the slit Dispersion and Absorption 27 C27.4. In a “direct-vision” prism, a combination of strongly and weakly refracting glasses is employed, so that at a certain wavelength, a light beam is noton deflected passing through the prism. lan Publishers, NY, 3rd ed. (1934), p. 492. C27.3. See R.W. Wood, Physical Optics R.W. W vapor prism (schematic in the inset at upper right) deflects waves with a refractive index of while waves with a refractive index of image improves the visibility Figure 27.4 540 Part II 27.3 The Special Status of the Metals 541 cally. On the red side, the deflection is downwards on the slit S2,that is the refractive index is > 1. On the violet side of the band, the deflection on the slit S2 is upwards, that is the refractive index is < 1. The spectrum thus forms a colored curve consisting of two branches (Fig. 27.5, lower image).C27.5 The shape of this curve corresponds to C27.5. A colored image the dispersion curve of Na vapor on both sides of the extinction (ab- can be found in the book sorption) band. The section of the curve within the band is missing. by R.W. Wood; see Com- It can be seen only with moderate absorption and then only by direct, ment C27.3. subjective observation. Part II 27.3 The Special Status of the Metals

We return to the important Fig. 27.1. The smallest values of the extinction constant K, i.e. the greatest penetration depths w, can be found in the visible and the neighboring spectral regions, in particular in the infrared. In these regions, the mean penetration depth can be up to many meters and greatly exceeds those for all other radiations, in particular for X-rays. An exception to this rule is provided by the metals. This is shown in Fig. 27.6 for silver. The ﬁgure spans a wavelength range of 16 orders of magnitude. The extinction constant K has very high values over the entire infrared and visible spectral regions; the extinction processes acting there extend into the ultraviolet. Details will be given in Sect. 27.17. Values for n and k are shown for two important metals in Fig. 27.7. The absorption coefﬁcients k increase to high values from the ultra-

107 106 105 104 103 102 101 110–1 10–2 10–3 10–4 10–5 10–6 eV 10–10 10–9 10–8 10–7 10–6 10–5 10–4 10–3 10–2 10–1 1101 102 103 104 cm 105 106 mm–1 hard soft Visible

X-rays Absorbed by air 10 μm w K 4 10 α 0.1 μm NI-VII MI-V 1 μm 102 LI-III 10 μm

Ag 100 μm K 1 1 mm Extinction constant constant Extinction

1 cm Mean penetration depth

10–2 10 cm 0.1 pm 1 pm 10 pm 0.1 nm 1 nm 10 nm 0.1 μm 1 μm 10 μm 100 μm 1 mm 1 cm10 cm 1 m 10 m 100 m 1 km Wavelength λ

Figure 27.6 The extinction (absorption) spectrum of a metal (silver) between D 1013 m(0.1pm)and D 1 km (the scale of the abscissa is half that of Fig. 27.1). The gap in the extinction curve for NaCl as seen in Fig. 27.1 between 0.2 m and 20 m is not present in this curve. The small minimum ˛ at D 0:32 misby no means comparable to that gap. The mean penetration depth w attains a value of only 50 nm here. The points marked with ‘x’ are calculated. For Al, the minimum of the extinction constant is at D 6 1014 m. There, the penetration depth 1=K D w D 17 cm. (Upper scale: 1 eV D 1:602 1019 Ws)

k Absorption coefficient coefficient Absorption ) m/s 30 20 10 0 30 20 10 0 8 25.37 ( n 10 n here is thus is also often k w n k 1) which is char- m, for example for ). Metallic bonding Ag Cu 4 R Wavelength λ : 27.17 D 2 2 k k

m/s instead of only 3

C 0.2 0.5 1 μm 5 μm C

3 2 1 0 3 2 1 0 n Refractive index index Refractive 8 as germanium, silicon, stibnite 2 2 on the order of 1 in the ultravio- / / 10 k 1 1 shows some examples which are of llic bonding in the visible spectral C . In the case of dyes, e.g. cyanine or n n ), and for some semiconductors, the ab- . . 27.8 27.2 D 25.1 at perpendicular incidence, we can apply R .Valuesof k attains large values already in the visible. As R 30. The mean penetration depth k C27.6 is predominant in the numerator and the denom- ). Figure 2 k k . The smallness of the refractive index 25.11 for silver and The optical con- k 1 400 . and ) etc. are indistinguishable from metals to the unaided eye. ’s formula: 3 n S 27.22 2 EER B for light in vacuum. If the summand copper. The scatter among the values is still rather large, even in the best measurement series available today. Fur- ther examples are given in Fig. However, the semiconductors lackin the large the absorption infrared coefficients which are characteristic of substances with metallic Figure 27.7 stants notable in metals. Forvelocity increases silver, up it to decreases nearly to 20 0.16. Thus, the phase inator, then we observe the high reflectivity ( equal to silver, we find 27.4 ‘Metallic’ Reflectivity For the reflectivity violet towards longer wavelengths. At acteristic of materials withregion (Sect. meta practical importance (more on this in Sect. is however by no meansabsorption the coefficient only reason for very large values of the let can be found for theexamples majority of are solid given and liquid in substances. Fig. Some brilliant green (see Table sorption coefficient a result, some semiconductors(Sb such . c . 23.3 c is the deci- ), and as we have Dispersion and Absorption 12.11 27 . This does not mean that information can be transferred at a velocity faster than C27.6. The phase velocity of light within materials can indeed become higher than the vacuum velocity of light, c For information transfer, the group velocity and sive quantity (see Sects. seen, it is always less than 542 Part II 27.4 ‘Metallic’ Reflectivity 543

100 % 80 R Rh 60 Ag Au 40 Reflectivity 20

0.1 0.2 0.4 0.6 0.8 1 μm 2 4 6 8μm

Wavelength λ Part II

Figure 27.8 The wavelength dependence of the reflectivities of gold, silver, and rhodium. Rhodium is particularly suitable for mirror surfaces without glass protection, due to its chemical inertness. In addition, thin, transparent rhodium mirrors attenuate all the wavelength regions of the visible spectrum (400 to 700 nm) to practically the same degree (they act as “grey filters”). In the minimum at D 316 nm, R D 4:2 % for silver. The values for the alkali metals are still smaller: at D 254 nm, R D 2:6 % for potassium and 1% for rubidium and cesium. bonding, and which are caused by their special type of electronic conduction (Sect. 27.17). In the case of germanium, for example, k is already vanishingly small at D 3 m. Thus, blocks of germanium a few cm thick look like pieces of metal. Nevertheless, they are transparent to infrared radiation, apart from the considerable reflection losses due to their large refractive index of n D 4. This can be demonstrated by a very surprising experiment. It makes use of the setup shown in Fig. 25.1, and shows quite clearly that whether metallic bonding is present or not can never be determined by the unaided eye; only absorption measurements in the infrared are conclusive. Finally, crystals with typical ionic bonding, such as the alkali-metal halides, exhibit extreme values of n and k in the infrared (compare Fig. 27.1, bottom). As a result, such crystals have a rather large reflectivity there. Figure 27.9 shows four examples. The wavelength scale is three times larger than in Fig. 27.1. These reflectivity maxima are called residual rays. Their position is determined by both n and by k. Therefore, their maxima are only approximately at the same positions as those in a plot of the absorption coefficient k.

The unusual name ‘residual rays’ is related to the method of their first observation. HEINRICH RUBENS (1865–1922) passed the radiation from an incandescent light heated by a gas burner back and forth several times with reflections between two crystal plates and then detected it using a radiometer (radiation thermopile). The remaining (“residual”) rays consisted of practically only those waves from the spectral regions of the reflection maxima. These “residual rays” are absorbed by thin mica or glass plates, ). 2 Op- hyperphysics.phy- experience no diffuse https://en.m.wikipedia.org/ ). Scattering plays a significant or m). It is caused in this spectral , X-rays 26.8 11 – 10 rnal boundary surfaces. The ‘head’ Wavelength λ 26.6 ). In all other materials (for example The refractive index for X-rays is very < 27.6 LiF NaF NaCl KCl As a result effect (see for example the 13th edition of “ ), and at still shorter wavelengths, by nuclear quite opaque to visible light, but completely ). (Fig. ”, Chap. 17. English: See e.g. ), X-rays do not have anything like the enormous 27.9 OMPTON 0.1 0.05 0.02 0.01 eV 27.1 10 20 40 60 80 100 200 μm

80 60 40 20

Residual rays exhibited by four alkali-metal halide crystals (the (Sect. R Reflectivity in cloudy, inhomogeneous materials such as muscle tissue, 100 % role even for hard X-rays ( The lack of diffuse reflectionis no in scattering the there X-ray (see range Sects. does not imply thatregion there by the C but can pass through thick layersa of demonstration paraffine, experiment, etc. most (This simply is using readily shown thin as slabs of LiF or CaF tik und Atomphysik wiki/Compton_scattering processesaswell. astr.gsu/hbase/quantum/comptint.html transparent to X-rays. bands are not simple Gaussian curves, in contradiction to earlier reports) The penetration depths oflight only X-rays in are metals greater than those of visible Figure 27.9 27.5 The Penetration Depth of X-rays reflection bones, wood etc.aries They between the are individual unaffected componentsals. by of inhomogeneous the Visible materi- innumerable light, bound- is in exceedingly contrast, sensitive with toon refractive inte light indices Pilsner of beer is about 1.5, NaCl; cf. Fig. penetration depths which are observed with lightthe from neighboring the infrared visible spectral and regions. The importance of X-rayshowever for not medical based and onsomething technical their quite applications great is different: penetration depths, but rather on close to 1 Dispersion and Absorption 27 “The ‘head’ on light Pilsner beer is quite opaque toible vis- light, but completely transparent to X-rays”. 544 Part II 27.6 The Explanation of Refraction in Terms of Scattering 545 27.6 The Explanation of Refraction in Terms of Scattering

From reading Sects. 27.2 through 27.5, we are now familiar with the more important empirical facts about refraction and extinction. Here, we want to interpret them and describe them quantitatively. In Sects. 27.6 through 27.11, we treat refraction; and in Sects. 27.12 through 27.21, we deal with that part of the extinction which is due to absorption. Part II We return to Fig. 26.12 d. There, the model scattering object is transparent. We can follow the course of the waves within it, although with some difﬁculty. We observe a picture like the one sketched in Fig. 27.10: The waves propagate more slowly in the region where secondary-wave sources are present than outside the object; the wave crests are noticeably delayed within the object. Or, expressed differently, the circular region has acquired a refractive index unequal to 1 due to the presence of secondary-wave sources in its interior. This basic fact will be veriﬁed by an impressive demonstration experiment. The best-known effects of refraction are exhibited by lenses. Thus, in Fig. 27.11 a, we show a group of “secondary-wave sources” arranged in a lens-shaped pattern. The scattering “atoms” are again small steel balls below the surface of the water in a wave trough. They are disordered, and their diameters and spacings are again smaller than the wavelength. In Fig. 27.11 b, water waves with straight wavefronts pass at a small angle of inclination through a wide slit. The slit cuts out a collimated (parallel-bounded) beam of waves; diffraction can be clearly seen. In Fig. 27.11c, the “lens” has been placed in the opening of the slit. The result is that the previously collimated wave beam is now focussed onto an image point. Now, there can be no further doubt: The waves pass through the region containing secondary sources with a reduced phase velocity. The region with secondary sources has a refractive index n. We compute it using the elementary lens formula 2 1 .n 1/ D (16.12) R f 0

(R is the radius of the lens surface; in Fig. 27.11a, R D 7cmand f 0 8:5cm) and obtain n D 1:4.

Figure 27.10 The occurrence of a phase shift due to secondary waves (sketched from Fig. 26.12d) The rigid vibra- and also the 0 l exhibits a very . The incident 0 n

). We repeat it 27.2 and and excite the resonators to

27.1 ve produced by scattering must relative to the primary waves. 0 ı . This dependence of the refractive index through phase-shifted secondary waves 27.12 ; for example, by “breathing spheres” (Vol. 1, refraction of the secondary waves generated by scattering causes . , the secondary-wave sources consisted of small 0 ı of Dispersion resonators 27.11 ,i.e. phase differences between the resonators and the primary waves are primary waves have a frequency of forced oscillations. Then both the forced amplitudes phase shift the refraction The wavelength dependence of the refractive index have a negative phase shift 27.7 The Qualitative Interpretation The explanation is readily found. Theand waves which behind propagate the within which lens are are produced the by scattering,The resultants primary together waves of generate with secondary the waves, all and primarytiary these the generate waves. waves, ter- secondary etc. waves Thewhich resultant is waves have less an thanfore, overall that phase each of velocity individual the secondary individual wave wa components. There- characteristic form inwavelengths the or neighborhood frequencies (Figs. of certain distinguished schematically in Fig. on the wavelength or thetative way. frequency To is this readily end, interpreted wemechanical in turn waves. again a to quali- our model experiments with In Fig. tions balls beneath thesecondary-wave surface sources of are the replaced by water. objects capable Now, of imagine that these Sect. 12.25). Their eigenfrequency is denoted by bc Water waves exhibit the occurrence of Dispersion and Absorption 27 Figure 27.11 a 546 Part II 27.7 The Qualitative Interpretation of Dispersion 547

Figure 27.12 Schematic of a dispersion curve in the β neighborhood of an opti- n γ α 1 cal eigenfrequency (band ε maximum; the points ˛ index Refractive δ to " correspond to the im- λ small λ large ages A EinFig.27.13; Frequency ν of the light see the text) Frequency ν0 of the band maximum

Figure 27.13 The occurrence of disper- AB sion as a result of phase-shifted secondary

δ' δ' Part II waves (the time increases in the clockwise sense) E Ep δ Ep r Er δ l0 l0 Ep E E s s δ l Er 0 C

Es D E δ' δ' E Ep p Er Er δ δ Es Es l0 l0

determined by the ratio = 0 (Sect. 26.2, see also Vol. 1, Fig. 11.42). Furthermore, the amplitude of each secondary wave is shifted rela- ı2 tive to the amplitude l0 of the secondary-wave source by 90 . We thus arrive at the simple vector diagrams shown in Fig. 27.13 A– E. The notation there is:

Ep is the amplitude of the primary wave; l0 is the amplitude of the forced oscillations; their relative values can be read off from Fig. 11.42 in Vol. 1 (e.g. for D 1);

ı is the phase angle between l0 and Ep; it can likewise be read off from Fig. 11.42 in Vol. 1 (with D 1);

Es is the amplitude of the secondary waves emitted by the resonators;

Er is the resultant wave amplitude from the superposition of the primary and secondary waves; and 0 ı is the phase angle between Er and Ep. The time and the phase angles ı and ı0 are taken as positive in the clockwise sense in the diagrams.

2 This is a simplifying hypothesis. In fact, this phase difference of 90ı results from the summation of all the secondary waves along the path of the primary waves. . . ı D ˛ (the 15 0 0or takes takes r F (27.1) 0 E 0 D ı ı 0 ı )tobepro- has remained 0 as the point . 26.1 ı . Then, . ı leads the primary .Then ı p is approaching the r ) corresponds in the 27.12 E E 27.12 140

n 180 is very small. . It can undergo forced . / 0 D ı f ).

in Fig. " 2 ı m e ˇ 0 reality, however, the secondary has increased to around and E . The resultant amplitude ı 0 ı is almost periodic force of amplitude ; is found from Eq. (

D ,and 0 shes only the exciting primary wave ı 0 0 90 0

l , there is one oscillating, bound elec- 1 2 3 of the electron and furthermore depen- e 25 D : D

,and m 0 ı D , of the primary waves. An oscillating dipole

0 p

value. This means that the resultant waves are , but its magnitude is larger. This means that the 27.13 ,e.g.1 ) has the same direction as

,e.g. s 0 is also larger: Point ,thatis 0 E 0 n

, the amplitude of the primary waves, and inversely ). value. The resultant amplitude 0 illustrates a typical dispersion curve. It exhibits qual-

. This means that the refractive index is smaller than 1 E D p p negative > negative of Dispersion E < E

). , but its magnitude has decreased; 27.12 ı positive . Its oscillation amplitude 1(Point 0 E remains D is somewhat larger than 1. It is drawn in Fig. Here, as always, ‘molecule’ refers to the smallest independent unit; this could 0 Finally, in Part E, amplitude positive value 1, but is still smaller than 1 (Point difference In Part B, slightly delayed relative to the primaryn waves, or the refractive index on a small on a ı e tron; its eigenfrequency is denoted by (Point oscillations under the action of a optical measurements to the maximum of an extinction band. itatively the same featuresiments. as The the distinguished curves wavelength observed (point in optical exper- Figure n is thus produced, and its electric dipole moment has the amplitude proportional to the mass dent on the frequency often be also atoms or ions. 3 27.8 A Quantitative Treatment Quantitatively, the treatment in the previousfactory. section In is particular, it rather distingui unsatis- from the excited secondary waves.waves In for their parttirety excite of tertiary all waves these and waves sois ﬁnally leads forth. computationally to not the Only simple, resultant. the buthowever, one The en- it avoids summation that can difﬁculty be by using carried the out. followingWe procedure: assume In general, that per molecule In Part A of Fig. portional to refractive index In Part D, In Part C, Dispersion and Absorption 27 548 Part II 27.8 A Quantitative Treatment of Dispersion 549

(the frequency-dependent expression f . / follows from a comparison with Eq. (26.1)). The quotient

p e2 0 D f . / D ˛ (27.2) E0 m is the electric polarizability of the molecules (see Sect. 13.9)atthe high frequency of the light waves.

In our discussion of RAYLEIGH scattering, it was assumed that Part II 0. This made the polarizability ˛ independent of the exciting frequency (Sect. 26.5). Therefore, ˛ could be computed from the statically measured (i.e. at D 0) dielectric constant ".In Sect. 26.5,weusedtheCLAUSIUS-MOSSOTTI equation for this purpose (Eq. (13.28)): Â Ã p 3" " 1 D ˛ D 0 (27.3) E NV " C 2

("0 is the electric ﬁeld constant, and NV is the number density of the polarizable molecules).

It takes into account the inﬂuence of the surroundings on the polarizability of the molecules.

Now, we follow the reverse path. We drop the limitation 0 and thus make ˛ dependent on (Eq. (27.2)); we insert the ˛ values into Eq. (27.3) and then calculate for each value of the exciting frequency a particular value of ". Thus we obtain a dielectric “constant” which depends on the frequency .

Finally, we come to the decisive step: According to MAXWELL,for long-wavelength electromagnetic waves (Sect. 12.8), p n D " I (27.4) here, " refers to the statically-measured dielectric constant, i.e. at D 0. We now apply this same relation to light waves, but at every frequency , we use the frequency-dependent dielectric constant especially calculated for that frequency, in order to compute the refractive index n for light of the frequency . In this manner, the dependence of the refractive index n on or can be quite satisfactorily reproduced. We will now carry out this plan quantitatively. We again take Eq. (27.1) for the induced dipole moments of the molecules, but actually compute l0 using Eq. (26.1). We avoid the frequency range close to the eigenfrequencies 0; the regions <0:7 0 and >1:4 0 sufﬁce. In these regions, the forced deﬂections l0 are practically 0 ). m

b 2 5 1.519 1.521 ) (27.8) (27.6) (27.7) (27.5)

As, is now 1 19 > 2 0 ˛

10 2 1.527 1.522 m. It could . For liquids Hz was em- mand5 Hz 0 V 6 :

15 15 N is the number 1 3 : : 1). Thus, we can : 10 10 0 V D 105 1 1.532 1.528 2 3 : 2 N 2 e

s Ä 0 m

), namely 85 D i 1 : 85 9 :

2 : b D 2 0 2 2 0.7 1.539 1.535 2 0i

D 26 . 27.7

kg, and D 0 1 As/Vm,

in place of the frequency- 31 D 2 0 i 2 0 m 12 e

2 1, 2 0.5 1.552 1.550 X

10 n 2

0 V 10 D 1 E 1 N refraction i 4 e 3 11 m 2 2 0 : 86 4, : 0.4 1.568 1.567 s 9

m 8 D D 2 9 0 : D 0 D V 1 b p E , 4 N is well fulﬁlled for gases and vapors, of 26 0.3 1.607 1.610 3 4 D ) D 2 D ). We write m e 0 0 0 l 2 is known as the 1 E 27.8 0 frequency-dependent polarizability 0 el " R 27.3 C 2 2 2 1 D , the logarithmic decrement ( contains a numerical example for rock salt, NaCl. D ion pairs/m n n

2 ˛ 1 ) assumes that there is only one eigenfrequency 12 28 C 2 2 n n and ﬁnally arrive at: 27.1 10 The dispersion of NaCl between D " 27.7 2 1 electron per molecule. In reality, each molecule possesses ) 28 : 2 C m is the electric ﬁeld constant is the mass of the electron dispersion formula 2 2 27.1 0 one D n " n m density of the polarizable molecules). ( in measured computed from Eq. ( V The quotient N n n ( This inserted into Eq. ( This value of the or (Fig. dependent The discrepancies between the calculatednever greater and the than observed 5 values unitsalthough is only in a the single third eigenfrequency place after the decimal point, and solids, it is however nothingmula. more than Table a useful interpolation for- course apart from the regions of their eigenfrequencies ployed. It corresponds to the wavelength a whole series of opticaland eigenfrequencies (numbered by often the also index i) several effective electrons (number denoted by 4 Table 27.1 Equation ( and independent of Thus, we must write a sum in place of Eq. ( neglect the second summand in the denominator and obtain Dispersion and Absorption 27 550 Part II 27.10 The Refractive Index and the Number Density. Light Drag 551 be called the “center of gravity” of the k curve in the ultraviolet (Fig. 27.1). Of course, by using i D 3 or 4, we could improve the agreement between the calculation and the measurements even further; but that would be rather unproductive.

27.9 Refractive Indices for X-rays

As an additional test of the dispersion formula (Eq. 27.8), we

determine the refractive index for X-rays. With X-rays, the chem- Part II ical bonding of atoms into molecules plays practically no role (Sect. 27.12). NV in Eq. (27.8) thus refers to the number density of the atoms,i.e.

NA% NV D Mn

23 1 (NA is the AVOGADROconstant D 6:02210 mol ,andMn is the molar mass, M=n).

Furthermore, b is the number of all the electrons in an atom, b D ˙bi. This number is the same as the atomic number Z, which can be expressed in terms of the molar mass Mn D Ar kg/kmol (see Sect. 26.7); for atoms whose molar mass is not too large, Z D 0:5 Ar (Eq. (26.21)). Therefore, Number of electrons 1 N b D D 0:5 6:022 1026 %: V Vo lu m e kg

In addition, we consider only one resonance frequency 0; and ﬁnally, for hard X-rays, 0 .Using D c=, we obtain from Eq. (27.8) Á n2 1 m3 N b m3 2 D 26:9 V D0:81 1028 % : (27.9) n2 C 2 s2 . 2/ s2kg c

Numerical example: % D 104 kg/m3 and D 1010 m, refractive index n D 0:999986, thus somewhat less than 1. This is conﬁrmed by observations; see Sect. 27.2.

The dispersion equation (27.8) thus encompasses the whole spectrum from the infrared up to the X-ray region. It also functions in the region of the longest waves; however there, we must take into account not only the secondary radiation from electrons, but also that from ions or from still larger objects.

27.10 The Refractive Index and the Number Density. Light Drag

We have interpreted refraction in terms of the secondary radiation from the irradiated molecules, and thus obtained the dispersion for- with ˛ . Then, (27.10) / 40 40 40 40 1 ˛ 10 10 10 10 (solid, liquid, m n = . , we have listed V Asm 2 3 on the light fre- 77 78 64 68 as well as for the : : : : : The propagation 0 Molecular polarizability in 1 1 1 1 n R 27.2 . Second, it describes 2 1 1 4 1 ; IZEAU 4 C V 27.9 2 2 10 10 10 – n n N 10 D 41 82 06 7 0 : : : : liquid transition, that is in spite 27.7 Refraction R 1 1 2 1 is nearly equal to 1, and thus the is nearly equal to . const of light”, was predicted in 1818 by / n 2 , D 5 / C drag n 1 2 n 589 nm which is derived from it. Both quantities are n /=. . , and it thus represents the refractive index that is due ). Compare the values determined here for D to be constant, and thus consider observations ˛ 1 independent of the aggregate state

), we have 27.4 1) is proportional to the number density, and in 1.00027 1.334 1.000255 Measured refractive index 1.222 at , the number density of the molecules, on the value 2 solvent solution V n n n )). It provides two kinds of information: First, it chemical bonding . N V 27.8 n N D and found in 1851 by A.H.L. F )and( D . n 27.8 0 . R 25 28 25 28 27.3 . That was shown in Sects. of single molecules in AC ﬁelds at the frequency of yellow light,

10 10 10 10 3 20.31 13.5 ˛ RESNEL 69 36 69 14 : : : : 2 3 2 Number density of molecules in m 2 In solutions, some numerical examples, both for the refraction of a change in density of around 1:1000. In Table i.e. in gases, ( solutions, it is proportionalferently, to each the molecule concentration. makes its Or,independently contribution expressed of to dif- all the the refractive index otherrefractive molecules. index and This the relation gasexperiments. between density An the experiment is for well a suitedin teaching Fig. for laboratory demonstration was illustrated The independent contributions of thetion individual molecules remains to valid refrac- even for the gas of a light wave is inﬂuenced by the motion of the molecules. A liquid electric polarizability A. F thus to a large extent gaseous) and of A strange effect, called the “ refraction instead of Eq. ( of the refractive index. We wantTo to that consider end, this we point in take more detail. solely to the dissolved molecules. 5 mula (Eq. ( describes the dependence of the refractive index quency using monochromatic light. For gases and dilute solutions the inﬂuence of % 3 kg m 1.43 1000 0.805 Mass density in 1130 C ı C, ı Hz, calculated using Eqns. ( 183 14 The electric polarizability Pa Pa Cand 5 5 10 ı 10 10 1 : 5 Dispersion and Absorption liquid, gas, 0 013 013 2 2 D : : 27 Water, liquid Water vapor, 0 O O Substance referred to 1 1 Table 27.2

those found for low frequencies in Table 552 Part II 27.11 Curved Light Beams 553 which is moving in the direction of the light propagation at the velocity u has a smaller refractive index for an observer at rest outside the liquid, i.e. the velocity of the light is greater than for the same liquid at rest. The velocity u of the liquid thus changes the phase velocity c=n of light within the liquid, but not by its full magnitude ˙ u; instead, by approximately ˙ u .11=n2/. (The experimental arrangement is the same as shown in Fig. 20.31, except that both light beams pass through chambers filled with water flowing in opposite directions.) We can understand the ‘light drag’ qualitatively from Eq. (27.8): The Part II number density NV of the molecules acted upon by the light increases when the liquid is flowing towards the light source, and vice versa. Quantitatively, the drag is calculated from the LORENTZ transformations in the special theory of relativity.

27.11 Curved Light Beams

The refractive index which holds for a particular monochromatic radiation depends on the number density NV of the active molecules (Eq. (27.8)). This density can be varied continuously within a certain volume, thus producing a gradient in the refractive index. In such a volume, curved light beams can be observed, as seen for example in Fig. 27.14. The boundaries of curved beams as well as their axes are drawn as curved light rays. In general, the radius of curvature of a ray changes along its path. At each position x,wehave n r D (27.11) dn=dr .derivation in Fig. 27.15/:

Here, dn=dr is the gradient of the refractive index at the position x in a direction perpendicular to the beam. Video 27.1: Experimentally, gradients in the refractive index can be produced by “Curved light beams” concentration gradients in solutions. The most suitable method is http://tiny.cc/wfggoy A laser is used as light source, so that there is no fanning-out of the beam due to dispersion. To produce the gradient in the refractive index, seven layers of sugar solution, each 1 cm thick and with decreasing concentra- Figure 27.14 A curved light beam in a liquid with a vertical, approximately tion, were laid on each other. linear gradient in its refractive index. The fanning-out of the beam at the right While each layer was being is due to dispersion: The path of the waves with the shortest wavelengths is poured, a thin cork sheet was the most strongly curved (this is also a model experiment for the appearance used to keep mixing to a min- ofthe“greenﬂash”(seebelow))(Video 27.1). imum. r n d d has r ) n ): At the .Atthe –d n · n 2 ( φ 1 s d 63); at its top, s 27.11 27.14 d : d 1 r r D d n Gradient the of refractive index, , / r d )by C r ). . . Above it is a glycerine/alcohol mixture, roughly 16.5 3 . At the top is water with ca. 10 % alcohol, density 50), and the transition between the two con- ' 3 : d .Furthermore, 27.11 1 C27.7 n . D D 2 1 04 g/cm s s : OOD d n 01 g/cm to the direction of the light beam is maximal. : , the gradient in the refractive index is also vertical, but A light beam with a wavy shape. The refractive index has its The derivation of D . Combining the three / r n . All the solutions contain quinine sulfate and sulfuric acid, and ). The three arrows indi- d 3 ' 27.16 d n . 27.11 D 2 1 98 g/cm s : s d cate the tipping of thetheir wave ends. crests The at opticalare path given lengths according to Eq. ( 0 equations leads to Eq. ( Eq. ( from the sketch, we canmetrical see relation d the geo- Figure 27.15 maximum value at the centerwith of a the density tank. of 1 Below is a saturated alum solution 1:1, of density 1 their boundaries were smearedis out due by to several R.W. W hours of diffusion. The recipe in layers withsurfaces appropriately which chosen are compositions.smeared initially out present The by between diffusion. boundary dient the In in this layers way, the are an refractive approximately rapidly linear index gra- was produced in Fig. Figure 27.16 to use two liquids which are completely miscible, and to add them d bottom of the tank is pure carbon disulﬁde ( pure benzene ( sists of around 10strongly layers, curved each at its 1 cm highestits thick. point, minimum i.e. value The its there. radius lighttop of beam of This curvature the is corresponds curve, most to theperpendicular gradient Eq. of ( the refractive index in theIn direction Fig. it changes itsdemonstrate sign a in light beam the with a center wavy shape. of the tank. In this way, we can (McMillan Dispersion and Absorption 27 Publishers, NY, 3rd edition 1934), p. 90. C27.7. See R.W. Wood, Physical Optics 554 Part II 27.11 Curved Light Beams 555

Figure 27.17 The lens action of α a cylindrically-symmetric gradient in the refractive index (‘snapshot’). Top: The cross-section of the arched rectangular piece of sheet metal under the water Part II

Radially-symmetric gradients in the refractive index, some with cylindrical symmetry and some with spherical symmetry, play an important role in the eyes of animals. The most prominent example are the compound facette eyes of insects in their various forms. However, even in the lenses of the eyes of vertebrates, there are also gradients in the refractive index in combination with the curvature of the lens surfaces. Strictly considered, in a sketch of the human eye, “Strictly considered, in one should draw curved light rays within the eye’s lens. a sketch of the human eye, one should draw curved Owing to its importance, we want to translate image formation light rays within the eye’s with curved light rays into the wave picture. To this end, we show lens”. a model experiment with water waves in Fig. 27.17. We start from Fig. 27.11 b and set a flat, rectangular, cylindrically-arched piece of sheet metal under the surface of the water, between the two sides of the slit. Its cross-section is sketched in Fig. 27.17 (top). Its long axis is set perpendicular to the slit; this gives a shallow-water area of rectangular shape and variable depth. The depth of the water is smallest in the center at ˛, and greatest at the side edges. As a result, the waves propagate more slowly in the center than at the edges (Vol. 1, Eq. (12.36)). They are convergent when they leave the rectangular area and are focussed at an image point (Fig. 27.17 (bottom)). Gradients in the refractive index with spherical symmetry play an important role in astronomical observations. We mention just one example. The density of earth’s atmosphere decreases on going upwards from the surface. A light beam which is incident tangential to the earth’s surface arrives at the eye of an observer on a curved path. The sun, when it seems to just touch the horizon, has actually already set; the “atmospheric ray refraction” causes it to appear too high by 32 minutes of arc. This means that a surprising effect can occur during an eclipse of the moon: We can see the sun and the darkened moon opposite to each another, both just above the horizon, at the same time. At sunset, especially at sea, one can sometimes see the last part of the disappearing disk of the sun flash brightly with a green color. This phenomenon, known as the “green flash”, can be explained by deflec- . They consist of C27.8 . The extinction spec- atoms m/s. They propagate along 27.2 8 10 and 3 s gives rise to new, additional D , was already explained in 1801 by u 27.1 00 88 : of the earth is only indirectly involved in the by a contrast effect in the eye of the observer). : Light rays in gravitational fields act like projectiles not with his general theory of relativity. of Absorption OLDNER )(and INSTEIN 27.14 gravitational field of about 1.75 seconds of arc near the disk of the sun (it is visible emitted from the stars at a velocity of hyperbolic paths. The otherby half A. of E the observed deflection was predicted Half of this deflection, that is 0 J. G. v. S the sum of the extinction spectrabonding of and all the the atoms aggregate involved.conclusion: state Chemical have The practically extinction no ofelectronic influence. the shells radiation of Our the occurs atoms, inof which the their are surroundings. innermost protected from the influence In the region of softfelt, X-rays, along with chemical the bonding aggregate state: beginsare to Crystals exhibit lacking make new in itself bands which the spectranow, of the individual middle molecules. and We outer concludeportant electronic that shells role; of they the atoms aresurroundings play of no an the longer im- atoms. impervious to influencesIn all from of the the remainingthe spectral visible regions, to from the the infrared,to ultraviolet the through a extinction large spectra extent of on thebonding the atoms of aggregate depend state the of atomsbands. the Conclusion: material. into Here, molecule Furthermore, thethe extinction is outermost electronic produced shells, by which processes areical in also bonding, responsible formation for of chem- the liquid phase and of crystal structure. phenomenon of atmospheric ray refraction.mal molecular Together with motions, the itmolecules ther- produces and the thus density the gradient gradientsphere. of of the the refractive gas index in theSurprisingly, atmo- however, gravitational fieldsa alone gradient in can the alsointeracting refractive molecules. produce index The even light in from empty the space, stars without is any subject to a tion only during total solar eclipses). 27.12 The Qualitative Interpretation We begin by looking back at Figs. the stronger curvature of the(Fig. optical path for short-wavelength light tra consist in generalAs of a a rule, they number are often, not single individual completely narrow bell-shaped separated bands from bands. mergebands. each together other, to and give of- broad “unresolved” We note for a preliminary overview:extinction In spectra the are region determined of only hard by X-rays, the the The , Vol. III, also due to the 1911) obtained INSTEIN Dispersion and Absorption OLDNER 27 a similar value to that of v. S initially ( relativistic time dilation. However, after his theory was completed in 1915, now containing also a spatial dilation, he predicted an overall light deflection of 1.75 seconds of arc. This value wasconfirmed then by observations during the solar eclipse of 1919. See e.g. H. v.“The Klueber, determination of Ein- stein’s light deflection in the gravitational field of the sun”, Vistas in Astronomy p. 47 (1960). C27.8. E 556 Part II 27.13 A Quantitative Treatment of Absorption 557

Dispersion curves can be explained in terms of forced oscillations: We assumed that electrical resonators were present in the interior of the molecules. Their eigenfrequencies 0 coincide with the frequencies of the maxima of the absorption bands. Given these circumstances, we are led almost inevitably to an explanation of the absorption process:Thedamping of the resonators consumes a part of the energy of the incident light and converts it into other forms of energy, e.g. into heat. Here, again, we offer a simple analogous example from mechanics: Aglass filled with white wine makes a clear ringing sound when “A glass filled with white clinked with another glass. The glasses and their contents are subject wine makes a clear ring- Part II to oscillations (standing waves). These are formed by the superposi- ing sound when clinked tion of travelling waves which are constantly reflected from the walls. withanotherglass...But A glass filled with champagne, in contrast, cannot be made to ring by a glass filled with cham- clinking it with another glass; champagne contains bubbles of gas. pagne cannot be made to They themselves also act as resonators: They are excited to forced ring by clinking”. oscillations by the waves. Their damping consumes the energy of the waves.

27.13 A Quantitative Treatment of Absorption

In Sect. 27.12, we gave a qualitative explanation of absorption. It is supported by the shape of single absorption bands, i.e. those that are well separated from neighboring bands. They often exhibit a noticeable similarity to the energy-resonance curves of forced oscillations (Vol. 1, Fig. 11.44). There, the ordinate is a measure of the kinetic energy stored in the resonator, or the average power consumed by the P damping, W . The quantitative treatment is closely related to that in Sect. 26.5.We again assume that the electric resonators are dipoles. We initially 0 make no assumptions about their nature. Their number density is NV. The incident light is again taken to be in the form of a collimated beam. The absorbing substance is in a dilute solution and the solvent has a refractive index of n. In a segment of the beam of length x and cross-sectional area A, 0 there are NVA x damped resonators. They give rise to an absorption constant P W 1 K D : (Deﬁning equation (25.1)) P x Wp

P Here, W is the power consumed by the resonators, and " WP D n 0 E2cA (27.12) p 2 0 .

0 : E 2 ,but

0 2 0

(27.17) (27.13) eE (27.14) (27.15) (27.16) 2 Ã

1 (liquids and . 2 Â

> together consume , since, precisely C ). , we ﬁnally obtain T W x : n 0 2 0 / ;

: T ; 2 2 A kin is the sum of the power 27.13

kin kin :

W W 2 W P D W H kin 2 ; 0 0 H

D T W = 4 . P t W 0

x H e 2 W w D 0 C27.9 E 0 ˛ m and

A 2 4 e 0 V D W of the free damped oscillation follows an 2 T / N H / = t t t Ã D :

˛. 2 D 2

˛.

P 4 e P W ), and using 0 W P D Â W W 0 is the average value of the kinetic energy stored

11.2 D D P is the exciting amplitude of the light for a single W of the exciting force was not set here to 2 0 kin

/ 0 w is the period of the harmonic oscillator (Vol. 1, Sect. 11.10). 0 ), we ﬁnd the time-averaged value of the kinetic W W l ): F E 1 ): (not shown here). This leads to Eq. ( 0 . 0

w 26.1 m D 4 1 eE T D Exercise 27.1 kin where (the power here is the value averaged over a period ( considered, the energy consumptionthe is frictional ‘pulsed’ force due that toTogether the causes with dependence the Eq. of damping ( on the momentary velocity). For the energy, it follows that Derivation: The amplitude In the steady state,also this holds when power the must oscillator is be excitedfrequency continually at a replaced. frequency above its This resonance result exponential law: is the halfwidth of the energy-resonance curve, as deﬁned in Vol. 1, and excites the resonators there. W Exercise 27.2 The amplitude rather to With Eq. ( resonator. In materials with a refractive index crystals), it is larger than the ﬁeld-strength amplitude in vacuum, H Fig. 11.42, and consumed by all of theof damped them resonators consumes together. the Each power single one All of the resonators contained in the volume is the power of the radiationA which passes through the cross-section in a resonator when it is oscillating at the frequency energy of one( oscillator which is vibrating at the frequency the power denoted by ; 2 vec- = 1 )fol- SA / 0 D ; 0 27.12 A 0 A ." / 2 0 B OYNTING n 0 E = D ), we have E c c n 0 0 c n )), Eq. ( is the time averaged =. 1 0 2 2 S E 8.7 Dispersion and Absorption D D D p p 27 0 P P lows. where B tor (Comment C12.4). In a dielectric material (refractive index and with W W value of the P (Eq. ( C27.9. Derivation: In vacuum, we have and thus 558 Part II 27.14 The Shapes of Absorption Bands 559 p Equation (13.32) holds here (n D "): E E D 0 .n2 C 2/: w 3 Combining these equations leads to the absorption constant

N0 e2 .n2 C 2/2 H 2 D V Â Ã : K 2 (27.18) 2c"0m 9n . 2 2/2 C 2 2 0 0

This equation gives information about the shapes of the optical ab- Part II sorption curves (Sect. 27.14). In addition, it offers us the possibility of determining the number density NV of the resonators from an optical measurement: It leads to a quantitative analysis method for absorption spectra, as discussed in Sect. 27.15. In both cases, we must keep an essential point in mind: In deriving Eq. (27.18), we did not take a mutual inﬂuence of the resonators on each other into account. For this reason, this equation and its various rearranged forms (Eq. (27.19)) can be applied only to the limiting case of dilute solutions or gases of moderate density (see the footnote in Sect. 27.10).

27.14 The Shapes of Absorption Bands

For a given solution, the first two fractional factors in Eq. (27.18) contain only constants. (The weak dependence of the refractive index n on the frequency can be neglected within the width of a band). Then we can choose the maximum value Kmax arbitrarily by fixing 0 NV and obtain the shape of the absorption band from the fractional expression on the right. Figure 27.18 shows two examples. The upper part of the figure refers to a solid solution of potassium atoms in a KBr crystal. In this system, a small fraction of the Br ions in the lattice has been replaced 6 by electrons . Their number density is NV. This gives rise to new absorbing centers for which the name color centers (F centers) has become established. The lower part of the figure shows data for a vapor solution of mercury in compressed hydrogen. It contains about 6 one Hg atom for each 6 10 H2 molecules. Note the different scales on the abscissas in Fig. 27.18. In the upper graph, the absorption curve is a broad band with a halfwidth of H D 14 1:21 10 Hz (Q-factor 0=H D 3:8). The lower graph, in contrast, shows a spectral line whose width is determined by thermal collisions

6 F centers. Considered chemically, a KC ion together with an electron forms a neutral potassium atom. H. Pick, “Struktur von Störstellen in Alkalihalogenid- kristallen”, Springer Tracts in Modern Physics (Springer-Verlag Berlin, Vol. 38, p. 1 (1965)). , , - ). 0 0 V 3 N = 10 H D 3 :

3 H ). The calcu- D –3 exponentially H –3 m OOS = 20 m 0

22 1.3 200 °C 34.5 °C Hz ( 1.4 ·10 5∙10 = =

at 30 atm

11 atoms dissolved 2 V

V N Hg atoms in H K in a KBr crystal N 10 resonators, combined with the 54 of the of light : ν * 3 to the observed maximum, yields * H H V of the of band maximum D 0 ), we obtain the absorption constant N ν H –4 Frequency Frequency 27.18 atmospheric pressure) 0.654 μm 0.254 μm 0.83 = 9.4 · 10 9.4 =

Frequency Frequency

= 0.7 0.8 0.9 1 1.1 1.2 1.4 0 =

λ λ Λ Λ . In such cases, one can consider Pa in Eq. ( 5 9 6 3 –1 –1 0 10 0.9996 0.9998 1 1.0002 1.0004 exponentially-damped

0.15 0.10 0.05

The representation of optical absorption bands as energy- Absorption Spectra cm cm

27.18 K constant Absorption D 013 : 1

at the band maximum. At the same time, we can solve for D max 1atm lated curves are fitted to the measurements at the band maxima. ( In both examples, the calculated curvesthe agree measurements. quite Thus, satisfactorily the with basic assumptionlation, underlying this that calcu- of a useful model forfor the every real absorption situation. band.culated This The and is, systematic measured however, deviations not bandsthan between the are in cal- case in Fig. most cases considerably greater damped resonators (e.g. electrons whichto are positive quasi-elastically charges, bound dipoles) onlyhas however as the an advantage that approximation. it is This intuitively model understandable. fitting of the number density resonance curves (lower part from measurements by G. J Setting 27.15 Quantitative Analysis Using Figure 27.18 (thermal broadening). It has K Dispersion and Absorption 27 560 Part II 27.15 Quantitative Analysis Using Absorption Spectra 561 remove the logarithmic decrement by using the approximate relation D H= 0 (for Ä 1), and find

2c" m 9n N0 D 0 K H : (27.19) V e2 .n2 C 2/2 max

0 We now compare the number density NV of the resonators as determined from the absorption band with the overall number density NV of absorbing centers (“molecules”). To do this, we deﬁne the ratio

N0 Number of resonators V D D f : Part II NV Number of molecules

This number f is called the “oscillator strength”. In Eq. (27.19), only constants precede the product KmaxH. We thus obtain for the number density of the absorbing molecules:

NV D const Kmax H (27.20) with 2c" m 9n const D 0 (27.21) f e2 .n2 C 2/2

(c D velocity of light; "0 D electric ﬁeld constant; m D mass and e D charge of the electron; n is the refractive index of the solvent at the frequency of the band maximum (KBr in the example in Fig. 27.18, top); f D oscillator strength).

This constant can thus be computed from physical constants, from the refractive index n of the solvent, and from the oscillator strength f . Equation (27.20) offers the possibility of determining NV, the number density of the absorbing molecules, from optical measurements. This procedure, corresponding to the derivations of Eqns. (27.19)and (27.20), is limited to the case that the resonators are independent and do not mutually inﬂuence each other (BEER’s law, see Sect. 25.4). In particular cases, one can set f D 1. Usually, however, the constant must be empirically evaluated by a calibration measurement with a relatively large number density NV which can be determined chemically. Spectral analysis using optical absorption is superior to chemical analysis in terms of sensitivity. We estimate the orders of magnitude: The constant in Eq. (27.20) has a magnitude of about 6 105 s/m2. At a layer thickness of 10 cm, absorption constants down to K D 0:01/cm can be measured (e0:1 D 0:9). The decisive factor is now the halfwidth H. For solid and liquid solvents, H is seldom less than 14 10 Hz. With these values for Kmax and H, we can optically deter- 20 3 14 3 mine number densities down to NV D 10 =m D 10 =cm .In solid and liquid substances, the number density of the molecules is of the order of 1028=m3. Therefore, we can optically detect one dissolved molecule among 108 molecules of a solid or liquid solvent. ”, kg, ponds 31 inﬂuence 10 when the energy of the were carried out at a pres- h . One refers for short = 8 Optik und Atomphysik W , the negative charge is as- 27.18 D is considerably smaller; values atories where mercury is used 0

H of air as in a mercury droplet with As. 3 electron. This model was found in the Pa. limiting case 19 . Then the absorption in a 10 cm thick 3 7 10 6 of such a dipole (resonator or oscillator) corres . : m (cf. 13th edition of “ C27.11 0 . Optically, we can already detect 1 % of this 1 W 3

and for the physical investigation of photochemical ˙ 2537 : of Optically-Effective Resonators Pa. 0 5 C27.10 10 D . Such a value of the number density corresponds to a vapor Hz are not unusual 3 10 quasi-elastically bound /m 16 16 Pa. In poorly-ventilated labor : Note that the measurements on mercury in Fig. The eigenfrequency sumed to carry only the small mass of one electron, i.e. 9 and all of thethe remaining positive large charge. massas Then of a positive the the ion, molecule moleculetions and remains is the of attached practically dipole the is to at electron created around rest only its through rest the position oscilla- to a molecule changes by At room temperature,0 mercury has a saturation vapor pressure of The mass of thethe molecule two charge can carriers. be In distributed a in various ways over a volume of 1 mm amount. Forlength the absorption measurements, one uses the wave- Fig. 14.19). Absorption spectral analysis hassolid substances, also for example been to detect employedof and vitamins, in measure the liquid concentration and sample suffices to detect molecules10 with a number density of about reactions in crystals. sure of 30 of 10 in quantum mechanics to the frequency in molecules; they arecenters electrically of “deformed” gravity or ofrelative to “polarized”: their one positive another. The This and periodiction variation negative of is charges the usually charge are represented distribu- shifted bydipole. the Two schematic elementary picture charges of ofat an its opposite ends, oscillating sign that are is assumed to be 7 8 We have seen thatsorption the in classical terms interpretation of ofreproduce dispersion the the and observations forced to ab- oscillationswant a of to good resonators supplement approximation. is thisvarious We able model types therefore of by to resonators giving that may some play information aLight, role. on just the like an alternating electric field, gives rise to 27.16 The Properties In gases and vapors, the halfwidth without sufficient protection,vapor there molecules can in be each just 1 m as many mercury pressure of the order of 10 . Natur- , 433 27.14 15 Zeitschrift für Physik Dispersion and Absorption , 603 (1930); see also the 27 C27.11. A. Smakula (Dr. rer. nat., Göttingen, 1927), 59 footnote in Sect. C27.10. R.W. Pohl, wissenschaften (1927); A. Windaus, Nobel Lectures, Chemistry, 1927 (Elsevier Amsterdam, 1966), p. 105. 562 Part II 27.16 The Properties of Optically-Effective Resonators 563

Table 27.3 Velocity of longitudinal sound waves, lattice constant and frequency of the residual-ray bands of some crystals Crystal Sound velocity Spacing D of neighboring lattice Frequency of the residual-ray band in m/s components (positive alkali ion and Calculated from Observed negative halogen ion) Eq. (27.22) NaCl 3:3 103 2:81 1010 m 5:9 1012 Hz 5:8 1012 Hz KCl 3.1 3.14 4.9 4.7 KBr 2.3 3.29 3.5 3.6 KI 1.95 3.52 2.8 2.7 Part II preceding sections to give good results both for visible light as well as for ultraviolet light and X-rays. The situation is different in the infrared spectral region: There, we encountered the absorption bands belonging to the residual rays.They were observed for cubic ionic crystals (Fig. 27.9). A platelet made from one of these crystals can be thinned at most to the spacing D of two neighboring lattice components, for example a NaC and a Cl ion in NaCl. Such a (limiting-case) platelet of thickness D would have a mechanical eigenfrequency of u D : (27.22) 0 2D Here, u is the velocity of longitudinal sound waves in the crystal. This mechanical frequency agrees with the optical frequency of the residual rays. This is shown by the numbers9 in Table 27.3. Thus, for the case of ‘residual rays’, we can calculate an optical frequency from data obtained by non-optical measurements. This is the fundamental importance of this relation, which was discovered in 1908 by E. MADELUNG. This fact at the same time leads to information about the type of resonators that give rise to residual rays: Both of the elementary charges are bound to the large masses of ions. These ions, e.g. NaC and Cl, vibrate oppositely to each other and thus form an oscillating dipole. Here, the picture of a dipole is already more than just a model scheme. In the simplest ionic crystals, of the type NaCl, the molecules have lost every trace of an individual existence. This is however a limiting case. In many other types of crystals, whole molecules or parts of them maintain their own identities. In such independent molecules, and also in molecular crystals, pairwise oppositely-charged molecular components can form dipoles and can absorb infrared light

9 The oscillation period T D 1= for the fundamental mechanical vibration of arodisD 2D=u. This means that a longitudinal elastic perturbation passes along the entire length D of the rod twice during the time T, namely going forward and then returning. The velocity of sound within a solid body is nearly always quoted as the special case of its value along the length of a rod, without further comment (cf. Vol. 1, Fig. 12.42 and Eq. (11.5)). In Eq. (27.22), however, a valid average value for a three-dimensional body must be used. 2 and ). ). those 3 -- 2 AMAN 3 NO 13.11 8.0 μm 2 8μm -- 3 group. They lie NO 2 7.3 μm and KNO ions. For the right-hand were employed. For the ”, Chap. 15, ‘R molecules have no im- 2 2 7μm Thin films KNO of =

λ polar Arbitrary units Arbitrary are in both cases at practically and NO 2 3 and KNO oducing absorption bands (only 3 of many possible examples can be ions were replaced in part by NO -- 2 ed molecular components. Only oscillation frequencies which are char- group and to the NO and NO NO 3 3 8.00 μm inner Optik und Atomphysik m. The right-hand image refers to KNO -- 3 ions. In spite of the different crystal structures, . The two parts of the ﬁgure show absorption NO 7.9 μm www.tsi.com/basics-of-raman-spectroscopy/ 3 electric dipole moments of 7.2 μm in a KBr crystal 27.19 Anions dissolved Anions 7μm Absorption spectra of NO = λ

–1

crystals, and the left-hand image to a solid solution of these permanent 0.8 0.6 0.4 0.2 K constant Absorption 2 mm portance for the absorptionTheir and role dispersion begins in only theThere, in optical the liquids spectral range with regions. oflarge electric dipolar refractive waves indices. molecules (radio, A can well-known microwaves). example exhibit is water strong (cf. Sect. absorption and The KNO at around 7.2 and 8.0 acteristic of the individual molecules.to avoid We an must error: howevercan Large be molecules careful have built many up eigenfrequencies fromonly (compare many some components Vol. fraction 1, ofcillations Sect. those of 11.5), frequencies electrically-charg are but associated with the os- oscillations can bethey effective are in “optically active”). pr frequencies The must optical be detectionin carried of the out the 13th remaining by edition a of “ different method (described the same positions. Thus,us to the the absorption recognition of of infrared radiation leads graph, thin crystalline layers of KNO left-hand graph, a solid solutionconcentration of was the ca. ions 0.1 % inIn in a the the KBr crystal, melt crystal the from was concentration which used. is the roughly crystal The ten was times prepared. lower than in the melt. the absorption bands of NO compounds in a KBr crystal. Ingrown, this in second case, which mixed individual crystals were Br scattering’, or e.g. through forced oscillations. Two found in Fig. andinpartbyNO Figure 27.19 bands belonging to the NO Dispersion and Absorption 27 564 Part II 27.17 The Mechanism of Light Absorption in Metals 565 27.17 The Mechanism of Light Absorption in Metals

The absorption spectra of metals show a special feature: In all the non-metallic substances, the bands from “bound” electrons are followed by an absorption-free region (Fig. 27.1). Only at longer wavelengths, in the infrared region, do we begin to see absorption due to ions. In metals, in contrast, an additional absorption is observed, beginning in the ultraviolet, which initially increases in strength with increasing wavelength. Usually, it overlaps with the longest- Part II wavelength bands that result from bound electrons (Fig. 27.6). There is thus no absorption-free region in the spectrum, and the absorption constant in the infrared attains values of the order of 105 mm1. This additional absorption, which is lacking in all other materials but is seen in metals, is due to their electrical conductivity ; it thus has its origin from “free” or “conduction” electrons. At >10 m, practically the only absorption is that due to free electrons. There, just as in the range of electric waves (radio, microwaves), we can calculate the absorption from the conductivity . The magnetic ﬁeld of the penetrating waves generates eddy currents which convert the energy of the waves into heat. The relations derived for electromagnetic waves apply, namely s 1 p p n D k D D 5:47 ohm (27.23) 4"0 and s r 4 p K D D 68:8 ohm (27.24) "0c

(here, n is the refractive index, k the absorption coefficient, defined by Eq. (25.3), K is the absorption constant, defined by Eq. (25.1), is the vacuum wavelength, the specific electrical conductivity, and "0 is the electric field constant, 8:86 1012 As/Vm).

Derivation: The third MAXWELL equation (see Sect. 14.5) states that

curl H D j C DP : (27.25)

P P Here, D D ""0E is the displacement current density and j D E is the conduction current density. For an undamped travelling electromagnetic wave of amplitude E0, we ﬁnd from Eq. (25.26)that

iEP E D E i!.tzn=c/; P D ! D : 0e and thus E i E or E !

Inserting this into Eq. (27.25) yields Â Ã P i curl H D "0E " : "0! m, 04 : (27.28) (27.27) 1 10 (27.26) relation D )) for spec- D : : ), we obtain as 2 27.7 ! ). For mercury, 0 i

" ’s formula for the .At AXWELL ; 1 27.6 : . . We obtain / 25.37 . ! 1 " 0 0 m EER i ik m "

causes only a (usually 1 0 D " 2 ,B k mm 2 n p k k 366 4 D 6 : D ) was based on the following ˝ 0 10 0 1 10 12 nik n and p nik 2 = 2 : 2 Free Electrons n (compare Fig. V 2 62 1 27.28 N 2 1 D 2 D e n . K D mm ’s rule: . It is linked via the M 5 )and( k 2 ! i.e. 2 0 10 0 0and

6, and : 7 ; D RUDE 27.7 : ) follows. =" 1 i 1 2 17 ! k m 0 i D V 0 D D " : For silver, " " 27.23 N k R 2 2 K 2 n e D " D , obtaining 4 2 0 n D " n , ). can be simplified. Instead of Eq. ( 1 2 with Weak Absorption (Plasma Oscillations) C , we derived the dispersion formula (Eq. ( / 0. Finally, we can equate the real and the imaginary terms in to the complex refractive index ) individually, so that R 136 and 1 0 ik m " D 27.8 1 )for D D 27.8 " D n k 27.26 2 . 2 / n 6 0 D 27.7 n Eq. ( n The expression in parentheses canstant, be regarded that as a is complex dielectric con- and from this, Eq. ( Numerical examples In metals (anddisplacement sometimes current; also that in is,stant semiconductors), we set we the can real neglect part the of the dielectric con- a poorly-conducting metal, the corresponding numbers are: 10 . considerations: 1. In acharged neutral “remainder” molecule, one of negativeelastic the electron oscillations with and molecule an the eigenfrequency can positively- of undergo mutual quasi- The derivation of Eqns. ( reflectivity 2. This oscillatory structurecident is light excited wave. to forced vibrations by an in- small) deviation of the reflectivitypare from Fig. the “ideal” value of 1 (com- In this formula, the absorption coefficient a good approximation D In Sect. 27.18 Dispersion by With such large and equal values of tral regions in whichEq. the ( absorption can be neglected. We can solve Dispersion and Absorption 27 566 Part II 27.18 Dispersion by Free Electrons with Weak Absorption (Plasma Oscillations) 567

3. With closely-packed molecules (in liquids and solids), the amplitude of these forced oscillations depends not only on the amplitude of the incident light waves, but also on the electric dipole moments p which are produced in the neighboring molecules under the inﬂuence of the light waves. In this derivation, it was left completely open as to how the quasi- elastic oscillations with an eigenfrequency of 0 were produced. One can treat them as a circular oscillation, in which the electron orbits on a circular path around the positive charge. The required radial !2 acceleration 0 r is produced by the attractive force of the positive charge e (COULOMB’s law, Eq. (3.8)). We then ﬁnd Part II

e2 !2r D (27.29) 0 2 4"0r m

12 19 ("0 is the electric ﬁeld constant D 8:8610 As/Vm;e D 1:610 As; m is the mass of the electron D 9:11 1031 kg).

A circular orbit of radius r requires a volume

4 e2 V D r3 D : (27.30) " !2 3 3 0m 0 This volume V necessary for the circular orbit cannot become larger than 1=NV, i.e. the reciprocal of the number density NV of the molecules. Then for the largest possible radius, we ﬁnd 3 3 D : rmax (27.31) 4NV At this magnitude of the radius, the circular frequency of the orbiting electron has its smallest value, !0;min.Itisgivenby

e2N !2 D V : C27.12. The derivation of 0;min (27.32) 3"0m the plasma frequency,i.e.the frequency of oscillations of equal to a cloud of electrons within 2 the lattice of positively- 2 e NV ; D : (27.33) charged ions (as in a metal), 0 min 122" m 0 can be found in Feynman’s Below this limiting frequency, the electron is free. It can no longer be Lectures on Physics,Vol.II, associated with a particular positive charge carrier or ion. We are then Sect. 7.3 (available online, no longer dealing with oscillations within a single neutral molecule, see Comments C6.1. and but rather with oscillations of a whole set of electrons relative to all C7.1., and the solution to of the positive counter-ions, that is with plasma oscillations. 0;min is Exercise 26.1.), or also in the eigenfrequency of a (transversally) oscillating plasma.C27.12 We F.S. Crawford, Waves,Berke- insert it in place of 0 into Eq. (27.28) and obtain the dispersion for- ley Physics Course, Vol. 3 mula of a plasma for the region of weakly-absorbed waves: (McGraw Hill, New York 1968), p. 87. It is given by 2 3 e2N 2 e NV m NV !2 D V : n D 1 D 1 80:6 : (27.34) p 2 2 2 2 " m 4 "0m s 0 . Hz 6 27.4 travel- 10 (27.35) electric 3 ;no 32, and thus : D 10 0 that the reﬂectiv- ;

n , i.e. the refrac- 1 D 2 n n m 15 10 )gives layer) are of great impor- ). ; it seldom occurs, and then 3 3 12 : m 1 /m = 27.34 12 28 > ) gives a refractive index which (which occurs only in solids and 2 10 10 , and is thus very small compared to 8 3 V EAVISIDE : 4 Hz). However, in the region of : N 1 27.34 /m 8 m/s. For -H 30 m. The necessary number density of 12 8 > ). For an electron number density of 11 D or V -10 10 ) for an imaginary refractive index 10 100 m), Eq. ( N Cu 15 < ; produced by these free electrons can be cal- 4 3 27.34 V : D D 2 ENELLY s N 1 (numerator and denominator have the same magnitude). 25.15 m V 6 N D , equation ( : ) even yields negative values for metallic bonding 3 2 ) cannot be applied to metals due to their strong light / i 80 Electromagnetic Waves by Free Electrons in the Atmosphere /m !) leads to the formation of the best-known plasma: E = < V 11 r 27.34 E N 27.34 2 . V 10

N (‘radio’ waves), the situation is different: At D 30 m makes it possible for short-wave electromagnetic waves refractive index wave can penetrate into the ionized layer. Making use of this D R It follows from Eq. ( V the electrons would be (corresponding to equation ( tive index becomes imaginary. Thenperpendicular even to waves the which layer are are incident subject to total reflection usually at altitudes of around 250 km. The free electrons in thesphere’, upper e.g. layers the of the K atmosphere (the ‘iono- tance for communications bybands. radio They reflect in the electromagnetic the waves andcurved medium- guide paths) them and (along to long-wave their< distant targets. The lackthat of are total reflection emitted for by the sun and more distant astronomical objects A cloud of freelyBut mobile Eq. electrons ( within a lattice of positive ions. absorption. Electrons can however alsoclosely-packed be atoms, released for without example thetions. by the interactions Thus, of effects especially oftrons through ionizing the are radia- action set of freesphere’). ultraviolet in Their light, the number elec- density upper at anof layers altitude magnitude of of of 100 the km is atmosphere of (the the order ‘iono- culated using Eq. ( The that in metals (e.g. waves “Echoes” are rare for ling total reflection, we can determine theat number density various of the altitudes. electrons A numerical example is given in Table 10 27.19 The Total Reflection of In the case of ity is N is very nearly equalfrared to light (around 1 10 for the frequency range of visible and in- a phase velocity of 9 in liquids), the strongvalue interactions of of closely-packed atoms (a high Dispersion and Absorption 27 568 Part II 27.20 Extinction by Small Particles of Strongly-Absorbing Materials 569

Table 27.4 The total reﬂection of electromagnetic waves in the atmosphere A signal of wavelength D 125 m 102 m will, according to Eq. (27.35), be totally 0:7 1011/m3 1:1 1011/m3 reﬂected at a number density of electrons NV D Its transit time t for outgoing and return 6:33 104 s 1 103 s paths is measured to be

The number density NV leading to total 95 km 150 km reﬂection thus lies at the altitude Hr D 1 D 2 tc Part II to reach the surface of the earth. They can be captured with large parabolic antennas (the concave ‘mirrors’ of radio telescopes). Radio astronomy has made many important contributions to our knowledge of the galaxy and the cosmos, e.g. the detection of the spiral structure of the Milky Way galaxy.

27.20 Extinction by Small Particles of Strongly-Absorbing Materials

In the cases described thus far, we have been able to treat separately the two contributions to extinction, i.e. scattering and absorption; the former in Chap. 26, and the latter in the present chapter. How- ever, this separation is not always possible. This is the case for example with extinction by small particles which consist of strongly- absorbing materials. Organic dyes and metals exhibit strong absorption even in the visible spectral region. When finely divided, they have quite different extinction spectra than as bulk materials. A long-known example is ruby glass. It contains very finely-divided gold particles; however, it does not allow green light to pass through, like very thin gold leaf, but instead red light (Fig. 27.20). The diameter of the individual gold particles is below the resolving power of light microscopes, but each particle produces a colored diffraction disk under dark-field illumination in the viewing field of a microscope. Light is thus scattered from each particle11. The proportions of scattering and absorption are found from experience to depend strongly on the size of the particles: Very small particles scatter only weakly; they attenuate the light mainly by absorption. For a quantitative investigation, a solid solution of sodium in an NaCl crystal is suitable. A hot NaCl crystal in Na vapor takes up excess Na atoms. The mechanism of this process is known: A small fraction of the negative chlorine ions in the lattice is displaced (and replaced) by thermally-diffusing electrons. The resulting absorption centers are

11 This detection of single particles is called “ultramicroscopy”. . m μ Hz Hz 14 14 4·10 particles absorption 5 .Inthermal Finest colloid 3 + 20° would be found /m 3 22 of the of light /m ν 10 6 of the of light 11 0.5 0.6 0.7 ). Therefore, one has ν 5 – 254° C, on the other hand, the 10 3 ı Wavelength λ /m D 3 Wavelength λ 27.15 22 + 20° Gold ruby glass Atomic V Frequency D 7 ). In equilibrium, the number 7 6 5 5·10 N (color centers) Frequency Frequency ensities cannot be detected even V 4·10 N =

0.4 0.45 0.5 0.6 0.7 μm 0.4

2 1 –1 V

27.14 K Extinction constant constant Extinction Na in NaCl N mm 8

–1

1.5 1.0 0.5 2.0 K constant Extinction mm is observable. (see Sect. C, for example, it is ı , left, shows the extinction spectrum of such a “frozen The The extinction of Na atoms in the crystal is about the same as in the scattering V ). N 27.21 color centers 27.37 in the crystal. Suchby small absorption-spectrum number d analysis (Sect. equilibrium at room temperature, diffusion rate has becomelattice measurably can large. precipitate As a a portion result, of the the crystal excess sodium, allowing it to extinction spectra of atomic and colloidally-dissolved metal particles (Na in an NaCl crystal). The dashed curve for the finest colloid, which shows no scattering, was calculated using Eq. ( spectrum of gold ruby glass to “quench” the crystalat and “freeze a high in” temperature, theFigure number so density that obtained it remains stable at room temperature. Figure 27.20 density called Figure 27.21 vapor; at 500 in” atomic solid solution oftwo Na temperatures. (color The centers) extinction in is an due NaCl crystal, here at entirely to No trace of At room temperature,crystal can the be maintained frozen-in for years. number At 300 density in an NaCl- Dispersion and Absorption 27 570 Part II 27.20 Extinction by Small Particles of Strongly-Absorbing Materials 571 condense into fine colloidal particles. This lowers the color-center band; at the same time, a new extinction band with a maximum at 0.550 m appears (at the right in the figure). The extinction in this band is also practically due only to absorption and not to scattering. Its position remains nearly unchanged with changing temperature, in contrast to that of the color-center band. Upon longer heating, the particles increase in size, and their extinction bands are shifted and extended to the region of longer wavelengths. Only then does scattering begin to play a role in the crystal, initially weakly and then more strongly.

The maximum of the new band (measured at room temperature) al- Part II ways lies at least 0.08 m to the long-wavelength side of the maximum of the color-center band. There is thus no gradual transition of the color-center band by a continuous shift into the new band. As a result, we must ascribe the new band to the smallest colloidal particles that remain stable. For the atomically-dissolved metals in alkali halide crystals (color centers), the shape of the band can be represented by the model of damped resonators (Fig. 27.18). The position of the band is determined by the lattice constant a of the crystals (cf. the footnote at the end of Chap. 22). The frequency of its maximum at 20ıCisgivenby an empirical relation12:

2 4 2 max a D 2:02 10 m =s : (27.36)

For the colloidally-dissolved metals, in contrast, the shape and position of the band are determined by the optical constants of the metals (indeed, by the values of n and k measured on bulk samples), and not by damped resonators. With these constants, we can calculate the absorption constant K for the smallest colloidal particles (diameter ) at various wavelengths. The following equation is used for the calculation: nk 1 n K D 36N V " # u" # V Â Ã Â Ã 2 Â Ã Â Ã n 2 k 2 n 2 k 2 C C4 C 1 nu nu nu nu (27.37)

(Here, nu is the refractive index of the solvent, the wavelength in air, NV the number density of the particles, and V is the volume of the individual particles. For a derivation, see earlier editions of this bookC27.13). C27.13. See the 9th edition (1954), or the 10th edition For our ‘standard example’, the smallest Na colloid in an NaCl crys- (1958) of “Optik und Atom- tal, the optical constants of sodium are collected in Fig. 27.22 (top). physik”, p. 206. nu, the refractive index of the surroundings, that is of the NaCl crystal, is practically constant at 1:55 (Fig. 27.1, bottom). There are no

12 E. Mollwo (Dr. rer. nat. Göttingen, 1933), Zeitschrift für Physik 85, 56 (1933).

k coefficient Absorption 0.50 2.5 2.0 1.5 1.0 0.5 0 0.6 μm ) do we ob- . 0.5 k k are hardly tem- K Na k and k n are not “optical resonance 0.4 and in each case. The re- Wavelength λ n 13 n 27.21 n ). . This leads us to the dashed 0.3 0 0 0.25

ttering in addition to absorption. 27.37

0.5 1.0

0.05 0.04 0.03 0.02 0.01 0.750.25 0.5 n index Refractive ; thus, we just compute the right-hand prod- 1 : V m, 0 at bout < n and 31 : . Its maximum value is fitted to the observed value k V 0 ) for various values of N is still a Colloids. Artificial Dichroism, Artificial Birefringence < 27.21 The optical K . For rubidium 1 27.37 m. ); but the extinc- mm 25 25.5 : 3 0 10 The extinction curves of the fine colloids in Fig. maining discrepancies are notby small alarming. changes in They the interpolation could curves for be eliminated < and cesium, the curves are similar to those for potassium. Therefore, for sodium also, we can expect a steepin rise the refractive index 2 tion constant constants of sodium and potassium. At potassium has a region of weak extinction, i.e. Figure 27.22 uct in Eq. ( curve in Fig. by choosing the constant in Eq. ( serve secondary radiation and sca In this case, the individual parts of the colloid particles are not all curves”; their shapes are insteadof determined the by material the of the curves particles. of the optical constants 27.21 Extinction by Large Metal For very fine metal andobserved, dye only colloids, absorption. no Only secondaryparticles when radiation the (diameters can colloids or be consist circumferences of comparable larger to 13 reliable data for (Sect. perature dependent, and theResult: same The holds calculation can for correctly the reproducetures the computed of two essential function. light fea- extinction by verythe fine small colloidal widths metal particles, of namely dence. the Furthermore, bands the and respective frequency their ofnearly each lack coincides band of maximum with temperature depen- the measured value Dispersion and Absorption 27 572 Part II 27.21 Extinction by Large Metal Colloids. Artificial Dichroism, Artificial Birefringence 573

Figure 27.23 Wavelength λ Pressure-induced 0.45 0.50 0.60 0.70 μm dichroism in NaCl mm–1 xz containing colloidal 1.5 Na particles Pressure y

K Light

1.0

E║ E┴ to the direction 0.5 of the pressure Part II Extinction constant constant Extinction

7 6 5 4·1014 Hz Frequency ν of the light excited with the same phase by the primary waves. As a result, interference occurs, giving preferred directions to the secondary waves; in particular in the direction of the primary waves, so that forward scattering predominates (compare Fig. 26.10). We can thus no longer assume a model employing the simple electric polarization of small spheres for a quantitative description of this process. Instead, we must use a similar description as in the calculation of the harmonics of antennas. In this calculation, the essential quantity is the shape of the particles; but precisely this is usually unknown for large colloidal particles. We of course cannot explore these complicated matters in detail here; rather, we must content ourselves with a qualitative treatment of ar- tiﬁcial dichroism (Sect. 24.3). As an illustration, we use a large- diameter Na colloid in an NaCl crystal. The crystal appears violet in transmitted light, and yellow-brown in reﬂected light. Its broad extinction band has a maximum around 0.59 m, independent of the orientation of the plane of oscillation of polarized light. Now we subject the crystal to pressure parallel to one of its cubic edges. The result is that it becomes dichroitic, i.e. it now exhibits two overlapping extinction bands in polarized light (Fig. 27.23). Ex- planation: The pressure causes the colloidal particles to take on an elongated shape (inset in Fig. 27.23). In the case of E?, the amplitude oscillates parallel to the longer particle axis x, while in the case of Ek, it oscillates parallel to the shortest axis y.ForE?, the long diameter of the particles preferentially determines the wavelength, while for Ek, the shortest diameter is determining. All birefringent substances are dichroitic; this follows inevitably from the general relation between dispersion and absorption. This connection is represented schematically in Fig. 27.24. The full curves refer to one of the two polarized oscillations, and the dashed curves ) 5 O 2 24.16 2 is replaced n 0.6 0.8 μm in 1875. Experi- 24.16 1 n ERR in Fig. 2 k G Wavelength λ 1 k ). Therefore, we cannot detect . 24.9 V/cm are employed. This form of

0.2 0.4 4 2 1

k coefficient Extinction

27.25

, one finds a difference of the refractive n index Refractive . Still more impressive is the demonstration G on the order of 10 E experiment described in Fig. just like a crystal plate in water between two glassother plates and by slide several the millimeters. platesbirefringent relative Immediately, to (“flow the each birefringence”). liquid layer It becomes acts in the setup of Fig. by a parallel-plate condenser filled withdirection one is of these adjusted liquids, to the be field strengths perpendicular to the light beam, and field Artificial birefringence can also beas produced well using polar as molecules, nonpolarformed by molecules electric which fields. however Theand can best-known carbon examples be disulfide. are nitrobenzene strongly The crystal de- platelet mentally, at a wavelength artificial birefringence was discovered by J. K Figure 27.24 to the other, which is perpendicularstances (calcite, to the mica, first. quartz), both Withthe transparent absorption ultraviolet spectra sub- spectral end region, already before in the beginning ofThe the fabrication of visible. very thin birefringentficult. crystal Therefore, the platelets relevant absorption is bands rather forbeen birefringence dif- have measured only in a fewcentration cases. of With the artificial light-absorbing dichroism, particles is thebother con- low, so with that one very need thin not that crystal now, samples. the birefringence Itsand produced disadvantage by furthermore is the it however particles isof is the seen quite strained on solid small, solvent the (Sect. background of the birefringence A schematic sketch illustrating the dichroism of all birefringent substances the birefringence from parallel-oriented, elongated particles withtainty cer- by simple means.cases. That can however be accomplished in other A parallel orientation ofmerous small ways particles even for can large also particle-numberby be densities; using among achieved electric others, in fields nu- can or place laminar a flows few drops in of fluids. a suspension of For vanadium example, pentoxide (V we Dispersion and Absorption 27 574 Part II 27.21 Extinction by Large Metal Colloids. Artificial Dichroism, Artificial Birefringence 575

Figure 27.25 A demonstration experiment showing flow birefringence (photographic positive). A glass cuvette about 4 cm deep and 1 cm thick is filled Part II with a suspension of V2O5 in water and observed between crossed NICOL prisms (Sect. 24.3;seeFig.24.16). When a glass rod is dipped into the liquid, the layers which flow around it glow with a bright red color. Similarly, when the liquid is stirred, the turbulence becomes visible; and likewise, in a tube with a laminar flow, the boundary layer at rest on the walls of the tube can be seen. indices for the extraordinary and the ordinary light beams equal to

2 neo no D B E : (27.38)

In this equation, the “electric KERR constant” B is given by

n n 1 1 B D eo o D ; (27.39) E2 l E2 when the light traverses a path l in the electric ﬁeld and thereby experiences an optical path length difference of D .neo no/ l.

Explanation: The molecules which exhibit a KERR effect have an asymmetric structure. They have a preferred direction of polarizability. The dipole moments produced by the field are proportional to the field strength E. Furthermore, the polarized molecules are progres- sively oriented as the field is increased, gradually overcoming their random thermal motions. Therefore, the birefringence increases proportionally to E2.

4:3 1010 Numerical example: For very pure nitrobenzene, B D . cm.V=cm/2 Then, for l D 1cm and E D 104 V/cm: D B l E2 D 4:3 1010 1cm 108.V=cm/2 D 4:3 102 or 0:04 . cm.V=cm/2 The KERR effect is employed technically for constructing control devices (‘switches’) for light. The radiant power transmitted by the device increases initially as E4. 0 l https:// vibrates at and the log- m H ) for a linear mass- 27.16 . Find the mean value of , as can be easily shown 0

l 4 0

2 ). Also, take into account that 0

) can be approximated as )). ) contains supplementary material, which The online version of this chapter ( . =/ 26.1 oportional to the square of the ampli-

C 27.17 2 / . Its amplitude is in Eq. ( (Eq. ( ! whichisusedinEq.( 1), and thus a narrow resonance curve. In this 0 0 l kin

/. W 2 0

< 2 . ) ) q m 2 Derive the relationship between the halfwidth A linear mass-and-spring pendulum with a mass 27.13 27.13 4 Ä = 0 (for a narrow resonance curve, F the energy of the oscillator istude. pr (Sect. its kinetic energy (Sect. arithmic decrement Electronic supplementary material doi.org/10.1007/978-3-319-50269-4_27 is available to authorized users. Exercises 27.1 27.2 case, the amplitude and-spring pendulum (Fig.weak 4.13 damping in ( Vol. 1). Consider the case of the circular frequency Dispersion and Absorption 27 576 Part II ahohri iia ools B colorless similar beside in objects equal-sized other but each different object reflecting heat strongly we a this, demonstrate or To transparent a than radiation thermal 4. hot’. yellow, ‘white glowing glowing red, finally glowing infrared then invisible heat; First as only sequence: perceptible the radiation, demonstrate can changes we also wire, spectrum metal thermal the in tensity 3. hra Radiation Thermal hroee,wihi lcda rdaigeitr tadistance a at emitter” “radiating a 0 around as of placed is which thermometer, a ature radia- 2. ‘cool as known in (jokingly and ice cooling warming, with indicate tion’). indicate filled will will beaker it radiometer a second, the the then case, the of finger, first focus warm the the a In At hold water. thermopile). first radiation can mirror, a one we ( of other, point radiometer focal a the At place meters. we several thermal of avoid distance first a at must other we radiation, via heat conductivity of warmed transfer are this objects strate cooler while cooled, thus 1. briefly: summarized be can results experimental fundamental The Results Empirical Basic 28.2 and molecules of excitation for possibilities atoms, various the Among Remark Preliminary 28.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © a aito a n10,adwt ttedsoeyo h physical the of discovery the it with and constant 1900, in law radiation mal fti okwstefruainaddrvto fP of derivation and formulation the was achievement work crowning this The of investigated. extensively been has tion”) thermally-excite Therefore, times. al ovsbelgtadeisol eky o aeo colored practi- of absorbs made glass rod A clear weakly. of only made emits and rod light A visible no emit: cally they that light the l bet ait nrytwrsec te.Wre bet are objects Warmer other. each towards energy radiate objects All tagvntmeaue nojc hc bob ih mt more emits light absorbs which object an temperature, given a At h ain nest nrae togywt nraigtemper- increasing with strongly increases intensity radiant The stetmeaueicess the increases, temperature the As hscnb eosrtduiga lcrccoigptwith pot cooking electric an using demonstrated be can This . hra excitation thermal h ,P : rmardoee htsre s“receiver”. as serves that radiometer a from m 5 ti xein ouetocnaemrosfcn each facing mirrors concave two use to expedient is It . LANCK scntn rP or constant ’s olsItouto oPhysics to Introduction Pohl’s a lydaseilrl ic ancient since role special a played has UNSEN aito o tmeaueradia- “temperature (or radiation d LANCK distribution bre ae n observe and flames -burner sqatmo action. of quantum ’s ihasol heated slowly a With . nodrt demon- to order In . , https://doi.org/10.1007/978-3-319-50269-4_28 fterdatin- radiant the of LANCK sther- ’s . 28 577

Part II , . M inci- absorbs less than scatters emits the two objects. absorb visible light between -burner flame in the well- , the objects 1 and 2 represent UNSEN 28.2 por is placed in front of the condenser ’s Law ). The innumerable fine carbon particles (soot) ’s law. We explain its content by considering 28.1 IRCHHOFF A brightly glowing, strongly turbulent IRCHHOFF towards 2, and in addition it reflects the non-absorbed fraction 1 P known manner; i.e. allThen the we carbon no is longer burnedlonger and absorbs see visible no an light. soot image At isishes. the of formed. same the time, A its flame flame emissionvisible on also which light. van- the A can candle screen, absorb flame likewise it nojector. produces a no In visible dark general, image light in thermal also ais emission pro- cannot based of on emit incandescent the light presence of by solid a particles flame which namely soot particles. Figure 28.1 flame from natural gas containingcasts benzene a vapor deep dark shadowcondenser when of placed a in projector front of the W a thought experiment.small In sections Fig. of twoferent, large, flat arbitrary objects. materials.towards They the are other, Each made and of ofture. in two them The equilibrium dif- radiated they radiates power are which thermalreflected is at completely emitted energy the and from same without their losses tempera- backThus by surfaces we is the need two consider idealIn only mirrors a the steady radiation state, objectobject 1 2 must as radiate object just 2 the receives. same Object radiant 1 power to radiates its own radiant power 28.3 K Quantitatively, the experimental factsby listed above K can be described glass absorbs part of the visibleglass spectrum tube, and emits filled strongly. A with clear finely-powdered colored glass, the screen (Fig. which float in thethe gases light from of the the projector lamp. flameconvert Now, by absorb the increasing a flame the air noticeable to flow, we portion a of colorless B dent light. Only ainterior of small the fraction tube of andless the be than absorbed light the there. can solid The penetratethe colored powder into rod. thus rod, the and therefore, it also Another example: A brightlygas glowing which flame contains benzene of va “carbureted”of natural a projector: A deep, dark image of the flame then appears on Thermal Radiation 28 578 Part II 28.3 KIRCHHOFF’s Law 579

Figure 28.2 Illustrating KIRCHHOFF’s law M M

P .1 A1/ of the power W2 that it receives from 2. Here, A1 is the absorbance of the object for non-monochromatic radiation, deﬁned Part II by the equation Absorbed radiant power Absorbance A D : (28.1) Incident radiant power The corresponding conclusions hold conversely for the radiation emitted by 2 and transported to 1. Therefore, in equilibrium, we have P P P P W1 C .1 A1/W2 D W2 C .1 A2/W1 ; that is P P W1 D W2 ; A1 A2 and, in general, .L / .L / e 1 D e 2 ; (28.2) A1 A2 where Le denotes the radiance (Sect. 19.2). This relation holds for any two arbitrary bodies. Therefore, the quantity Le=A must be independent of all materials properties. It can depend only on other quantities, such as the temperature or the wavelength of the radiation. This statement is KIRCHHOFF’s law. Continuing our thought experiment, we suppose that between the objects 1 and 2 there is now an absorption-free interference ﬁlter (Sect. 20.13) which allows only a very narrow interval of wavelengths to pass through. The radiance is given by Z1 @L L D e e @ d 0

.@Le=@ is the spectral radiance/; so that for the selected wavelength interval [ ! C](whichwe denote simply by its central “wavelength ”), we have ZC @Le L ; D I e @ d 1 / ; e . L (28.3) (28.4) . .Thenit all the infor 1, can be real- . , multiplied by the D black A : 2 / 1 thus absorbs : black-body radiation 2 A 2 / D . / ; 1 e 1 / A L / . . ; A tions. Following a suggestion by e . ;itistermed is called L D . 1 1 / / D ooking in through an opening, we can ; 2 e A ” for the wavelength / L . . opening ; ), we find e )that L . black 28.2 28.3 of the body which is not black at of wavelength (1859), such black bodies were heated to high, uni- 2 / of any arbitrary body is equal to the radiance 2 A / . of Black-Body Radiation ; e L . IRCHHOFF As a demonstration experiment, weplatinum electrically tube heat of a ca. roughly 2reflecting cm 15 diameter cm cross long in is air drawn untilNear it on it, begins the the to glow. wall tubehighly A of reflective wall weakly platinum the has tube wall a tube glowscross small the using glows least; more opening iron strongly, the but to weakly-reflecting oxide theing the completely paint. glows non-reflecting interior. the “black” most open- brightly. TheLarger polished, black bodiesIn can general, be a constructedIts long outer of tube wall fireproof with is coveredheating ceramic several with energy. cross-sectional materials. an For baffles insulating material thetungsten is in is purposes sufficient. order a of to suitable measurements material. conserve sten at It filaments is high in mounted temperatures, an andrequired. incandescent heated lamp; just like that the is, tung- no external insulation is An object 1 with the absorbance In words: Fordiance thermally-excited, monochromatic radiation, the ra- cident radiation formly distributed temperatures andas their emission sources openings (“black-body were radiators”). The employed which incandescent emerges light from the absorbance An essential feature of everyture usable in black its interior body must iscondition be that uniformly is the distributed fulfilled, and tempera- constant. then, If l this Zero light reflectivity, i.e. an absorbance of see no details or internal structure. Examples are the furnaces used to of a body which is “ 28.4 The Black Body, and the Laws and, instead of Eq. ( follows from Eq. ( ized in the formotherwise of opaque a to small light.a opening soot-coated in plate Such held the next anvia wall to opening multiple, it. of appears mostly It a absorbs blacker diffuse box all reflec than of which the is incident light, G. K Thermal Radiation 28 580 Part II 28.4 The Black Body, and the Laws of Black-Body Radiation 581

Frequency ν Wavelength λ 511.5 ∙1014 Hz 0.4 0.5 0.8 1 3 μm watt 2 m watt m sr 2 Hz 0.72 μm m sr 1.3 μm 12 ∙10–9 4∙1012 ν /∂ /∂λ visible e e L L 4 4000 K 8 visible 4000 K 2 1.6 μm Part II

4 Spectral radiance ∂ Spectral radiance ∂ 0.96 μm 1 3000 K 3000 K 2.5 μm 2000 K 1.44 μm 2000 K 0 1 29603 μm 3∙1014 Hz Wavelength λ Frequency ν

Figure 28.3 The spectral radiance distribution in the spectrum of a black body. At the left, it is referred to equal wavelength intervals, and at the right, to equal frequency intervals. These curves, as well as Eqns. (28.5)and(28.6), hold for unpolarized radiation. For the quantitative experimental investigation of the spectral radiance distribution, one uses the arrangement sketched in Fig. 19.1 (usually for the special case of # D 0, that is perpendicular emission). P P One measures the radiant power dW or dW which is emitted into the selected spectral interval and into the solid angle d˝. From the deﬁning equation for the radiance Le, it then follows that @ @ P Le P Le W D A ˝ W D A ˝ d @ d d P d or d @ d d P d

(dAP is the projected emitting area as in Fig. 19.3. For perpendicular emission, dAP D dA). melt glass (glass kilns) in a glass factory, or the coke ovens in a cok- ing plant. Every surface element in the interior of the black body, independently of its material or structure, emits at exactly the same radiance: Surface elements with a high absorbance (Eq. (28.1)) also emit a large amount and reflect very little of the radiation from all the other surface elements. For surface elements with a low absorbance, the converse holds: they themselves emit less strongly, but they reflect more of the incident radiation from the other surface elements. The distribution of the radiance over the various spectral intervals has been investigated quite thoroughly for “black-body” radiation, that is, for the incandescent light emerging from the opening of a black body. In particular, its dependence on the temperature has been very carefully determined. The results are collected in Fig. 28.3. The spectral radiance is plotted in the left graph against equal wavelength intervals, and in the right graph against equal frequency intervals. In the search for a formal description of these empirical results, a number of competent physicists invested considerable efforts. The final success was attained at the end of 1900 by MAX PLANCK, with his famous (28.5) (28.6) (28.7) ; LANCK m/s). 8 3000 K, the : , and with .P 10 k h ’sconstant

mK h D 3 2 sK D T h D 11 4 10 C : D Ã ’s formula. We refer E ; : 4 OLTZMANN 78 10 439 K : : 2 h 1 2 c W 4 1 2 1 is B m ). m, and up to k D D 1 8 D , 4

LANCK 1 2 4 4 2 T 3 2 T 8 T IEN C : C C C 10 C 0 e Ws ,thatis e A T 34 67 3 : The total power emitted outwards ; : 1 5

< 5 ; 10 C k 2 hc ; law D 3 D 4 3 C D 4 2 626 D of a black body increases proportionally h P k : is the vacuum velocity of light W 2 5 e 6 Wm m 2 c D c Ws L C @ 2 D @ 16 e 15 50 L C28.1 @ @ D ; A 10 2 10 : OLTZMANN Â 1 hc B Ws/K,and 2 - ’sconstant 191 47 : : 23 D 1 1 1 are empirical coefficients with the following values: 10 D D C wanted to refer these coefficients back to universal physical 4 ’s radiation formula contains two important laws as special TEFAN LANCK 1 3 C 38 : C C 1 in the denominator can be left off; the resulting error is less than 0.1 % 1 is P h ::: ( D 1 LANCK LANCK In the visible spectral range, i.e. for P P or C term (this then gives the radiation formula of W. W from a surface area constants. In this process,coveries: he He made found one the of new the universal greatest physical physical constant dis- to the details givenHowever, in independently all of modern the textbooks derivation,the the on empirical relationship theoretical constants between physics. inphysical constants the remains. We radiation find: formula and the universal to the 4th power of its temperature cases; these had both been discovered previously: 1. The S 1 was the first to introduce the energy equation radiation formula it, he opened thecalled door quantum to physics). the world of atomic-scaleToday, there processes are (now several derivations of P is used both for the A Thermal Radiation 28 C28.1. In this chapter, the letter absorbance of a body and also for a surface area.should This however be clear from context, so that the danger of confusing the two quantities is minimal. 582 Part II 28.5 Selective Thermal Radiation 583

The sun emits radiation to a good approximation as a black body. At its surface, the emitted power per surface area (Sect. 19.3)is:

WP W D L D 6:3 107 : A e m2 According to Eq. (28.7), this corresponds to a temperature of 5700 K. See Fig. 28.6.

In practical applications of this equation, it is often desirable to determine the net power which is lost by a body through radiation. Then, in addition to the power emitted by the body, we must also consider the power with Part II which the body is irradiated from its surroundings. This reduces the net power loss by the body. The result is

P D 4 4 W A T Ts (28.8)

.Ts is the temperature of the surroundings; e.g. of the receiver/:

2. The displacement law of W. WIEN: The wavelength max at which the spectral radiance has its maximum value is inversely proportional to the temperature T.Wehave:

hc T D D 2:88 103 mK: (28.9) max 4:97 k

In the solar spectrum, we can observe that the maximum value of the spectral radiance is at the wavelength D 0:48 m. For a black body, this corresponds to a temperature of 6000 K (see Fig. 28.6).

28.5 Selective Thermal Radiation

The absorbance A of a black body is 1 for all wavelengths. For all other bodies, A depends on the wavelength and is furthermore always less than 1. For this reason, at a given temperature and wavelength, instead of the radiance Le; of a black body, only the fraction A Le; is emitted by a non-black object. A is especially small for the limiting cases of very strong or very weak absorption (Sect. 25.5). With strong absorption (w<as in metals, w D average penetration depth of the radiation), the radiation cannot penetrate into the object; often, more than 90 % of the incident power is reflected. For “weak absorption” (w>), only a small fraction of the radiation is prevented from penetrating the object by reflection, and therefore, the major portion of the incident radiation can be absorbed. This however takes place at great depths, too thick for many technical applications. There is in addition a further complication: The optical constants also change with temperature. ), is solid R ough selective 27.17 can only be determined ), and the use of : Solid objects (kindling, on 28.3 ; burner, the platinum begins to e L (Fig. UNSEN heating. There are in principle two ways of carbon is formed as very fine particles, called ). incandescent gaslight OULE 28.1 Solid ): The zinc vaporizes, the vapor becomes oxidized, High temperatures ZnO smoke, glowing blue-green. In the 28.4 . Their absorbance must be as close as possible to that of This dependence is wellspectral ranges, understood e.g. in for metals only in a the few infrared. cases There, the and reflectivity for limited experimentally, and only approximately.large variations Very of few their materialsalways, temperatures can the without survive internal permanent and changes. surfacehistory of Nearly structures the depend object. strongly A onmosaic microcrystalline the of texture strongly thermal is reflecting converted into single a crystals, r etc. determined only by theand electrical its conductivity temperature of dependence thebodies, is metal the well (Sect. dependence known. of In the general, radiance for non-black shifting a large fraction ofspectral range: the emitted radiant power into the visible pletely burned. emitters a black body inspectral the regions, visible especially range, in the and infrared. as smallThe as flames possible which in other produce have a been used typical torches) as or light liquid sources fuels,heat since soaked of up antiquity combustion by into a gaseous hydrocarbons. wick, are These are converted by not the com- glow red, but the ZnOcrystals glows blue-green. absorb only The reason the is short-wavewith that portion a hot of very ZnO steeply the rising visibleonly absorption spectrum, this curve; part as of a the result,visible spectrum they thermally. to To can make a emit this large demonstration cally audience, (Fig. we heatand a the zinc-coated hot iron ZnO wirea smoke considerable electri- glows distance. like a blue-green torch, visible from Figure 28.4 light of an arc lamp,shadow, the just smoke like casts the a sootcandle deep particles flame black in (Fig. a glowing gas or Thermal light sources make exclusive usebodies. of the These radiation are from heatedor either electrically by by chemical J processes (in a flame), covered with a thin layera of Pt ZnO, film. and On the heating other over half a is B covered with 28.6 Thermal Light Sources In the visible spectral range,ily selective demonstrated thermal experimentally: emission A can be small read- quartz-glass plate is half Thermal Radiation 28 584 Part II 28.6 Thermal Light Sources 585

Figure 28.5 The spectral radiance. The solid curve is for an AUER mantle, /∂λ while the dashed curve e L applies to a black body at ∂ the same temperature Spectral radiance 0 5 10 15 μm Wavelength λ

soot (Fig. 28.1). These solid, very hot carbon particles produce the Part II radiation. Only at the beginning of the 19th century was the production of gas for combustion separated from the locality of the consumer. The gas was produced centrally from solid or liquid fuels and distributed to the consumers through pipes. The last decade of the 19th century “The last decade of the then saw the second leap of progress since PROMETHEUS:C28.2 The 19th century then saw the carbon particles in the flame, which radiate nearly as “black bodies” second leap of progress in the infrared as well as the visible range, were replaced by a mantle since PROMETHEUS”. which radiates selectively. It is heated by a colorless BUNSEN-burner C28.2. According to an- flame and emits predominantly visible radiation. cient Greek mythology, PROMETHEUS brought The mantle, a small bag of silk mesh, is soaked with a suspension of very selectively absorbing cerium oxide (ca. 1 %) in a thin and therefore not fire back to mankind using strongly absorbing layer of thorium oxide. Fig. 28.5 shows the spectral a torch ignited secretly on radiance @Le=@ from a commercial AUER mantle (C. AUER, 1885; T the Sun Chariot; it had pre- 1800 K), and above it, dashed, the form of the @Le=@ curve of an ideal viously been taken away by black body at the same temperature. In the blue spectral range, the curves ZEUS. coincide; there, the absorbance A of the AUER mantle is nearly equal to 1. Thus, the mantle radiates nearly like a black body in this range. Between D 1and7m, however, the absorbance A of the mantle is small, and therefore only a small portion of the radiance is emitted in this spectral region, which is useless for illumination purposes. For >9 m, the radiance again approaches that of a black body.

When JOULE heating is to be used to raise the temperature, today we make use of metal wires with a high melting point. Metals have a high reflectivity R in the infrared region (Fig. 27.8) and thus a small absorbance A D 1 R there. As a result, their thermally-emitted radiation also predominates at shorter wavelengths. Temperatures of the order of 6000 K would be desirable (Fig. 28.6). But even tungsten, with its melting point of Tm D 3700 K, can tolerate at most temperatures of only about 2700 K over longer periods of time due to increased evaporation of the metal. This is the usual operating temperature of gas-filled tungsten-filament lamps with a double-wound filament (Fig. 28.7). The windings of the filament radiate nearly as black bodies in the visible. Their lifetime is longer than 1000 hrs. For tungsten lamps which produce a particularly high radiance Le, the temperature is increased up to around 3400 K. But the operating lifetime of the filament is then only 1 to 2 hrs. In both types of lamps, evaporation of the metal filament must be reduced by using an unre- 2 μm C. Ther- ı λ C, optical ı around five 1 6000 K light-emitting 0.48 μm visible

2 1

13 , and thus e /∂λ L ∂

In contrast to thermally- sr radiance Spectral 2 C28.4 ). 3·10 C. Above 2600 watt m ı C28.3 hnology replace thermal excita- , 1705 AUKSBEE H RANCIS . 2 The distribution of the spec- A double-wound lamp filament The Black-Body Temperature and the Color Temperature (LEDs). Their performance is improving rapidly; their light emis- C. ı times greater than thatalso of mention here gas-filled the tungsten-filament electroluminescencediodes from lamps. solid-state We should sion efficiency is currently over 100 lm/W. Modern developments in illumination tec tion by electrical excitation. The well-knowntubes” and in widespread “fluorescent use todaygases) are filled and with are mercury aillumination vapor further (F (or development in of some theexcited cases very sources, noble first these form fluorescentviolet of tubes radiation, electrical produce which a isis large useless converted amount to to of the visible ultra- tubes human radiation with eye by for coating a illumination.emits the chemical visible inner It compound fluorescence walls light. that ofthe light the absorbs For emission glass the almost efficiency colorless is ultraviolet (80–100) fluorescent light lm/W tubes, and Gas thermometers with iridium vessels can be applied up to 2000 thermometry is in factmeasurement the only practicable method of temperature Figure 28.6 tral radiance of a blackthe body temperature at of 6000 the K, sun’s surface moelements made of tungstento and 2600 a tungsten-molybdenum alloy can operate up 28.7 Optical Thermometry. Black-body radiation and its laws find anmeasurement important of application in temperatures the above 600 2 Figure 28.7 active gas atmosphere (Ar, Kr) or alamps”). halogen-containing gas (“halogen - . AUKS 29.1 H , No. 303, Electricity in Philosophi- RANCIS ”, Univ. of Califor- Thermal Radiation , F.R.S. (died around 28 BEE Chap. VIII.) the 17th and 18th centuries: A study of early modern physics nia Press, Berkeley (1979), cal Transactions Vol. 24 (1705); see also J.L. Heilbron, “ 1713). He produced light by shaking an evacuated glass tube partially filled with mercury. ( C28.4. The unit ‘lumen’ (lm) is defined in Table C28.3. F 586 Part II 28.7 Optical Thermometry. The Black-Body Temperature and the Color Temperature 587

Red filter Screen

Figure 28.8 Optical temperature measurement using a pyrometer. In this demonstration experiment, the radiance of an arc-lamp condenser is compared with that of a tungsten-ﬁlament incandescent lamp. At the correct value of the lamp current, the ﬁlament of the tungsten lamp becomes invisible. Part II

In such an application, we want to compare the radiance Le; within a narrow spectral range of a body of unknown temperature with the radiance Le; of a black body at a known temperature T. The simplest method for all comparisons is a null method: We vary the known temperature of the black body until its radiance is the same as that of the body whose temperature we wish to measure. Then we deﬁne the known true temperature T of the black body as the “black-body temperature” Tb of the body of unknown temperature. The black-body temperature Tb of a body thus means that within a limited spectral range which should always be quoted, the body radiates with the same radiance as a black body at the true temperature T. The true temperature of a body must always be higher than its black-body temperature. Otherwise, the body could not emit the same radiance Le; as a black body with A D 1, in spite of its absorbance A < 1.

Based on this definition, practical and convenient pyrometers are constructed. Their main component consists of a tungsten-filament lamp with variable current, an ammeter and a red filter. The tungsten filament is placed in front of the image of a radiating surface and its radiance is varied by changing the current. When the radiance of the filament and of the surface coincide, the filament becomes invisible (demonstration experiment in Fig. 28.8). The instrument is calibrated using the surface of a black body, and then the true temperatures of the black body are marked on the scale of the ammeter.

The discrepancies between the “black-body temperature” and the “true” temperature are often considerable, even for materials with little selective absorbance, e.g. tungsten, which is technically very important. This is shown in Table 28.1.

Table 28.1 Optical temperature measurements (in K) with tungsten True temperature T of the tungsten 1000 1500 2000 3000

Black-body temperature Tb, measured 964 1420 1857 2673 with the radiance Le; in the region around D 665 nm Color temperature 1006 1517 2033 3094 (The ratio of the true to the black-body temperature is not constant, because the absorbance of the metal varies with temperature.) color =@ 1; it e body. 1, are L . For its D (28.10) @ < . A demon- A A F temperature of 6 was chosen. : D 0 1 2 , with D true )). For our sensory of the emitting A of the two objects, one 28.5 hue .The color equality color temperature fluxes 28.9 ack-body temperature, another ). In that case, color equality means , and conversely, the same color hue const is not fulfilled for the non-black D 28.10 determines the fluxes A Iris visible radiation is used, that is without SiC-rod diaphragms ratio F , two solid curves of the spectral radiance A I 28.10 unresolved A demonstration experiment for the measurement of the color . Spectral radiance in the wavelength range around Spectral radiance in the wavelength range around 28.1 implies strictly equal true temperatures. In general, however, the case The hue is thusspite the of same their different for radiant the black body and the non-black body, in body. The lower curve thendashed has or a the shape dotted such curves as (Fig. shown for example by the perceptions of light, the characterizes the temperature range used (Eq. ( only approximate equality of thelower temperatures. than the The true color temperaturehigher temperature in in is the the case case of ofonly the the for dotted dashed bodies curve. curve, with But extremely and such selective it deviations absorption. is become serious are shown for the visibletemperature. range; For both, both the hold absorbancebeen for over assumed the the to same whole be visible arbitrarily-chosen spectrum constant. has For the upper curve, we have set Rationale: In Fig. The ordinates of the two(bodies curves with thus differ an only absorbance by which a is constant factor independent of of 0.6 thus applies to a black body. For the lower curve, frequently called “grey”). The ratio temperature. The body of unknownSiC temperature rod. here is an As electrically-heatedfor a this comparison demonstration, object, a tungsten-ribbon ais lamp black quite with body sufficient. a should variablesame A current in irradiation definite source intensities fact color on be the comparisoniris used; screen; requires diaphragms they but approximately are the obtained by adjusting the Figure 28.9 For this reason, intemperature addition has to been the defined, bl called the stration experiment is sketched in Fig. the black body which applies at color equality is defined as the temperature of the bodytemperature which deviates is in being general much comparedthan less the to from black-body it. the temperature true does.Table The temperature An color example is also given in definition, the compares their hues (red, yellow-red, etc.).is Here, again also, the a simplest, null method that is adjustment to a color filter, and instead of the radiant Thermal Radiation 28 588 Part II 28.7 Optical Thermometry. The Black-Body Temperature and the Color Temperature 589

Figure 28.10 Measuring the color 3∙1011 temperature. The values shown = watt m Aλ 1 apply to the direction perpendicular m2 sr to the emitter surface. 2500 K /∂λ e

L 2 = Aλ 0.6

1 Part II Spectral radiance ∂

400 600 800 nm Wavelength λ

The blue of the sky corresponds to a color temperature of around 12 000 K; in April and May, even up to 27 000 K. This means that in the visible spectral range, the distribution of the radiance of the light from the sky, which results from diffuse (RAYLEIGH) scattering, is the same as that from some extremely hot stars (e.g. Sirius, 11 200 K, or ˇ Centauri, 21 000 K). hr scnandwti oi nl d angle solid a within contained is there sal ntredmnin,a oeo esbih n fe shiny. often and bright less or them, more see as Sections We dimensions, surfaces. three monochrome in or usually variegated con- bodies, colored, own of our including sists eyes, our by seen is which Everything etetdi Sects. in treated be n Photometry and Perception Visual npyis ecasf aito codn oisradiant its to according radiation classify we physics, In of sense their of properties important perception. more visual the of much physicists aware of be Nevertheless, object also the must research. been has psychological organs, and sensory physiological other our like eye, The Need The Remark. Preliminary 29.1 .Ldr,RO ol(Eds.), Pohl R.O. Lüders, 2018 K. AG Publishing International Springer © W locl hsqatt h radiant the quantity this call also (We n fteuuluis ..i at h ain oe d power radiant The watt. in e.g. units, usual the of one feeg are yaba frdain ..twrsarcie;see receiver; a towards e.g. radiation, of beam Chap. a by carried energy of our vlae nyi em fisefc ntehmnee ..o u vi- our is on power i.e. eye, radiant human the the ( on which perceptions effect in sual its found of terms be in to only evaluated mea- had of radiation method the a small Therefore, a suring in spectrum. selectively electromagnetic very the only of power region radiant to responds sense sual sa as o h es fvsa ecpin h hsclrdatpwrand power (Chap. radiant it physical from the derived perception, quantities visual the of sense the For h ain nest o simply (or intensity radiant The ecpin fclradbrilliance and color of perceptions derived 19 ain intensity Radiant .Figure ). 29.8 o Photometry for uniywt h nt1W/sr. 1 unit the with quantity to 29.13 photometry 15.4 29.2 r neddt show to intended are hwdtemaueeto ain oe in power radiant of measurement the showed olsItouto oPhysics to Introduction Pohl’s through I # .Tefnaetl fpooer will photometry of fundamentals The ). D intensity ain oe d power Radiant 29.7 oi nl d angle Solid . 19 occur. flux ˝ stu esrdi physics in measured thus is ) r o eeat u vi- Our relevant. not are ) hnw en the define we Then . oepaiete“flow” the emphasize to ne hc conditions which under ˝ W P , https://doi.org/10.1007/978-3-319-50269-4_29 # : W P sdefined as power ( 19.2 W P ) . 29 591

Part II ) : .It 0 # 19.4 P A ( W d d 29.1 D from the source R 2 ). / ˛ ( 29.2 (previously known as the ir- of the light source # to the source of the source in the numerator. I R # I αβ irradiance Distance (at normal incidence) in the denominator, or . 0 A Radiant intensity D e E Schematic drawings of two groups of technical methods for Rotating sector wheel (“chopper”) for varying the time average the Irradiance Irradiance to the irradiated area d This deﬁning equation (cf. Eq.the two (19.4) possibilities: and Either Comment we C19.4.) change the distance shows completely independent of the spectral range employed.drawing of A schematic this method can be seen in Fig. changes only the time average of the radiant intensity, and thus is Among the available experimentalthe possibilities, following: two will sufﬁce for 1. A rotating sector wheel (“chopper”), as shown in Fig. of the radiant intensity.are no More longer than perceivedblack about by circle the 30 indicates eye to the (cf.the 60 cross-section chopper motion dark of to picture phases be the camera!). shifted light per to beam. The second the side small A in carriage the allows direction of the double arrows. we change the radiant intensity Figure 29.1 Figure 29.2 29.2 Experimental Aids for Varying For the demonstrationsconvenient in variations this of the chapter,radiation we intensity): will require rapid and ing prisms. varying radiant intensities. Left a rotating sector wheel, right two polariz- Visual Perception and Photometry 29 592 Part II 29.3 The Principles of Photometry 593

2. Grey ﬁlters (see Fig. 27.8) of variable thickness, or two polarizing prisms or foils, one behind the other (see Sect. 24.3). They can be used only within limited spectral ranges. A schematic of this method isalsogiveninFig.29.2 (ˇ).

29.3 The Principles of Photometry

The fundamental principle of photometry is simple: We measure the

radiant intensity using our visual perception sense as a new base Part II quantity, termed the “luminous intensity”. The unit of this new base quantity is realized in terms of the intensity of an internationally agreed-upon standardized light source and is called 1 candela1 (abbreviated cd). The meaning of this statement is explained by Fig. 29.3. Its upper part shows an incandescent lamp whose luminous intensity is to be measured; its lower part shows three standard lamps, sketched for simplicity as candles. A section of the same printed text is pasted onto the two areas d A0. The number of standard lamps has been empirically chosen so that the lower area appears to be “illuminated” by them in just the same way as the upper area is “illuminated” by the incandescent lamp: This means that one can read the text equally well in each of the two areas. Then for our eyes, we could replace the incandescent lamp by three standard lamps at its position and would see the text equally clearly. If each standard lamp has a luminous intensity of 1 cd, then the luminous intensity of the incandescent lamp is thus equal to 3 cd. For the luminous intensity as a new base quantity, we then ﬁnd the following juxtaposition of photometrically- and physically- determined quantities:

dΩ dA'

Separating partition dΩ dA' = dΩ Solid angle Disks with the same printed text

Figure 29.3 An example of the principle of photometry: For the eye,the incandescent lamp could be replaced by three standard lamps at the same position, represented here as candles. In the technical version, one employs only a single standard lamp and reduces the luminous intensity from the incandescent lamp on the upper disk to one third. The method of accomplishing this was described in Sect. 29.2.

1 Candela (the second syllable is stressed) is the Latin word for candle. The luminous intensity is thus denoted by the same word that refers to a commercially- available object, for example a light from burning wax. . / Ã 2 tity tity ) per- ,as 0 A A D / D lx . names ,wehave nous density Solid angle / 2 = 1lux 29.1 steradian steradian D 2 Source area d meter 2 Power Power Power/Solid angle . . . ) Â ”), i.e. the radiance as meter lumen candela candela meter 1lumen(lm) Unit candela (cd) candela or tities which can be related to : (Power/Receiver area d sufﬁces in order to determine illuminance is sometimes called the luminance C29.1 2 receiver photometry as derived quantities Radiant energy Radiant ﬂux Radiant power Radiant intensity Radiance

Solid angle Irradiation intensity Irradiance 9 candela. : D 0 and for the )( simpler to develop the photometric measurement proce- instead of instead of instead of instead of : ) ‘luminous intensity’ : source or and a temperature of 1770 °C, the solidification tem- 19.3 instead of . Then the luminous intensity would become the derived quan (Distance to the source) 2 source (an Austrian physicist, 1845–1904) was used. Its radiant Receiver area source area, etc. The use of the luminous intensity as base quan receiver Luminous intensity of the source Luminous flux cm Definition Base quantity Luminous intensity/ Apparent source area (Fig. Luminous intensity / experimentally 60 photometry, and their SI units = base quantity EFNER 1 . Of course, one could introduce other physical quan Illumination intensity Luminous flux Luminous power Luminance Luminous intensity collected the names of some of these quantities and their units. is often the case, thenon this the harmless measurement aspect specification of takes a truly esoteric doctrine. In Table 2. For the Illuminance If the units of these derived quantities are given special 3. For both the ( intensity in the horizontalOne direction Hefner candle was is termed a “Hefner candle”. Luminous energy Up to 1979, the standard lampsof were “black bodies” with an aperture ceived by the eye our visual perception sense asor base photometric quantities, brightness for (now example the called lumi the “ luminance all the other quantities needed in The 2 1. From the perature of platinum. Previously, before 1942,F. a H gas lamp named for makes it dures. 117 Specially- has been Luminance Luminous flux Illuminance Quantity Luminous intensity Some quantities relevant to candela PTB-Mitteilungen Visual Perception and Photometry 29 For the Receiver For the Source Table 29.1 C29.1. Since 1979, the following definition of the base unit “illumination intensity”. See Comment C19.2. and the reference there. Note: The luminance is also sometimes called the “luminous density”, and the in effect: “The candela is the luminous intensity in a particular direction of a radiation source which emits monochromatic radiation at the frequency 540 THz and whose radiant intensity in that direction is (1/683) W/sr”; see e.g.: (2007), No. 2; English: see http://physics.nist.gov/cuu/ Units/current.html constructed incandescent lamps are employed in practice as secondary standards. 594 Part II 29.4 Definition of the Equality of Two Illuminances 595 29.4 Definition of the Equality of Two Illuminances

All of photometry stands and falls with our ability to identify two areas or “ﬁelds” which are perceived as “equally illuminated” by different light sources; or, more precisely stated, we perceive their illumination intensities, or illuminances, as being equal. In comparing two light sources of similar construction, for example a large and a small tungsten-ﬁlament incandescent lamp under normal current

strengths, the approach to equal illuminances is readily detected. We Part II let the two areas dA0 in the schematic of Fig. 29.3 be adjacent to each other. At a precisely equal illuminance, the boundary between them vanishes, and we can no longer distinguish the two illuminated areas from each other. The situation is different in the comparison of different types of light sources, e.g. a yellow sodium-vapor lamp and a blue-green mercury- vapor lamp, or two arc lamps with colored filters F1 and F2, one a red filter and the other a blue filter. In such cases, the concept of equal illuminance or illumination intensity must first be defined.Thereare several possibilities for accomplishing this: 1. Visual sharpness or acuity. This possibility was already explored in Sect. 29.3. Now, we suppose that on a sheet of newsprint, there are two rectangular fields next to each other. Each one is illuminated with an arc light, one with a red filter, the other with a green filter (Fig. 29.4). The illuminance of the one field can be continuously varied in a quantitatively measurable manner using the apparatus ˇ. With a remarkable accuracy, one can adjust the two fields to have the same legibility or the same visual sharpness. Therefore, for any color, we can define the visual sharpness or acuity as characterizing equal illuminances. 2. Delay time. The two rectangular, colored fields are projected onto a wall screen side by side with a vertical boundary between them, but interrupted by the shadow of a horizontal rod. The rod is moved up and down. Its moving shadow in general doesn’t appear to be a horizontal straight line, but instead it seems to show a dislocation at the boundary between the two fields, as shown e.g. in Fig. 29.5.That is, our sensory system perceives the motion only with a certain delay

F + 1 − + − F2 β

Figure 29.4 The definition of equal illuminances using visual acuity (sharpness). The surrounding area in this and in the following photometric demonstration experiments is lit with an illuminance of about 10 cd/m2.Then it radiates diffuse light itself, with a luminance of about 3 cd/m2. ), al- ˇ 4m) Demon- limiting frequen- to indicate equal ), the higher the ˇ ) and presented (the bifilar suspension . The higher the illu- 3 29.7 defined each other, but are rather equal illuminances. with radial sectors opens both . Intermittent illumination just it to eliminate the appar- β S define in one plane beside e fields continuously (apparatus . The two fields illuminated with col- . (In motion-picture films, about 0.01 s. Every S limiting frequency , causes flickering. This flickering appears on each other (Fig. equal 29.6 4 s). The observer views it from the side with both 1 2 F F : Hang a metal ball by two cords (bifilar, length T llipse depends on whether the reaction time of the left eye or − − + + The definition of equal illuminances by equal The definition of equal illuminances superimposed to the eye of the observer, around 10 times per second. In limiting frequency of flickering , which is dependent on the illumination intensity. Again, we can of flickering. The rotating sector disk In technical photometers, stereoscopic effects candiffering be lengths. produced When by they delays disappear, of thestration experiment illuminance is equal. as a gravity pendulum andguarantees let this; it swing eyes, but holds a piece ofball dark appears or colored to glass follow in an frontaround elliptical of the path. one e eye. The Then direction the the in right eye which is it delayed moves by the dark glass. Flicker-free field exchange The limiting frequency is, from experience, smallest when the durations of the precisely ternately mination intensity (variable usinglimiting frequency. the When apparatus thesame illumination limiting is frequency of of flickering different canilluminance. colors, be the 4. ored light are no longer projected cies light sources simultaneously and for equally long times. Figure 29.5 through equal delay times image is projected twice and onlythe every next second image; dark the interval image is used frequency to is change thus to 25 Hz.) Figure 29.6 3 dark and light intervals are 3. The (stroboscopic light), produced e.g. bydial a sectors rotating as sector in disk Fig. to with vanish ra- above a certain vary the illuminance of one of th time and then with good accuracy,ent we dislocation can of ad the rod’s shadow. Thus,we independently can of use the color, the equal delay times to Visual Perception and Photometry 29 596 Part II 29.5 The Spectral Distribution of the Sensitivity of the Eye: The Luminosity Function 597

+ F1 −

F2 + − β Part II Figure 29.7 The definition of equal illuminances by means of a flicker-free field exchange general, one sees the exchange of the two images as a flickering of the color hue. By changing the illuminance of one of the sources (apparatus ˇ), one can suppress the flickering. The eye then sees the fields in an unchanging, mixed color. This flicker-free field exchange can be defined as an indicator of equal illuminances, independently of the colors used. These different definitions for the equality of two illuminances lead to moderately good agreement among the results4.Makinguseof them, the luminous intensities of various sorts of light sources can be compared and measured in multiples of the conventional unit, the candela. The numerical values obtained from photometry naturally hold only for an “average normal human”; and even for such a person, they hold only under normal conditions, and not when some sort of stress-related disturbances have affected the subjective well-being of the person.

29.5 The Spectral Distribution of the Sensitivity of the Eye: The Luminosity Function

According to the considerations in the previous section, luminous intensities for any color can be measured in candela. As a result, we can determine experimentally how the ratioC29.2 C29.2. It is equivalent to the Luminous intensity, measured photometrically in candela definition E D Luminous flux Radiant intensity, measured physically in watt/steradian E D (29.1) Radiant flux (unit: lumen/watt), as plotted on the right ordinate axis of 4 One would have to give preference to the definition whose results best obey Fig. 29.8. an additivity rule. Illuminances are additive; for example, following one of the methods described, we determine two illuminances A and B. When added, they yield the illuminance C D ACB. When a direct measurement by the same method also gives the value C, we can say that the method obeys the additivity rule. In this sense, definition no. 4 appears to be the best. . 2 as . E 555 nm; 3cd/m luminosity

> Luminous flux/Radiant flux flux/Radiant Luminous Finally, the . W lm –3 3 2 –1 –2 watt lumen 10 10 10 10 1 10 10 002 : If the 10 % of all male 683 C29.3 . Or, still more simply: We D . It holds for an eye which sr = cd W 29.8 averaged over a year for hundreds 555 nm = 002 λ as a function of the wavelength : Wavelength λ 683 D of the eye, or alternatively as the max E is plotted in Fig. 3 2 300 400 500 600 700 800 nm 1 –1 –2 –3 10

candela 10 10

10 10 10 Luminous intensity/Radiant intensity intensity/Radiant Luminous of the eye, for an eye adapted to bright light, as defined by watt∙steradian The spectral distribution of the luminosity function or luminous luminous efficiency E or spectral sensitivity remove the sector disk andIt hold divides a the needle spectrum horizontally in along front its of whole the slit. length horizontally by a straight, The position of the maximumdemonstrated of qualitatively the by spectral sensitivity very ofspectrum simple the from eye experiments. an can arc be light Weintensity onto project a of the wall the screen different andciently assume wavelength good that regions the approximation. is radiant path roughly We and constant, then gradually a place suffi- increasespectrum flickers, a its then sector rotational its disk ends frequency.region which (violet in is and still At flickering the red) becomes narrower first, become optical andlimiting narrower. flicker-free. the frequency of The whole flickering is reachedtrum; in this the green is region the of the region of spec- highest sensitivity at this wavelength, The maximum of the curve then lies at a wavelength of the is adapted towhen bright the light, luminance i.e. ofreflected for light light a sources (room state (lamps) walls, of or furniture, the printed the eye pages) illuminance is which of applies of individuals the current internationally agreed-upon values. observers with minor color-vision disturbancesimum are would eliminated, then be thewavelength max- range shifted from to 400 to asomewhat 750 nm arbitrary. wavelength is termed of “visible”. 565 This nm. is, however, Conventionally, the efficiency depends on the wavelength of the radiation. One can denote Figure 29.8 function The methods of measurement arethe sufficiently well previous sections. known The to result us from , for 28.3 which are ,CIE)in )isacomplexpro- intervals 29.8 Visual Perception and Photometry https://en.wikipedia.org/ http://www.cie.co.at/ 29 Commission Internationale cess. Here, as in Sect. we are dealing with wavelength wiki/Luminosity_function and small and are denoted by their central wavelength. The curve was established by the International Com- mission on Illumination ( de l’Eclairage 1931 for a bright-adapted eye (“photopic curve”). See e.g. more details. C29.3. The measurement of the luminous efficiency curve (Fig. 598 Part II 29.5 The Spectral Distribution of the Sensitivity of the Eye: The Luminosity Function 599

dark streak. We then move the needle slowly up and down; the dark streak appears to be bent into a curved arc, its ends in the red and violet regions lag behind. The top of the arc is in the green region, i.e. again in the region of maximum sensitivity, where the delay time of the eye is shortest.

At low illumination intensities, the light receptors of the bright- adapted retina of the eye, the “cones”, are no longer functional. Instead, other receptors, the “rods”, take over. At illumination intensities of < 3 103 cd/m2, only the rods are active. The spectral sensitivity distribution of the eye is then shifted towards shorter wavelengths. Its maximum lies around 510 nm. The eye still reacts to an irradiance of around 6 1013 W/m2, i.e. its pupil, with an area Part II of 5 105 m2, must pass a radiant power of about 3 1017 Wor a luminous ﬂux of around 2 1014 lm5. With the rods alone, the eye can no longer see objects in color “At night, all the cats are grey”. “At night, all the cats are The rods are not present in the angular region of greatest visual acu- grey”. ity (see the last paragraph of Sect. 18.14). Therefore, objects seem to disappear when looked at directly (ﬁxed), and they reappear when one looks past them. We see “ghostly lights” and will-o’-the-wisps.

To demonstrate these effects, we completely darken the lecture hall and project a spectrum onto the wall screen while varying the illumination intensity from the slit using two crossed NICOL prisms or Polaroid ﬁlters (Sect. 24.3). After a few minutes, the observers are all dark-adapted. The spectrum appears to be a silvery-shining band, with a clear maximum intensity in the region that was previously “blue”. When one ﬁxes an object directly, it is invisible; one has to “look past it” in order to see it.

Having determined the two spectral sensitivity distributions for the bright- and the dark-adapted eye, we have laid down the physiological basis of photometry. For technical and economic purposes, aptly-chosen average values (e.g. as in Fig. 29.8) can be agreed upon as internationally binding. Based upon them, all the necessary practical measurements of light can be carried out with instruments only, without reference to human visual perceptions. It is not difﬁcult to provide a photoelectric radiometer (a photocell and an ammeter, see Fig. 15.6) with the same spectral sensitivity as that of the eye. The selective photoelectric effect of the alkali metals, especially cesium, is very well suited for this purpose (cf. 13th edition of “Optik und Atomphysik”, Chap. 18); it can be combined with special ﬁlters. Such arrangements are often called objective photometers. They indicate the radiant power (watt) with the same measure, which varies with wavelength, as a conventional, average “normal eye”. The scale of the ammeter can be directly calibrated in a photometric unit, e.g. candela. In this and other forms, technical photometry solves by convention the task of providing economically usable data and avoiding controversies. For the vision of a single individual, these data are of “Where the results of such course not strictly applicable. Where the results of such data contra- data contradict visual per- dict visual perceptions, the eye is always right! ceptions, the eye is always right!”

5 Corresponding to roughly 100 light quanta/second. en- of the (29.2) , mea- : product 2 photo detector / from two stars luminance 2 ; of its radiant area. L : of the star E L I size and 1 to the star ; L itional flashes of light, for s without perceiving it to R E 5 of a lamp, of a firefly, etc., illuminance ). 10 eryday language uses the word 2 ochemical reaction products are . Thermal processes and biological Distance cd/m . illuminances 6 . As long as the irradiation time is short 10 energy Luminous intensity D luminous intensity , both for primary light sources and for reflection rise time 2 . Our visual sensory perceptions are initiated by photo- )foratimeofe.g.5 , finally, the word ‘brightness’ is used in three different 2 Time of the Eye Receiver area Luminous flux cd/m 9 D input. 10 as registered on the earth. Then they define (on the basis of a long historical Astronomers compare only the Application: We can allow a lampof to glow brief continuously, and overload then by pulses, means we can produce add example to serve as signals.but they The can be human registered by eye a does receiver which not makes use perceive of the a signals, with a very short rise time. L astronomy 05 s. Therefore, one can look at the disk of the sun (luminance E : ways; most frequently in the sense of radiant power and the irradiation time is relevant; it measures the Example: For the bright-adapted eye, the accumulation time is ergy compared to this riseaccumulated. time, During the the phot accumulation time, only the regeneration in the livingthe cells concentration effect never exceeds ais a reconstitution. proportional steady-state to limiting As the valueonly a radiant which after power. result, a This certain value is however reached be brighter than aribbon continuously-observed, lamp weakly (luminance glowing tungsten In (secondary) sources. In addition, ev ‘brightness’ for the measured in candela, without considering the 0 29.7 Brightness This often-used word fromIt everyday language denotes is for rather example ambiguous. ‘violet’ the can quality never of be a perceived‘yellow’. sensation: to Usually, the be brightness chromatic is as used hue bright in as the the sense chromatic of hue 29.6 The Rise Time and Accumulation The results of the previousof section the hold eye. only forradiant Only power steady then illumination is thechemical processes luminous in intensity the proportional retina of toreaction the the products eye. The is concentration by oftional their no to the means absorbed light always and continuously propor- sured in candela/m Visual Perception and Photometry 29 600 Part II 29.7 Brightness 601

development) the visual magnitudes mi by the equation

EL;1 m2 m1 D 2:5log : (29.3) EL;2

The difference between two magnitudes m2 and m1 is thus proportional to the logarithm of the ratio of the corresponding illuminances from the two stars. The value m1 is taken to be C 2:12 for the north polar star, Polaris (an arbitrary convention). On this scale, the visual magnitude m2 of a star which can just be discerned by the naked eye is equal to C 6. Its value for ˛ Cygni (Deneb) is C 1:3; for Sirius, it is 1:6; and for the sun, it is 26:7. (Compare the deﬁnition of the phon in Vol. 1, Sect. 12.29).

In Eq. (29.3), the illuminance EL is used. With known distances R,as- Part II 2 tronomers instead use the luminous intensity IL D ELR (Eq. (29.2)), and deﬁne the difference between two numbers M1 and M2 by the equation

2 I ; E ; R D : L 1 D : L 1 1 : M2 M1 2 5log 2 5log 2 (29.4) IL;2 EL;2R2

These numbers are called the absolute magnitudes or absolute brightness. Combining Eqns. (29.3)and(29.4) yields

R1 M2 M1 D m2 m1 C 5log : (29.5) R2

6 For a fixed star at a distance of R1 D 10 parsec which has a “visual magnitude” of m1 D 0, M1 is assigned the value 0. Thus, for a fixed star at a distance R2 and with a visual magnitude of m2,wefindtheabsolute magnitude to be given by the number

10 parsec M2 D m2 C 5log ; (29.7) R2

7 or, if we use Eq. (29.9) to express its distance R2 as a parallax from the earth, replacing the distance by ˛2: ˛ M D m C 5 C 5log 2 : (29.10) 2 2 100 The brightness or magnitudes which are termed ‘absolute’ are thus those which would be observed from a distance of 10 parsec.

6 In astronomy, the distance unit ‘parsec’ is equal to that distance R0 from which the radius of the earth’s orbit r would be seen to subtend an angle of 100,thatis

00 16 R0 D r=1 D 1parsecD 3:08 10 meter (29.6)

.100 D .1=3600/ı D 4:85 106 rad; r D 1:49 1011 m/:

7 The parallax ˛ of a ﬁxed star is deﬁned as the angle

Radius of earth’s orbit r ˛ D : (29.8) Distance to the fixed star R From Eqns. (29.6)and(29.8), we find that a fixed star with a parallax ˛ is at a distance

100 100 R D R D parsec (29.9) ˛ 0 ˛ 8 8 , 6 8 10 6 10 3 3 10 10 ond, 2 (29.12) 8 11 5 10 / 4 6 10 10 .Forev- ’s cosine 3 / / (29.11) 2 2 a 10 / numbers : 10 10 8 : 10–15 ) < ca. 10 ca. 5 ca. 3 Luminance in cd/m ca. 10 ca. 10 6 (2–6 (5–35 1 (4–12 up to . cd/m 5 tities: First, the AMBERT 19.3 10 2 of the star / I , just like a white ball and 2 disk 3820 K) 6 8). These numbers lie between D : . Therefore, we see the sun’s 4 10 lamp) ), which are termed “photographic (for dwarf stars). , the astronomers then define DC 6 29.3 ECK L ˇ E or temporary blindness occurs, i.e. the Luminous intensity M reflectors . refers to the ratio of the star from the earth R ); that is, both for primary light sources and Projected source area (Fig. instead of 0 e dazzling itself to an astonishingly wide range of lumi- E D emitted by a star (measured e.g. in watt); and sec 19.2 L Distance .The L . in astronomy refers to two different quan , namely the ratio of the absolute magnitude of a star to the adapt (for giant stars) and 10 ). 5 Examples of luminances Photochemically-evaluated radiant intensity 19.3 radiant power as if it were a uniformly-emitting luminosity D photographic brightness this desolate muddle, we should avoid the word ‘brightness’ 0 e Luminance pure number E Using the quantities around 10 The magnitudes”. The using an equation analogous to Eq. ( total absolute magnitude of the sun a Mercury arc light Tungsten filament lamp (gas-filled) Carbon arc crater (black-body temperature ditto, with added cerium fluoride (B High-pressure mercury arc lamp (quartzSun bulb, 45 bar) Reflective sources (secondary sources) Objects in illuminated working and living rooms Objects in workplaces for veryObjects fine or on precision the street, work sunObjects behind outside the observer in cloudy weather Primary light sources Night sky Neon lamp Gas mantle lamp For short times (fractions of a second), this maximum value may be greatly ex- nances, namely the range between 2 that is illuminated uniformly from allin sides Sect. (see however the footnote The eye can for ideally diffuse-scattering sphere law applies (Sect. ery state of adaptation, aexceeded; certain otherwise maximum luminance should not be Table 29.2 is independent of the direction of emission when L Given a ceeded, up to a large multiple of the luminance of the sun. as far as possible Visual Perception and Photometry 29 “Given this desolate muddle, we should avoid the word ‘brightness’ as far as possible”. 602 Part II 29.8 Achromatic Colors. The Conditions for Their Formation 603 visual acuity of the eye and its ability to distinguish colors are greatly reduced. At the upper limit of the eye’s adaptation range, first dis- comfort and then pain warn of the impending permanent eye damage. The luminances of many light sources exceed the adaptation range of the human eye. This is shown in Table 29.2. The influence of optical instruments such as telescopes on the luminance of the objects being observed is very important. But here, psychological aspects also play a significant role.

29.8 Achromatic Colors. The Conditions Part II for Their Formation

The achromatic colors can be arranged along a series, called the grey scale. At one end is white, at the other end is black. The transition between them includes all the grey colors. It can be continuous or in discrete steps, e.g. in 10 steps of equal change in contrast. To demonstrate the grey scale, we first make use of commercially-available white, grey and black paper (Fig. A1 on Plate 1 at the end of this chapter) and illuminate it with sunlight or with natural light from an incandescent lamp (see the end of Sect. 16.1). Physically, all these achromatic surfaces have one thing in common: In the visible spectral range, their reflection coefficients depend only weakly in the ideal case not at all on the wavelength of the light. The different achromatic surfaces differ only in their overall reflection coefficients. This is around 90 % for white paper, but only about 6 % for black paper. For this reason, all achromatic surfaces (white, grey, and black) reflect visible radiation with the same spectral distribution; only their radiance differs: . Power ; W / Solid angleArea unit: srm2 , or, measured photometrically, their luminance (in cd/m2). This corresponds to a very important empirical fact: Every achromatic surface appears to have the same color, white, when it is illuminated by itself in a darkened room with natural light.

Demonstration experiment: In Fig. 29.9, a circular aperture is placed in front of the condenser lens of an arc lamp and imaged on a screen. In the well-lighted lecture room, a white cardboard sheet is hung up, and then the lights are dimmed: One then sees a luminous white circular disk. Then the arc lamp is switched off, the white cardboard is secretly replaced by a black cardboard, the power to the arc lamp is increased and the observation is repeated. The observer again sees a luminous white disk, something like the luminous white moon against the dark sky.

I + − Video 29.1: α “The color white” Figure 29.9 How the color white comes about (Video 29.1) http://tiny.cc/dgggoy I visible .Lamp two radiation, but ). lack of variety 29.10 itself, the other without changing own 29.1 illuminates only an II object achromatic surface and remains. Now, the inner one changes at all in the illumina- frame is surrounded by a luminous outer α can thus never be seen as grey or black. . This can be arranged in two different alone . The ratio of the two luminances determines (e.g. sector wheels; cf. Fig. ˛ I How the colors grey and black are formed II Their Tinting and Shading + − − : A surface becomes black not through its + surroundings 29.9 Chromatic Colors, Their Hues, and With natural light, wewe produce look a at continuous it spectrum.of only different fleetingly, Even colors. we when A are large surprised group, by the purple the hues (wine red, etc.), is Figure 29.10 An achromatic surface illuminates a circular “inner field”, while lamp At first, only thescattered inner radiation field is isappears adjusted illuminated to and to be the a pure luminance medium white. of value. Then, its the The inner field inner field To see grey orluminance black, in the its eye field requires of a view second area with a higher ways: Either we usedifferent at reflection least coefficients two and differentsource illuminate achromatic of them surfaces natural with with light; a common or we use only illuminate two separate fieldsminous on intensity. The it latter with method two is shown lamps in of Fig. different lu- “outer field” which is bounded outsideillumination by intensities a can be rectangular border. varied Both continuouslyusing over the a apparatus wide range rather as a result of the radiationwe from see its surroundings. nothing Without at light, all;surroundings. black Grey can colors be are formed seen by only the through presence light of from the anything in its own illumination field. Immediately, the color ofmore intense the the inner illumination field of changes thethe to outer inner field, grey. field. the The We darker can theblack, pass grey without, through of we the repeat, entire makingtion grey any of scale the up inner to field. deep removed, At the so end, that the only illumination its of the illuminated inner field is field appears to besoot. blacker than the best matteSummary black paper or even the degree of greyness (the grey scale value). from its radiations. One of them is radiated from the Visual Perception and Photometry 29 604 Part II 29.9 Chromatic Colors, Their Hues, and Their Tinting and Shading 605

+ I −

II α + −

III + − Filter

Figure 29.11 Tints and shades of chromatic colors (Video 29.2) Video 29.2: Part II “Tinting and shading of chromatic colors” completely missing. We search in vain for the most frequently-used http://tiny.cc/sgggoy colors of our clothing, our furniture and our wall coverings. There is no brown, no pink, no dark green, etc. Alongside a commercial color chart, the range of colors in the spectrum appears limited and paltry. We can, to be sure, quite properly refer to radiation within a very narrow wavelength range as monochromatic, that is as a ‘pure color’: We see it as a characteristic chromatic hue or ‘chroma’. But we cannot turn this sentence into its converse: Only in rare cases is monochromatic light, that is light that we see as a ‘pure’ chromatic hue, identical to the radiation from a narrow wavelength range! (Sect. 29.10). It is difficult to bring order into the sheer unmanageable variety of chromatic hues. Every chromatic color exhibits a certain hue which cannot be precisely defined, namely red, yellow, purple etc. In addition to this hue, another characteristic may be present: The color may be tinted or shaded, i.e. a red can be whitened (tinted) or darkened (shaded); it may furthermore often show a grey shading. ‘Pure’ colors which are not tinted or shaded can be produced using filters and can be ordered on a circle according to their similarities (cf. Fig. A2 in Plate 1 at the end of this chapter). In this color circle, every hue is more similar to its two immediate neighbors than to any other hue on the circle. In order to demonstrate tints and shades,Fig.29.10 is complemented by a third projection lamp (Fig. 29.11). It illuminates only the inner field, initially through a red filter. Then four experiments are carried out, one after the other: 1. Only lamp III is turned on; the inner field appears as a pure or free red hue. 2. Now we want to add a white tint to the red in the inner field. To accomplish this, the inner field is additionally illuminated with light from lamp I, and its illuminance is gradually increased using the apparatus ˛. This leads us from the pure red through pink to an achromatic white. 3. In the third experiment, the red of the inner field is to be given a dark shading.LampI is switched off, and in its place, the outer . , 29.12 of black, ı C29.4 thus corre- grey shading .InFig. Then the light beam brown – white 8 cardboard disks ) and also the outer field I (1834–1918). or circle. In this manner, all ark shading is produced, and the ERING H tinting and shading triangle ’s triangle” (Fig. A3 on Plate 1 at the end WALD ERING Iris diaphragm Iris , we can vary the illuminance from both lamps and of yellow. It is rotated rapidly. The motion causes the three ˛ ı Hues Black shading of a chromatic color is possible only with the from the spectrum of natural light must such a filter trans- ) illuminate the screen with their natural light. Making use . We pass from the pure red via pleasant dark red hues to an II II of red, and 60 ı The formation of brown shades can be demonstrated by quite simple means. which illuminates the diskalso is illuminated. broadened so Immediately, that a the d screen behind is reddish-yellow becomes a typical brown. can pass from a pure orof free achromatic red grey via in greyish-red to the any inner arbitrary field. shade All the tints and shadesdimensionally of by a “H given color hue can be represented twoof the apparatus a brown paper disk isthe placed darkened in lecture hall, front it of issees an first no achromatic illuminated brown, screen with but natural and, instead light. only in corresponds the One to corresponding brown, unshaded a hue reddish-yellow that color. A black- or grey-shadedgrey chromatic shade, color never can, appear alone just in like the a field black of or view of this chapter). One such sponds to each individual huethe on manifold the colors col in nature andtions, art be can, catalogued through and the right designatedused combina- by in letters commercial color and charts, numbers.on which the This are classic is based works without of exception E 90 (lamp A circular disk has three glued-on sectors of colored paper, around 210 individual colors to vanish and ‘merge’ together into a unified brown. To accomplish this, itand has black, to that is be both overlaid the inner simultaneously field with (lamp white achromatic black in the inner field. 4. Finally, the pure red of the inner field is to be given a 29.10 Color Filters for Producing Pure In the previouscolors section, free we of saw anyrange that white tint we by could using produce suitably-chosen luminous filters.8 What field is illuminated with increasinglamp intensity by the natural light from Figure 29.12 aid of a second illuminated area 1940. ” beginning Optik ’s “ Visual Perception and Photometry 29 OHL P with its 1st edition in See also the addendum toComment this at the end ofchapter. this C29.4. This disrespectful experiment was described in 606 Part II 29.10 Color Filters for Producing Pure Hues 607

Spectrum and template F L2 a Image of the Aperture red L 1 violet a aperture S A Illuminated B slit C D violet red Part II Figure 29.13 The spectral distribution of pure hues with a high luminous intensity. With a small modification, this arrangement is also suitable for the demonstration of complementary colors. For this, we replace the template by a narrow prism with a small angle of refraction. Then we can obtain two images of the aperture on the screen, preferably partially overlapping. These two images are colored with complementary hues; in the overlap region, an achromatic circular segment (‘lune’) is seen. Tilting the auxiliary prism along the spectrum allows us to image a number of different pairs of colors (Video 29.3). Video 29.3: “Complementary colors” mit? We wish to answer this question experimentally as shown in http://tiny.cc/4gggoy Fig. 29.13, in particular, as an example, for a pure, untinted and un- In the video, some parts of shaded green hue (chroma). the spectrum are not blocked The incandescent light of the arc lamp emerges from the slit S and off by the template, but rather illuminates the aperture F. On the way, it is spectrally decomposed are deflected perpendicular to the spectrum by a thin quartz by the prism and the lens L1. The image of the aperture is projected prism. onto the wall screen by the lens L2. The continuous spectrum appears in the plane aa. In this plane, a template made of opaque cardboard is placed in the optical path. We first use a very narrow slit (A in Fig. 29.13) and locate the desired chromatic hue in the spectrum. We then widen the slit gradually on both sides until it is a broad rectangle. By trial and error, we find the largest width that is still permissible (template B); it gives the maximum possible luminous intensity without white tinting. The ratio = approaches a value of several tenths, so that the light is by no means monochromatic (in the sense of ‘single-frequency’)! Finally, the steep sides of the template can be tapered off somewhat (C). This changes neither the luminous intensity nor the purity of the colors. In the same way as for the green hue here, we can fabricate a broad template for every other pure hue in the color circle. The template D in Fig. 29.13 shows an example for a purple hue9. Templates of this type, with tapered and bent flanks, can be replaced by filters containing selectively-absorbing substances. The filter must allow the same broad regions of the spectrum to pass through as does the template. For demonstration experiments, we proceed as shown in Fig. 29.14. We image the spectrum of the incandescent light (arc lamp) on the wall screen, placing a filter cuvette in front of the slit,

9 For the purple hue, we need two narrow slits, one in the blue or in the violet region, and the other one in the red. itted color chromatic 05 g/liter). After violet red : 0 c , itted region of the spectrum. ı 20 each filter; those that are suitable show one CKSLP A demonstration experiment on the selective light absorption occurs rather sharply at a layer thickness of about 3.5 cm of the circle as bright, and the other half dark. The brightest sector of itself. To demonstrate this, we image a tungsten-filament lamp on Starting with a largeproducing selection pure, of free light hues filters,observe can those a color be that circle selected are through suitable with for relative ease.the We color simply circle shows the hue that can be selected by that filter. half The densely dotted area indicatesspectrum a has high been luminous decomposed intensity. intocorresponds Imagine horizontal to that the stripes. the thickest The layera uppermost of solution narrow stripe region (the of image the is spectrum upsidethe is down!); spectrum transmitted. only corresponds A to middleregion horizontal a stripe represents medium in several layer tenths thickness; of here, the the whole spectrum; transm and so forth. by dye solutions.layer The in front solutions of are theshaped placed entrance layer in slit of of the the waterrefraction. form spectral prevents At of apparatus. the a a upper A disturbing wedge-shaped right, similar deflection we wedge- see of the the transm light due to transition passing through a short filterappears layer thickness, blue-green; the light a from longer the lamp thickness gives winefor a red. lamp operating at The the usual temperature. The concentration and layermeans only thickness the degree of of color whitehue tinting, filters but rather select often the bythe no screen and allow itsfor light example to containing pass an throughlongitudinal a direction indigo (wedge wedge-shaped solution, angle filter, that can be slid in the with a diagonal separation partition.transparent contains The cuvette pure which water;solution. is the shown as Then shaded the cuvettewavelength region, spectrum contains as appears if a it on dye weresition the bounded of by screen the a dye with template. solutionoutlines The only as of compo- one the well spectrum as are itstemplate. identical thickness with are the varied opening until of the the desired Finally, we remove the prism, openit up by the slit itself fairly onimaged wide, and the in image pure screen. chromatic hues.hues Behind are With the tinted decreasing more filter greatest and thickness, filter more the with thickness, white. it is Figure 29.14 Visual Perception and Photometry 29 608 Part II 29.10 Color Filters for Producing Pure Hues 609

Figure 29.15 The dependence of yellow- the chromatic hue of a ﬁlter on its orange bluegreen red layer thickness d K –1 cm a 6

Absorption coefficient 1 b Part II

0.5

d = 0.3 cm

0 1 c Incident radiant power radiant Incident Transmitted radiant power radiant Transmitted = 0.5 D d = 5.8 cm

0 500 600 700 nm Wavelength λ

The explanation can be seen in Fig. 29.15. Part a of the figure shows the dependence of the light absorption constant K on the wavelength. It is small in the red wavelength region, then passes through a maximum in the orange ( D 605 nm), but still remains larger in the blue than in the red region. Parts b and c of the figure show the transmission coefficients D,defined by the equation Transmitted radiant power D D ; (29.13) Incident radiant power for a small and a large layer thickness. At the small layer thickness (b), the short and the longer waves are transmitted equally well: The broad region of shorter wavelength predominates and therefore the light from the lamp appears blue-green. At a larger layer thickness (c), the shorter waves are much more strongly absorbed than the longer waves; thus, the red spectral region predominates and the light from the lamp appears to be wine red.

The layer thickness which corresponds to the color transition varies with the temperature of the lamp (demonstration!). We can thus recalibrate the thickness scale as a temperature scale. This provides us with a very convenient instrument for measuring color temperatures (Sect. 28.7). of of twice white absorp- of surface . Usually, fraction cloudy concentration elimination loudy. One can make out details of are made artificially , the solution is free of any sort of inhomo- (e.g. in numerous articles of clothing) can be obtained , and often, grey shadings appear rather “dirty”. . A black shading can also be obtained by using high concen- contrast between varnish and opaque paints can be demonstrated opaque or topcoat paints . Selective reflection plays practically no role, except in the colors enamel or varnish paints grey shading with any cup ofleaves tea. at Clear the teadrops bottom forms of of milk a the converts “varnish theThe cup layer”, varnish bottom layer and are into of the clearly a the tea a cloudy visible. strong cup topcoat white layer. is tinting Adding can no be some longer observed. visible,In and general, the at colors theduced of same by objects, pigment time, both coatings, natural are colors produced and through those selective pro- tion of metals. Theflected light from from the an surface archues of lamp when a which projected gold onto has or a been screen, copper multiply with object very re- exhibits little reddish white tinting. reflections. This isBlack most shading successful with textilesin of any the desired velvet degree; type. one need only increase the 29.11 Dyes and Pigments The filters containsuspended selectively-absorbing solids substances or in inthe solution. the technical form Among dyestuff such of and solutionsapplied pigment are to an materials. most object of in These two layers ways: In can be geneities which would make itthe surface c through the paint layer.passes Light through coming the from paint a layer to lighttered source the there. surface of the It objectbefore thus and reaching is passes the scat- through eye the ofconcentration, whole the rather layer observer. pure thickness As chromatictinting a hues or result, can shading. with be the Thepainted proper obtained, reflected layer free may light cause of from a themade weak smooth front white as surface a tinting, mirror, of but thus thisrange the limiting surface this of can disturbance specular to be reflections. the Still angular better is the the absorbing substances. The the selectively-absorbing substance is not dissolved, butpended rather as is a sus- fine powder inreach a binding the agent. surface The of incidentthe light the cannot fine painted particles. object; Thus, itessential the is portion surface scattered of of backwards the the by object light is passes covered, and through an only some tinting trations of the absorbing substance, buthas the a ever-present white disturbing tinting effect.a The tinting and the shading together lead to the paint layer thickness. As a result, there is a considerable The Visual Perception and Photometry 29 “The contrast between varnish and topcoat paints can be demonstrated with any cup of tea”. 610 Part II 29.12 Brilliance and Shine 611

dirty red-orange wire mesh matte paper

Figure 29.16 The appearance of brilliance or shine (Video 29.4) Video 29.4: “How brilliance comes about” 29.12 Brilliance and Shine http://tiny.cc/zgggoy

Often, we see objects as not only colored, but also as shiny. As exam- Part II ples, we could mention polished wood and smooth metal objects, e.g. a copper teakettle. Shine or brilliance can be seen whenever light is scattered in strongly preferred directions. In this type of scattering, even small motions of the object or the observer change the perceived luminance strongly. The essentials can be shown in a simple demonstration experiment. In Fig. 29.16, on the right we see a piece of non-shiny, matte red- orange paper. At a few millimeters distance in front of it is a dirty and therefore likewise non-shiny, slightly beaten-up wire sieve. At the left is the light source, an arc lamp. The paper and the sieve can be rotated or swung around together. Every observer believes that he or she is seeing a somewhat dented but still very shiny copper sheet. The reason: The shadow of the sieve falls on the paper surface. At certain angles, the observer looks through the mesh of the sieve at the shadows of its wires; the luminance in such directions is small. From a somewhat different angle, he or she looks through the meshes at the unshadowed parts of the paper, so that the luminance is large. ‘Shine’ is thus seen; it is no more a physical property of the object than is its color.

29.13 Shimmering Colors

Shimmering colors are shiny colors in which a small change in the angle of observation or of the incident light causes a strong variation in the hue. Their hues alone, that is without any directional dependence, can be produced for example with the arrangement shown in Fig. 29.13.We need only replace the template by a screen and use it to mask the main portion of the spectrum, beginning either at its violet or red end. Shiny, that is strongly directionally-dependent, shimmering colors can be seen for example on the wings of butterflies and beetles, and on some ornamental vases. In these cases, we always find substances which form layered structures. In them, either a selective reflection of light occurs on a series of equidistant planes (Vol. 1, Fig. 12.41c), or else a selective transmission of light occurs; the latter is found Die 6/77, (1975), in ”, 2nd edition, 1998, Die Universität Göttingen Elektrizitätslehre ecame clear only after the War, Praxis der Naturwissenschaften . onalsozialismus 20.13 ”, 2nd edition, 1998, page 572f). 1944. As b iticized for using a black-white-red flag (the Praxis der Naturwissenschaften Physik if they had, it would probably have cost him chapter is intended only to stimulate the reader Instead, their mistrust of Pohl, according to statements by the : We have not attempted or claimed to give a complete ”, 6/77, June 15th, 1977, page 159). 29.1 . June 15, 1977, page 159). 3. The local leadershipsuspected Pohl of of the being NSDAP aon (National sympathizer Hitler following Socialist on the or the assassination Nazi 20th attempt party) of July, former Rector of the Universityambiguous of comments Göttingen, was on duenot only the a to color his color critical- brown; atUniversität Göttingen for all”. unter dem example, Nati (Source: thatpage H. 572). “brown Becker, is H.-J.4. Dahms, During C. the Wegeler, War “ pitulation Winter of of Germany while 1944/45,a demonstrating Pohl small an white called experiment. flag indirectlythat to He for mark in attached the a 1919, movable ca- he object had in been the cr experiment, and noted colors of the Oldcolors Regime); of and the later, Weimar for Republic).(Source: using H. Now, Becker, a H.-J. he black-red-gold Dahms, would C. flag simply Wegeler, “ (the use a white flag. unter dem Nationalsozialismus Chap. 25, Sect. 22. Source: Addendum to Comment C29.4: Concerning the formation ofbolic the of color the National brown Socialistswhich (the were in color collected Germany), by which we Mr. E. was add1. Sieker: the sym- When following the quotes first (brown)was SA not uniforms pleased: appeared “Either in youyou his participate won’t lecture have in room, time the Pohl for SA, both!” or (Source: you study “ physics2. – When Pohl was demonstratingment his well-known of experiment on colored theyou move- ions can in see, KClis the crystals, described brown e.g. he in masses the commented are 21st on edition shuffling of the along”. Pohl’s side, (This “As experiment Physik the NSDAP had not the slightestGroup inkling of of Pohl’s civil contacts to resistance the Goerdeler his life to engage in selfcolor study of the very diverse and delightful subject of even at the boundary betweenwave two portion or materials, when the either long-wavereflected. portion the Occasionally, of short- we the are naturalthose dealing light that with is were interference treated totally filters, in Sect. like To conclude, we emphasizeSect. once more the preliminary remarks in treatment here. This ”. color Visual Perception and Photometry 29 “This chapter is intended only to stimulate the reader to engage in self studythe of very diverse and delightful subject of 612 Part II Color Plates 613 Color Plates

Plate 1

Figure A1. The grey scale Part II

Figure A2. The color circle

Figure A3. Hering's triangle (E. Hering, 1874) https:// The online version of this chapter ( ) contains supplementary material, which is Electronic supplementary material doi.org/10.1007/978-3-319-50269-4_ available to authorized users. Visual Perception and Photometry 29 614 Part II Table of Physical Constants

Some important physical constants (CODATA values from Dec. 2014) 23 1 Avogadro constant NA, D 6:022140857 10 mol 11 2 2 Gravitational constant G D 6:6674 10 Nm /kg 12 Electric field constant "0 D 8:854188 10 As/Vm 7 Magnetic field constant 0 D 12:566370 10 Vs/Am 1=2 8 Velocity of light in vacuum c D ."00/ D 2:997925 10 m/s 1=2 Wave resistance of vacuum Zel D .0="0/ D 376:73 Ohm * Relative atomic mass of the proton .A/p D 1:007277 u * Relative atomic mass of the neutron .A/n D 1:008665 u 27 Rest mass of the proton mp D 1:672621 10 kg 8 Rest energy of the proton .Wp/0 D 9:382720 10 eV 31 Rest mass of the electron m0 D 9:101383 10 kg 5 Rest energy of the electron .We/0 D 5:10999 10 eV Ratio of proton mass/electron mass mp=m0 D 1836:152 Elementary electric charge e D 1:602177 1019 As 11 Specific charge of the electron e=m0 D 1:760366 10 As/kg Faraday constant F D 96 485:3329 As/mol Boltzmann’s constant k D 1:380648 1023 Ws/K D 8:617325 105 eV/K Stefan-Boltzmann constant D 5:670367 108 Wm2 K4 Planck’s constant h D 6:626070 1034 Ws2 D 4:135667 1015 eV s

Bohr magneton B D he=4m0 D 9:274010 1024 Am2 (D J/T) 2 15 Classical electron radius rel D 0e =4m0 D 2:820 419 10 m D 4 = "2 3 D : 15 1 Rydberg frequency Ry e m0 8 0h 3 289842 10 s * D 4 = "2 3 D : 1 Rydberg constant Ry e m0 8 0h c 10 973 731 6m 12 Compton wavelength C D h=m0c D 2:426310 10 m 2 Sommerfeld’s ﬁne structure constant ˛ D e =2"0hc D 1=137:036 v ˛ D Electron’s velocity 0 in the smallest H orbit Velocity of light c

* u is the atomic mass unit, u D 1:660539 1027 kg. See http://physics.nist.gov/cuu/Constants/index.html.

I. Electromagnetism

1.1. I D 4:98 A

1.2. 745 hours

1.3. The voltmeter is connected in series with a resistance of 99 .

1.4. Rshunt D 5:025 m

1.5. R D U=.I U=RV/ (If OHM’s law does not hold for the conductor, then R depends on I.)

1.6. Two values of R2 fulﬁll this condition: R2 D 1 and 4 !

1.7. The battery has an internal resistance Ri of 1 , UI D 1:25 V

1.8. N D 147

1.9. I D 0:167 A

1.10. U D 6:8kV P P 1.11. a) I D 1:25 A, b) Q2=Q1 D 3

2.1. a) % D=h;b)E D.1="0/%.h x/, up to the height h, E is negative, and for x h, 2 2 E D 0; c) Ux D.1="0/%.x =2 hx/;d)U D .1="0/%h =2

2.2. C D 356 F

2.3. C D 0:9995 nF

2.4. CV D 0:9nF

10 2 2.5. Ne D 5:54 10 cm

2.6. C=l D 93 pF/m

617 618 Solutions to the Exercises

2.7. a) 1.37 m from Q1, on the side away from Q2; b) 25 cm away from Q1, between the two charges R 2.8. U dt D 105 Vs

2.9. a) t1=2 D RC ln 2; b) C D 44:8pF

2.10. With I D dQ=dt and U DQ=C,weﬁnddQ=dt DQ=RC C Ug=R;.Ug D 157 V), t=RC with the solution Q D Ae C UgC;.A D Umax Ug; Umax D 220 V). It then follows that D 1:16 s. R D D D 2 ; D . = / 2 2.11. WBatt U0 I dt U0Q U0 C WC 1 2 U0C (Eq. (3.19)). Half of the energy supplied by the battery is therefore converted to heat in the resistor, independently of its resistance! This result however does not hold for R D 0, since then the condenser would be charged within an inﬁnitely short time without allowing any energy to be converted to heat. See Chap. 10, “Electrical Oscillations”.

2.12. 17 plates

2.13. " D 2:8

2.14. 6.5 % of the volume between the plates is ﬁlled with metal. This reduces the spacing of the plates effectively by 6.5 %, i.e. the capacitance is increased by about 7 %. This leads to " D 1:07.

2.15. Corresponding to Eqns. (2.11)and(2.25), the capacitance of the completely ﬁlled condenser 2 would be C D ""0.ab=l/, and thus the energy stored in it would be We D .1=2/""0.U=l/ abl. In the present experiment, the energy thus increases when h increases, by an amount We D 2 .1=2/." 1/"0.U=l/ hbl D Fh. The liquid is thus pulled up into the condenser with a constant force F. h is then found by setting F equal to the opposite force of the weight of the liquid which 2 has been pulled up: h D .1=2/." 1/"0.U=l/ =.%g/.

2 2.16. The electrical energy is We D .1=2/""0.U=l/ Al. This leads (compare Eqns. (3.17)and 2 (3.11) in Sect. 3.7) to the force F D .1=2/""0.U=l/ A.

3.1. The weight of the brass plate is F D 1:6 N. Using Eq. (3.12), this leads to l D 50 m (this corresponds roughly to the thickness of a human hair; see Vol. 1, Sect. 1.2). The roughness of the polished stone causes an average spacing which is however certainly much smaller, so that a considerable fraction of the potential drop must occur within the stone!

3.2. R D 80 nm

3.3. The capacitance C and thus also the energy We vary by a factor n.Ifn < 1, then energy will be pumped back into the current source. R 5.1. U dt D 3:2 104 V s, in good agreement with the observed 10 scale divisions R 5.2. a) u D 2:04 m/s; b) U dt D 3:9 105 V s, in good agreement with the observed 1.2 scale divisions (3:8 105 Vs) Solutions to the Exercises 619

5 2 ı 5.3. a) Bh D 2:5 10 Vs/m ;b)' D 66 (In Göttingen, the horizontal component of the earth’s magnetic ﬁeld points to the north and its vertical component points downwards).

5.4. D 1:06 Hz

5.5. The voltmeter indicates zero! (See also the ﬁrst paragraph in small print in Sect. 5.5). The pilot could indeed detect an electric ﬁeld, e.g. by carrying out the experiment of W. WIEN described at the end of Sect. 7.3, but he cannot detect the voltage of U D 0:8 V between the wing tipsasgivenbyEq.(5.9), since it will be compensated by the charges which collect on the wing tips. (The “insulated wire” corresponds to the conductor CA in Fig. 7.2).

5.6. The angle between the ﬁeld vector B and the surface vector A is ˛ D 2 t.Fromthis, it follows from Eq. (5.10)thatU Dd=dt.BAN cos.2 t// D 2 BAN sin.2 t/,i.e.anAC voltage, as described in Chap. 8 (Fig. 9.3). Note that this result does not depend on the frame of reference. R 6.1. a) I D 93 A ( H ds DR1:85 scale divisions); b) measured: H D 1360 A/m, calculated with Eq. (4.1): H D 1610 A/m; c) H ds D 500 A

6.2. a) Since the magnetic potentiometer measured outside all the remaining coils along a closed path (Fig. 6.7), we ﬁnd Umag D 0; b) From a), it follows that Umag;i DUmag;a.

6.3. If we combine the two methods of measuring the magnetic potentials Umag;a and Umag;i, a closed path results. Around this path, the magnetic potential is always equal to zero unless the path encloses a current. This is the case here, since, as mentioned in Sect. 6.3, each tunnel bored through the material does not pass through individual molecules but rather between them; i. e. no molecular currents can be included. We thus have Umag;i DUmag;a. The same result follows also from the MAXWELL equation (14.12). (See also Exercise 6.2.)

8.1. B D 3:37 104 T

8.2. Mmech D 0:1Nm

8.3. From the area which is marked with dashed lines, we obtain the ﬂux ˚ 1:5 106 Vs.The magnetic ﬂux measured in Video 5.1 is 8 106 Vs.

2 2 2 8.4. Wmagn D .1=2/0.N =l/I r . Wmagn increases when l becomes smaller. In order to prevent this, we require a force of F D 102 N.

10.1. L D 0:55 H

2 10.2. LN D N L1

10.3. R D 3:466

10.4. Ieff D 0:796 A

10.5. D 50 Hz

10.6. L D 123 mH 620 Solutions to the Exercises

10.7. D 97:5Hz

10.8. L D 158 mH

10.9. LC D 2:53 108 s2

ı 10.10. a) ZRL D 849 , '1 D 87:4 (the current in the RL branch “lags behind” the applied ı ı voltage); b) IRL;0 D 86 mA, IC;0 D 84:85 mA, I0 D 86 mA sin 2:6 D 3:9mA, '2 D 15:2 ; c) It follows from these values that Z D 73 V=3:9mA D 1872 , and from Eq. (10.33), we ﬁnd Z D 1840 , in agreement with the value found in b) (compare this value also with the one that can be read off at the maximum of the curve in Fig. 10.20).

10.11. D 0:139, in agreement with the value in Fig. 10.20. P 10.12. a) The active current is 3.77 mA and the reactive current is 1.0 mA; b) W D .1=2/ U0I0 ı P 2 cos 15 D 0:138 W, WRLC D .1=2/.IRL;0/ R D 0:141 W, and thus within the error limits, the values are the same.

D ! D =. ! 2 / 10.13. The current in the primary coil is Ip Ip;0 sin t, with Ip;0 Up;0lp 0 Np Ap (Eq. (10.4)). From this we find, with Eqns. (4.1)and(5.4), the flux density B of the magnetic field from the primary coil, and with it, applying the law of induction (Sect. 5.6), the voltage across the secondary coil, Us D Ns.dB=dt/Ap. This leads finally to Eq. (10.39).

10.14. a) Owing to the phase difference of 90ı between the current and the voltage, in the primary circuit we expect WP D 0. b) With a finite value of R, a current will flow in the secondary circuit. The thermal energy from JOULE heating in the secondary circuit then must come from the primary circuit. The mechanism for this is that the current in the secondary circuit induces an additional current in the primary circuit, so that the phase difference there is no longer 90ı, and a finite active current results.

11.1. The voltage across the condenser is UC D Q=C. The voltage across the coil is UL D 2 2 2 2 L dI=dt D L d Q=dt . These voltages are equal and opposite at all times: QC=C C L d Q=dt D 0. D ! This differentialp equation can be solved by the trial function QC QC;0 cos t. This then yields ! D .1=LC/. Thus, we ﬁnd that UC D .Q0=C/ cos !t and I D dQ=dt DUC;0!C sin !t.

11.2. UL D L dI=dt (Eq. (10.16)). Therefore, UL;1 D L1 dI=dt and UL D .UL;1 C UL;2/ D .L1 C L2/ dI=dt. From this, it follows that jUL;1=ULjDL1=.L1 C L2/. The coils thus act as a voltage divider. An application is shown by the experiment described in Fig. 11.7. With the correct choice of the inductance of the wire loop relative to the overall inductance of the oscillator circuit, a voltage will be induced which causes the lamp to light up without burning out.

11.3. With the values of the capacitance C D 3:2 nF and the inductance L D 0:048 mH, we ﬁnd the frequency to be 0 D 400 kHz.

12.1. The rectiﬁer allows only half of the sinusoidal current to pass. The throttle coils in Fig. 12.21 pass only the low-frequency FOURIER components of this current (for FOURIER analysis, see Vol. 1, Sect. 11.3). These components include a direct-current contribution which is proportional to the amplitude I0, and it will be indicated by the galvanometer. Solutions to the Exercises 621

12.2. a) Using the trial function Ex D Ex;0 sin !.t C z=c/, superposition with the incident wave gives Etot D 2Ex;0 sin !.z=c/ cos !t, that is a standing wave with a node at z D 0, as seen in Fig. 12.28. b) The magnetic component of the incident wave is described by the term By DBy;0 sin !.t C z=c/, and thus its amplitude points in the negative y direction (right-handed coordinate system, wave propagation in the negative z direction). The reﬂected magnetic component is given by By D By;0 sin !.tCz=c/ (in phase with the electrical component; the sign follows again from the right-handed coordinate system). Superposition of the two magnetic components gives Btot D 2By;0 cos !.z=c/ sin !t, i.e. again a standing wave, but now it has a maximum (wave crest) at the reﬂecting surface (z D 0), with the amplitude 2By;0.

12.3. a) E D 220 V/m, B D 1:8 10 4 Vs/m2, S D 31:5kW/m2;thePOYNTING vector points towards the interior of the spiral, as if the thermal power were deposited in the spiral from outside. b) b D 1:35 kW/m2, E D 713 V/m, B D 2:38 106 Vs/m2 (also effective values).

12.4. The electromagnetic wave which is radiated from the outside and is used to excite the small dipoles of the water molecules (Fig. 12.6) has a wavelength of D 3 m. With its velocity of propagation, c D 3 108 m/s, we ﬁnd its frequency to be D 108 Hz. The dielectric constant of water at this frequency is " D 81 (Table 13.2 and Fig. 13.11); thus its index of refraction is n D 9:0. It follows from this that R D 0:64 (at normal incidence). 36 % of the incident radiation thus penetrates into the water; this is sufﬁcient to excite the small dipoles.

13.1. The energy changes from .1=2/QU0 to .1=2/QUm,whereUm D U0=". The excess energy is taken up by the person who places the plate into the condenser.

13.2. The voltage is U0 D E.l d/ C .E="/d with the electric ﬁeld E D U=l. Solving for ",we ﬁnd " D d=.d C l.U0=U 1//. In this example, the result is " D 5.

13.3. The voltage between the points 1 and 2 is zero when the voltage drop over the capacitance C1 is equal to that over the resistance R1. Since the charges Q on the two condensers are then equal, we have Q=C1 D I=R1 and Q=Cx D I=R2. From this, it follows that Cx D C1R1=R2.

13.4. The electric ﬁeld E experiences no depolarization (Eq. (13.11)). It follows that D D "D0. The displacement density in the rod is thus larger than in the space outside it. The concept of “depolarization” refers to the ﬁeld E (see also “demagnetization”, Sect. 14.6)

13.5. P D 3.." 1/=." C 2//"0E0 (compare this result with the magnetization of a sphere, Eq. (14.18))

30 13.6. pp D 3:04 10 A s m, and thus only half a large as measured in the gas phase. The reason for this discrepancy is that in deriving the CLAUSIUS-MOSSOTTI equation within the “cavity”, the interactions between neighboring molecules were not taken into account; that is, for water, the hydrogen bonds between the water molecules (see H. Fröhlich, “Theory of dielectrics”, Oxford University Press (1949), p. 137).

13.7. In paraelectric materials, " depends on the temperature, both due to the equation of state of ideal gases and to the LANGEVIN-DEBYE theory. For the electric susceptibility, one ﬁnds e D " D . = " / 2=. /2 1 1 3 0 pp kT p (to distinguish it from the pressure p, the dipole moment is denoted here by pp). This explains the difference between the two values of e to better than 1 %. 622 Solutions to the Exercises

13.8. a) From Fig. 13.11, we read off the dielectric constant at this frequency, thus obtaining P the index of refraction n D 9:0. It then follows that W D 1:36 cm. b) W D 697 W. c) S D 2 11:6kW=m D 8:6 Ee (this is about the same as the value of the solar constant on the planet Mercury!) (For reﬂection losses, see Exercise 12.4).

5 14.1. m D1:42 10 . The permeability is thus only slightly greater than 1.0, so that the demagnetization effect can be neglected.

14.2. a) Analogously to Eq. (13.15), we have here H D H0 C Hm. Comparing with Eq. (14.18), we then ﬁnd Hm DM=3 D.. 1/=. C 2//H0.b)FromEq.(14.7), it follows that Bm D .2=3/0M.

14.3. Hi D 3aHa=.i C2a/; from this, we ﬁnd i D 1andHa D H0=a: Hi D 3H0=.1 C2a/, and for a 1: Hi .1:5=a/H0.

14.4. F0 D 600 N, Fd D 17:6 N. Due to the hysteresis of the iron (Fig. 14.7), we must consider these values to be only rough estimates.

14.5. a) and b): Following the rapid change of the current during the ﬁrst second, both measurements show an exponential variation with a relaxation time of r D 10 s. c) Here, also, following an initial rapid change, an exponential variation begins, but with a relaxation time of 110 s. Only near the saturation value does the rate of change become more rapid and again approaches a relaxation time of 10 s. For a qualitative explanation: During the initial rapid variation, of which we can see an inkling even in experiment c), the magnetic domains cannot follow. The coil thus has a small inductance during this variation. Thereafter, the reversals of the domains delay the rate of change of the current in the coil in the same manner as the changes of the current in the coil delay its rate of change due to induction. The coil now has a larger inductance because of the iron core. From experiments a) and b), we obtain L D rR D 1300 H. In experiment c), the delay of the current changes in the coil is even greater, since in the ﬁrst 60 seconds, the magnetic domains (remanent magnetization) rotate through 180ı, causing the induced voltage which delays the current change to be increased. This results in a still larger inductance: L D 14 000 H. Only near the end of the increase in the current, after the remanent magnetization has been completely overcome, does the rate of change again correspond to the same time constant as in experiments a) and b); L is thus again smaller.

14.6. In Exercise 14.2, it was shown for a magnetized ball that the field Hm in its interior is given by Hm DNM (N is the demagnetization factor). Starting from Eq. (14.14), it can be shown that the same result holds for every ellipsoid of rotation which is magnetized parallel to its axis of rotational symmetry. For the disk with l=d D 0:1, we find N D 0:863 (Table 13.4). For a long rod (l=d D1), N D 0, and the field Hm is thus zero. From this, with Eq. (14.7) we find for the field Bm in the disk the value Bm D 0.M 0:863M/ D 0:1370M, and for the rod, Bm D 0M. These flux densities continue into the space outside the material, since div B D 0(MAXWELL’s equation (14.13)).

II. Optics

16.1. From the expressions for the law of refraction on both sides of the prism, sin ˛1 D n sin ˇ1 and sin ˛2 D n sin ˇ2, we obtain from the sum and the difference: sin ˛1 ˙ sin ˛2 D n .sin ˇ1 ˙ sin ˇ2/. Rearranging, this gives sin..˛1 C ˛2/=2/ cos..˛1 ˛2/=2/ D n sin..ˇ1 C Solutions to the Exercises 623

ˇ2/=2/ cos..ˇ1 ˇ2/=2/ and cos..˛1 C ˛2/=2/ sin..˛1 ˛2/=2/ D n cos..ˇ1 C ˇ2/=2/ sin..ˇ1 ˇ2/=2/; dividing these two equations yields: tan..˛1 C ˛2/=2/= tan..˛1 ˛2/=2/ D ntan..ˇ1Cˇ2/=2/= tan..ˇ1ˇ2/=2/. With ˇ1 Cˇ2 D and ˛1 C˛2 D ıC ,Eq.(16.8) follows. b) ı ı Inserting the values of ˇ1 D 19:47 and =2 D 30 into Eq. (16.8), we obtain a quadratic equation for tan.ı=2/, and thus initially two solutions for ı. To ﬁnd the correct solution, we consider a sketch corresponding to Fig. 16.13. The result is ﬁnally ı D 47ı.

16.2. The point at which the incident beam in Fig. 16.24 is reﬂected by the mirror is called A,and the angle of reﬂection is ˛.ThenwehaveAF D FZ D R=.2cos˛/. For rays near the axis of the mirror (paraxial rays), ˛ is small and so cos ˛ 1. It follows that f D R FZ D R=2.

16.3. A scale drawing corresponding to Fig. 18.1 for a concave mirror with an object y at a distance a and the associated image y0 at a distance b allows us to compare similar triangles and obtain y=y0 D f =.b f / D a=b, from which it follows that 1=f D 1=a C 1=b.

16.4. A parallel shift of light beams, even when they have differing wavelengths, leaves the image in the focal plane of the eye unchanged (see e. g. Fig. 18.24, with parallel light beams between the lens and the eye).

17.1. a) d 100 m; b) B 11 cm

7 7 17.2. 2'min D 1:04 10 , ˛ D 2:28 10

18.1. Considering two pairs of similar triangles in Fig. 18.1,weﬁndy=y0 D a=b D f 0=.b f 0/. From this, it follows that 1=a C 1=b D 1=f 0.

20.1. a) mmax D 133, x D 21 cm, b) x10 D 1:19 m

20.2. The observation is carried out at an angle of inclination ˇm, which can be treated as constant for small changes in the order number. Then from Eq. (20.4), we ﬁnd d D .1=3/=2 D 0:1 m.

21.1. a) No change, since over the whole slit, the same path differences are found; b) likewise no change, since the path difference due to the glass, D d.n 1/, is an integral multiple of the wavelength ( D 145:8 , thus nearly the “order-one position”); c) ˛ D 8:2ı. since in general the quantities d, n and are not known with sufﬁcient accuracy, this calculation shows how one can obtain the two positions in practice.

21.2. Due to the phase difference of =2 between the two halves of the slit, in making the construction, we must rotate the arrows 7 to 12 by 180ı; we then obtain the diffraction pattern of the order-two position, that is with extrema at sin ˛ D N=B,whereN D 1; 3;:::for the maxima and N D 0; 2; 4;:::for the minima. (A more precise calculation shows that the maxima at N D 1 (Fig. 21.12) are slightly shifted.)

21.3. The angular range of the principal maximum determined by the width B of the slit ranges over sin ˛ D˙=B. The maxima which occur through interference with immediately neighboring slits are found at sin ˛m D˙m=D (m D 0; 1; 2;:::). Then for a), we observe only three maxima, but for b) there are 39 maxima.

21.4. a) In this case, the whole surface of the grating radiates with the same phase; diffraction occurs only at its edges. b) The diffracted waves which emerge from two neighboring pairs of beams and slits produce a diffraction pattern as shown in Fig. 21.12, lower left. The superposition 624 Solutions to the Exercises

of the waves emerging from all the beams and slits leads to a sharpening of the maxima (Fig. 22.6), but leaves their positions and relative heights unchanged.

21.5. Since in the order-two position (ˇ), the radiant power is zero for the non-deﬂected beam (˛ D 0), the power of the light which passes through the gaps and the “grating beams” must be the same. Thus, if we can neglect absorption in the grating beams, they must have the same width, i.e. L=D D 1=2.

21.6. c D m =.m sin ˛/ D 875 m/s

22.1. a) D 600 nm, b) n D 1:33

2 2 26.1. a) The equation of motion for free damped oscillations is given by FRCFD D m d l=dt , with ıt FR D˛ dl=dt (frictional force) and FD DDl (elastic force). That the expression l D l0e , with ı D .1=2/.˛=m/, is a solution of this equation of motion can be demonstrated by substituting it into that equation. For the relation between ˛ and , we obtain with this solution ˛ D 2ım D 2 0m. 2 2 b) The equation of motion is now given by Fp C FR C FD D m d l=dt , or, after inserting the 2 = 2 C = C 2 2 D . = / . / forces and dividing by m:dl dt 2 0 dl dt 4 0 l F0 m cos 2 t .Wenowinsert i.2 t/ the complex trial solution l D l0e together with its ﬁrst and second time derivatives into the 2Œ. 2 2/ C . =/ D . = / i' equation of motion, obtaining 4 0 i 0 F0 l0m e . From this, applying the mathematical relations aCib D r.cos ' Ci sin '/ D rei' (Eq. (25.27)) and sin2 ' Ccos2 ' D 1, we obtain equations (26.1)and(26.3). (For literature on “forced oscillations”, see e.g. The Feynman Lectures, Vol. 1, Chap. 21-5 (http://www.feynmanletures.caltech.edu/), or http://hyperphysics.phy- astr.gsu.edu/hbase/oscdr.html).

. = / 2 = 2 D . =/2 4=Œ. /2. /2 C . =/2 4 27.1. 1 2 l0;max l0;max 0 2 0 0 0 , . = /. /2. /2 D . = /. =/2 4 . / D D . =/ 1 2 2 0 0 1 2 0 ,2 0 H 0. This result holds for every harmonic oscillator, in particular for an electrical resonator.

27.2. Let the deﬂection be x D l0 sin !t; taking the time derivative gives: xP D !l0 cos !t. 2 2 The kinetic energy is thus Wkin D .1=2/mxP D .1=2/m.!l0 cos !t/ . With the average value of 2 2 cos !t D 1=2 over a whole oscillation, it follows that Wkin D .1=4/m.!l0/ . Index

A Alternating-current generator, 161, Avogadro constant, 274 Abbe, Ernst 164 Axis, optical, 471 imaging and diffraction, 340, Ammeter or ampere-meter, 11 434 Ampere second, 21, 47 B Aberration Ampere, unit of current, 12 Babinet’s theorem, 416, 436 astronomical, 460 Ampere’s law, 119, 122 Ballistic deflection or impulse spherical, 344 Ampere-turns, 96 deflection, 21, 52 Absorbance, 579 Amplitude, 181, 182, 208 Ballistic galvanometer, 145 Absorption, 492 Amplitude of a light wave, 318 Barkhausen effect, 285 and electrical conductivity, 565 Amplitude of alternating current, Barlow’s wheel, 109 and forced oscillations, 557 181 Beam boundary, 299 by atomic and colloidally Amplitude structure, 422, 436 Beam splitter, 384 dissolved metals, 570 Analyzer, 479, 481 Beer’s formula, 509, 542 by free electrons in metals, 565 Angle of inclination Drude’s approximation, 566 interpretation of, 556 of principal rays, 363 Beer’s law, 494 weak and strong, 496, 497 Angle of vision, 333 Belt generator, 44 Absorption band, 559 definition, 353 Birefringence, 478, 574 Absorption coefficient, 258, 494 increase (magnification), 356 due to flow, 574 Absorption constant, 492, 559 increasing, 354 Birefringence or double refraction, definition, 492 Angled mirror, 324 468, 472 in the X-ray region, 538 Angles of calcite, 473, 474 Table, 493 units, 314, 333 of mica, 475, 478 Absorption spectra Angular momentum, 97, 277 of quartz, 473, 488 analysis, 560 Angular velocity or circular Black body, 372, 379, 580, 594 AC (alternating current) frequency, 181 Black-body radiation, 580 generator, 195 Antenna, 31, 234 Black-body temperature, 587 resistance, 183 Anti-reflection coatings, 502 Blazed grating, 453 synchronous motor, 166 Aperture, 310 Bohr magneton, 277 Achromatization, 351 Aperture diaphragm, 299 Bohr’s atomic model, 277 Active current, 190 Aperture stop, 342 Boltzmann’s constant, 256, 275 Activity, optical, 486 Aperture, numerical (NA), 358 Bose, G. M., 75 Acuity Area turns, 107 Bound charges, 63, 245 visual (sharpness of vision), 363 Astigmatism, 311, 345 Boundaries of light beams Aepinus, F. U. T., 36 Atmospheric ray refraction, 555 and perspective, 367 Air Atomic model, Bohr’s, 277 Boyle, Robert, 33 in a magnetic field, 266 Attraction Bragg spectrograph, 455 molecular picture, 34 electric, 72 Braun’s tube, 22, 77 Air gap of an electromagnet core, electric, of current-carrying Brewster’s angle, 498 272 conductors, 7 Brewster’s law, 498 Alkali halides magnetic, 156, 273 Bridge circuit, 244 absorption of light, 537 Attractive force Bright-field illumination, 358, 433 reflectivity, 543 magnetic, of current-carrying Brightness, 600 Alternating current, 161 conductors, 139 Brilliance, 611 production, 195 Auer mantle, 585 Brownian molecular motion, 19 production, high-frequency, 198 Autocontrol (feedback) Buildup of a conduction current, Alternating current D AC, 180 with diodes, 204 177

625 626 Index

C Complex index of refraction, 508, inductive, 160 Calibration factor 509 Current transformer, 190 of instruments, 108 Complex numbers, 507 Curved light rays, 553 of measurement instruments, 13 Compound lenses, 335 Cyclotron frequency, 237 Candela (cd), unit of the luminous Concave gratings, 453 Cylindrical lenses, 310 intensity, 593 Concave lenses, 315 Cylindrical rotor, 162 Capacitance, 56 Concave mirrors, 315 of a parallel-plate condenser, 56 Concentration, definition, 494 D of a sphere, 57 Condenser, 30, 72 Damping, 496, 557 with a dielectric, 62, 243 charging and discharging, 59 Damping constant, 207 Capacitive reactance, 185 electrolytic, 61 Damping, aperiodic limit, 145 Capacitive resistance, 185 in an AC circuit, 184 Dark-field illumination, 358, 433, Carrier of electricity, 31 technical, 61 436 Cathode-ray tube, 22, 77 Condenser D capacitor, 28 Dazzling, 602 Cat’s eye, 325 Condenser lens, 342, 343, 358, 363 DC motor, 167 Cauchy’s formulas, 513 Condenser plates, attraction of, 72 Debye, cgs unit for molecular Central projection, 366 Condensers electric dipole moments, 246 Characteristic (I-V)curves,204 examples, 31, 56 Debye-Scherrer method, 440 Charge carrier, 42, 81 parallel circuit, 56 Debye-Sears experiment, 426 Charge, electric, 34, 37 series circuit, 56 Decrement, logarithmic, 207 bound, 245 Conductance Deflection of a current-carrying definition, 48 specific, 16 conductor in a magnetic influence, 41, 52 Conductivity, 16 field, 7, 138 measurement, 46 electrical, 36, 565 Delimiting of light beams, 343 moving; production of magnetic specific, 232 Demagnetization, 271 fields, 94 Conductor, 36 Demagnetizing factor, 251, 271 separation, 52, 159 Continuous spectrum, 320 Depolarization, 250 surface density of, 51 Control grid, 203 Depolarizing factor, 251 Charging a condenser, 59 Control of voice currents, 165 Depth of field or depth of focus, 365 Choke coil, 225 Converging lenses, 315 Depth of focus, 365 Christiansen filter, 514 Convex lenses, 315 Diamagnetic materials, 263, 264, Chroma, 605 Corner mirror, 325 279 Chromatic aberration, 350 Corona ring, 53, 73 Dichroism, 471, 573 Chromatic colors, 604 Coulomb D 1 ampere second, 48 artificial, 573 Circular vibrations, 170 Coulomb, Charles A., 52 Dielectric, 62, 243 Clausius-Mossotti equation, 254, Coulomb’s law, 52, 72 Dielectric constant, 62, 217 549 Creeping galvanometer, 145 complex, 566 Closure bar Critical angle for total reflection, definition, 243 for magnetic circuit, 147 306 frequency dependence, 257, 549 Coaxial cable, 237 Crystal measurement, 243 Coercive field, 268, 288 principal plane, 472 of dielectric materials, Table, Coherence condition, 385, 390, 407 Crystalline axes in mica, 475 246 Coherence, partial, 408 Curie constant, 264 Table, 244, 246 Collimated beams, 300, 381, 491 Curie temperature, 248, 264 Dielectric materials, 246, 252 of light, 301 Curie’s law, 264 Dielectric polarization, 245 Collimator, 444 Current Dielectric susceptibility, 245 Color centers, 559, 570, 571 as a moving charge, 96 Diffraction, 317, 318, 328, 413 Color filters for pure hues, 606 during electric-field decay, 46 by a step, 422 Color hue, 588, 605 during magnetic-field decay, 179 model experiments, 318 Color temperature, 588 effective value, 181 of X-rays, 438 Colors electrolytic effect of, 9 Diffraction pattern achromatic, 603 examples, 17 from a semi-plane, 416 chromatic, 604 time integral, 20 from a slit, 319, 407 of the spectral regions, 322 Current impulse, 20, 46 from a step, 422 tints and shades, 605 measurement of, 145 from a wire, 417 Coma, 348 Current meter D ammeter, 11 from circular disks and Compass needle, 4, 88, 92, 96, 155 Current resonance, 189 openings, 414 Complementary colors, 607 Current sources, 159 in model experiments, 318, 423 Diffuse reflection, 306, 517, 532 Index 627

Diodes, 204 theory of relativity, 103, 133 number density, 527 Diopter = unit of lens power, 314 Electret, 85, 132, 248, 251 quasi-elastically bound, 560, 562 Dipole Electric charges Elementary electric charge, 77 electric, 81, 213, 215 bound, 63 Elliptically-polarized light, 476, 479 electric field of, 223 free, 63 Emissivity in water, 217 Electric current, 4, 10 of the sun, 376 magnetic, 153 Electric dipole, 81, 213, 215 Emitter area, projected, 373 radiationfieldof,228 in an inhomogeneous electric Enamel or varnish paint, 610 Dipole field, 225 field, 83 Energy Dipole moment, 82, 245 Electric dipole moment, 82, 520 electrical, measurement of, 23 electric, 520 from influence, 84 of the electric field, 78 electric, of polar molecules, 521, of polar molecules, 564 of the magnetic field, 156, 179 564 permanent, 85 resonance curve, 209 molecular, 246, 257 Electric field, 27, 30, 114 Energy current, 401 Dipole radiation, 518, 521 decay of, 36, 211 Entrance pupil, 340, 343 Direct-current dynamo, 163 direction of, 49 Equipotential surface, 79 Direct-current generator, 162 energy of, 78 Equivalence principle Direction of the electric current, 93 force on a charge, 68 Faraday, 141 Direction of the magnetic field, 93 general definition of, 70 Faraday’s equivalence principle, Dispersion, 322, 443, 539 gradient of, 54 12 anomalous, 538, 540 measurement of, 49 Exit pupil, 340, 343 by free electrons, 566 of a dipole, 225, 228 Exit window, 363 curves, 538, 539, 541, 547 path integral of, 50 Extinction demonstration experiment, 539 shielding of, 42 by large metal colloids, 572 interpretation, 546 within matter, 249 by scattering, 492, 523 normal, 444, 538 Electric field constant, 52 by scattering of X-rays, 526 of a prism, 445 Electric field lines, 30, 71, 75 by small, strongly-absorbing of NaCl, 550 Electric field strength, 49 particles, 569 quantitative treatment, 548 Electric fields by very fine colloidal metal Dispersion formula, 550 dependence on the frame of particles, 572 for gases and vapors, 550 reference, 129 Extinction constant, 492, 523 Displacement current, 123, 184 in vacuum, 33 molar, 494 Displacement current density, 124 Electric flux, 55 of a metal, 541 Displacement density, 52, 123 Electric flux density, 52 of NaCl, 538 Displacement law of W. Wien, 583 Electric motor, 166 specific, 494 Distinct vision Electric polarization, 249 Eye distance of, 353 Electric potential, 80 acuity (sharpness of vision), 365 Diverging lenses, 315 Electric voltage, 13 adaptation, 598 Doppler effect, 463, 520 Electric waves, see Waves, as a radiation detector, 293 Double refraction electromagnetic, 211 fixing, 362, 599 due to inner strains, 489 Electric wind, 58 fovea centralis, 363 of an optically-active material, Electrical conductivity, 565 nodal points of, 338 487 Electrical oscillations, see range of accommodation, 353 Double-fiber voltmeter, 14 Oscillations, electrical, 196 resolvable angle of vision, 333 Drag of light, 552 Electrification, 84 retina, 332, 599 Dressing of glass surfaces, 502 Electrodeless ring current, 202 rise and accumulation times, 600 Drift velocity, 141 Electrolytic condenser, 61 sensitivity, 599 Du Fay, C. F., 34 Electromagnet, 143, 156, 272 spectral sensitivity distribution, Electromagnetic fields, 135 597 E Electromagnetic waves, see Waves, Eyeball Earth electromagnetic, 211, 230 turning while observing, 363 electric field, 53, 54 Electron, 77 Eyes magnetic field, 94, 108 mass of, 527 of insects, 555 Earth inductor, 109 Electron beam, made visible, 137 Echelette grating, 453 Electron volt (eV), energy unit, 538 F Eddy currents, 142 Electronic spin, 278 Farad, unit of capacitance, 56 Eddy-current damping, 145 Electrons, 276 Faraday cage, 42 Effective values, 181 free, 565 Faraday constant Einstein equivalence principle, 12, 141 628 Index

Faraday effect, 486 in an electric field, 68 Grazing angle, 439, 455 Faraday, M., 101 in an inhomogeneous electric Green flash, 555 Feddersen spark, 198 field, 83 Grey filter, 543 Feedback in an inhomogeneous magnetic Grey scale, 603 in a klystron, 235 field, 151 Ground, 31 with triodes, 202 Forced oscillations, 205, 517, 520, Grounding, 80 Feedback D autocontrol, 203 548 Group velocity, 238, 459 Fermat’s principle, 304 quantitative treatment, 207 Guericke, Otto von, 45, 76 Ferrites, 287 Forced oscillations D forced Guericke’s levitation experiment, 76 Ferroelectric materials, 247 vibrations, 566 Gyromagnetic ratio, 277 Ferromagnetic materials, 264, 266, Forces of an electron, 278 280 between two parallel currents, Field gradient, 55 139 H Field lens, 363 in electric fields, 70 Half-maximum width, 208 Field lines in magnetic fields, 137 Halfwidth, 448 electric, 30, 71, 75 inhomogeneous, 263, 265 of diffraction patterns, 444 electric, closed loops, 113, 201, on moving charges, 137 of spectral lines, 559 228 Forward scattering, 530, 534 of the energy-resonance curve, magnetic, 4, 87 Fovea centralis, 363 558 around and in a conducting Frames of reference, 103 Halo phenomena, atmospheric, 305 wire, 200 Franklin, Benjamin, 42 Hammer interrupter, 192 Field of view, 362 Fraunhofer’s observation mode, Harmonic oscillator, 517 Field of view aperture, 363 406, 416 Harmonic oscillator circuit, Field strength, 92 Free charges, 63 electrical, 212 electric, 31, 49 Fresnel zone plate, 428 Harmonics, 218 magnetic Fresnel’s formulas, 501 Hauksbee, F., 586 at the center of a cylindrical Fresnel’s observation mode, 416 Headlight, 378 coil, 120 Fresnel’s zone construction, 413 Heating by electric current D Joule Field, electric; see Electric field, 27, Frictional electricity, 45 heating, 8 30, 114 Frictional electrification machine, Heaviside layer, 568 Field, magnetic; see Magnetic field, 22 Hefner candle, 594 87 f-stop Helmholtz coils, 120, 137 Filter, 323 camera lens, 366 Henry, J., 173 for infrared, 323 Fundamental oscillation, 217 Henry, unit of inductance, 175 for pure hues, 606 Hering’s triangle, 606 for ultraviolet, 323 G Hertz, H., 227 Fixed stars, 333, 408 Galilean telescope, 360 Heusler alloys, 268 aberration, 460 or terrestrial telescope, 361 Hilsch, R., 204 color temperature, 589 Galvanometer, 7, 11, 19, 102, 145 Holography, 431 parallax, 601 Gas molecules Hysteresis loop, 247, 267 Fixing with the eye, 362, 599 average spacing, 33 Flame, light absorption, 578 mean free path, 34 I Flickering, limiting frequency, 596 Gauss (unit) D 10 4 T, 107 Illuminance, 594, 600 Flow birefringence, 574 Generator, 159 Illuminances Fluctuations, 410 Geometric distortion, 349 definition of two equal, 595 Fluorescence, 380 Gilbert, W., 84 Illumination intensity, 594, 600 Fluorescent tube, 586 Gold-leaf electrometer, 14 Image definition Flux Gold-leaf voltmeter, 14 aberrations, 349 electric, 55 Gradient Image distance, 335 magnetic, 106, 261 in the refractive index, 553, 555 Image formation, 327 Flux density Granulation, 419 by a pinhole camera, 427 magnetic, 107, 261 Grating constant by a steel sphere, 427 Focal length, 312, 336 optical, 446 by circular rings, 427 Focal plane, 313 Grating spectrometer, 446 by concave mirrors, 327 Focal point, 313 Grating, optical, 423, 451 by lenses, 327 virtual, 315 resolving power, 447 using water waves, 328 Focal ratio, 366 sine-wave, 424 Image point, 310 Force sinusoidal modulation by, 424 as a diffraction pattern, 328 Gray, S., 36 from a lens, 328 Index 629

real, 311 Interaction cross-section, 495, 522 Law of refraction, 302 virtual, 315 Interference, 238 Lead tree, 9 Image scale, 314 A. Fresnel’s biprism experiment, Lecher line, 220 Image-plane arching, 345 389 Lecher system, 220 Imaging curves of constant inclination, Lens and irradiance, 376 396 focal ratio Nf, 366 by a pinhole camera, 365 due to particles which redirect Lens formulas, 312 by a ring grating, 428 the light, 404 Lenses by lenses, 311, 331 experiment of T. Young, 388, for electromagnetic waves, 236 by zone plates, 429 406 image points from, 328 of invisible objects, 432 H. Lloyd’s experiment, 389 model experiment, 546 of non-emitting objects, 435 microscopy, 398 resolving power, 331 of three-dimensional objects, model experiments, 387, 389 thick, 336 365 of polarized light, 481, 483 with gradients in the refractive Imaging aberrations, 344 redirection of radiant power by, index, 555 astigmatism, 345 401 Lenses, “rubber”, 352 chromatic, 350 sharpening the fringes, 397 Lenz’s law, 142, 145, 177 distortion, 349 spectrometer, 454 Levitation experiment, 75 Imaging errors of FABRY and PÉROT, 454 Leyden jar, 61 coma, 348 with a continuous spectrum, 399 Lichtenberg, G. C., 34, 38 spherical aberration, 344 with planar, parallel plates, 390 Lichtenberg’s figures, 38 Immersion fluid, 357 Interference filter, 402, 579 Light Impedance, 183 Interferometer absorption of, 538, 541 or AC resistance, 183 after A.A. Michelson, 409 as a transverse wave, 468 Impulse, 107, 145 optical, 408 drag, 552 Impulse deflection, 21 Interrupter, electrolytic in a moving reference frame, 460 mechanical, 146 Wehnelt, 192, 204 in gravitational fields, 556 Incandescent gaslight, 584 Ionization chamber, 298, 529 monochromatic, 383 Incandescent light, 397, 481, 482, Iron core, 146 Light beam, 299, 300 580 Irradiance, 296, 318, 319, 374, 376, boundaries of, 343 Inclination angle, 314 592 principal ray, 310 Increase of the solar image, 379 Light density or radiance, 197 of the angle of vision, 356 Irradiation intensity, 374, 592 Light rays, 300 Index of refraction, 233, 302 curved, 553 Table, 303, 322 J Light source, 598 Inductance, 175 Joule heating, 8, 23, 177 in the moving frame of Induction, 101, 142, 265 reference, 461 in conductors at rest, 104 K linear, 342 in moving conductors, 108, 131 Kenelly-Heaviside layer, 568 measuring the diameter of, 407 Induction coil, 101, 116, 191 Kepler’s telescope or astronomical pointlike, 383 Induction damping, 145 telescope, 356, 360 thermal, 584 Induction furnace, 191 Kerr effect, 575 Light sources, 301 Induction phenomena, relativity, Kirchhoff’s law, 578 Light vector, 471 103 Klystron, 234 Light waves Inductive reactance, 180, 183 amplitude of, 318 Inductive resistance, 180, 183 L Light waves as electromagnetic Inertia Lambert’s cosine law, 372, 379, 533 waves, 233 mechanical, 176 Lambert’s law of absorption, 492 Light waves, standing, 403 of a magnetic field, 176 Landé g-factor, 278 Light-beam boundaries, 334, 340 Influence, 38, 71 Langevin-Debye formula, 257 and energy transport, 371 Influence machine, 19, 29 Larmor precession, 279 and perspective, 368 Holtz, 19 Laser, 301, 399, 430 Line grating, 423, 452 Wimshurst, 19 Lattice constant for X-rays, 453 Infrared, 323 crystallographic, 455 sinusoidal modulation by, 424 Initial curve D new curve, 267 Laue diagram, 439 Line spectrum, 448 Insulator, 36 Law of induction, 105, 106, 109, in the X-ray region, 456 Intensity, 373 174 Line structure, 454 of scattered radiation, 519 most general form, 110 Law of reflection, 302 630 Index

Logarithmic decrement, 187, 189, Magnetic polarization, 262 electric polarizability, 523, 552 207 Magnetic poles, 88, 152 non-polar, 246 Lorentz contraction, 130, 133, 139 Magnetic potential difference, 115 number density Lorentz force, 132, 137, 144, 160 Magnetic potentiometer, 116, 117 optical determination, 559 Loss factor, 190, 207 Magnetic shielding, 272 polar, 246, 521, 564 Loupe, 354 Magnetic susceptibility, 262 Monochromatic (single-frequency) Low-voltage transformer, 191 Magnetizability sound waves, 444 Lumen (lm), unit of luminous flux, molecular, 273 Monochromatic light, 309, 322, 594 Magnetization, 262, 270 344, 351, 396, 397, 419, 421, Luminance, 594, 602 remanent, 147, 268 422, 424, 425, 430–432, 443, Table, 602 Magnetometer, 93 449 Luminosity function, 598 Magneton, Bohr, 277 hologram, 431, 432 Luminous efficiency, 598 Magnetron, 236 laser, 399, 410, 430 Luminous energy, 594 Magnification, 354 single-frequency, 443 Luminous flux, 594 Magnification or power Monochromatic wave group, 449 Luminous fountain, 307 of a microscope, 356 Monochromatic wave train, 383, Luminous intensity, 593 of a telescope, 355, 361 429–431 Lummer-Gehrcke plate, 454 Magnifying glass, 354 Monochromator, 449 Lux (lx), unit of illuminance, 594 Magnitudes Motion pictures, image changing of stars, 601 without flickering, 596 M Malus, E. L., 498 Muller’s stripes, 399 Mach’s stripes, 294 Mass Madelung, E., 563 of the electron, 527 N Magnetic charge, 153 Maxwell equations, 124, 125 Nanometer (nm) D 10 9 m, 322 Magnetic constant, 106 in matter, 270 Natural light, 301, 320, 449 Magnetic dipole, 153 in vacuum, 55, 89, 114, 124, 125 spectrum, 320 Magnetic field, 87, 113, 153 Maxwell, J. C., 140 Néel temperature, 286 direction of, 89, 93 Maxwell’s relation, 233, 549 New curve, 267 gradient, 151, 263 Mean free path of molecules, 34 Newton’s rings, 396 homogeneous, 89, 91 Meissner-Ochsenfeld effect, 265 Nicol prism, 471 inertia of, 176 Meniscus lens, 339, 347 Night glass, 362 of a current-carrying wire, 5, Mercury vapor, optical detection, Nodal points, 338 118, 120 562 Non-reflective coating, 401 of a long coil, 89, 93, 120 Michelson interferometer, 409 Normal element, 15 of a permanent magnet, 90, 270 Microphone, 165, 248 Number density, 254, 274 of a ring coil, 91 Microscope, 356 of electrons, 527 of an oscillating dipole, 229 free object distance, 359 of molecules, 495, 567 of the earth, 108 immersion fluid, 357 Numerical aperture NA, 358 production of, 94 magnification, 356 switching on and off, 177 making phase structures visible, O unit of, 93 437 Object distance, 335 Magnetic field constant or induction numerical aperture, 358 Objective, 335, 358 constant, 106 oblique illumination, 437 Objective lens, 355 Magnetic field lines, 4, 87, 200 resolving power, 357 Ocular, 351, 355, 363 of the earth’s field, 94 telecentric optical path, 367 Oersted, H. C., 87 Magnetic field of the earth, 94 with dark-field illumination, 358 Ohm, unit of resistance, 16 Magnetic field strength, 92 Microwaves, 237, 258 Ohmic resistance, 16, 182 at the center of a cylindrical coil, Millikan, R.A., 77 Ohmic voltage, 182 120 Mirror charge, 71 Ohm’s law, 16, 17 Magnetic field, rotating, 170, 184 Mirror galvanometer, 11, 19 Opaque or topcoat paints, 610 Magnetic fields Mirror image, 302 Opening angle, 299, 314 dependence on the frame of Mirror prisms, 324 and radiant flux (power), 375 reference, 129 Mirror surface grating, 453 Opening-angle ratio = focal ratio, Magnetic flux, 106, 146, 152, 154, Mirror, concave, 328 352 261 Modulation length, 239 Optical activity, 486, 487 Magnetic flux density, 107, 261 Molecular currents, 97, 147 Optical axis, 471, 472 Magnetic moment, 148, 262, 275 Molecules Optical constants of a molecule, 275 and dispersion, 548 Beer’s formula, 509 of an electron, 278 measurement of, 500, 513 Index 631

of Na and K, 572 Permittivity of vacuum D electric Potentiometer, magnetic, 116, 117 Optical path, 304, 397, 444 field constant, 52 Pot-type magnet, 157 Optical temperature measurements, Perspective, 343, 366 Power of a lens, 314 586 Phase grating, 425 Power, electric, 25 Order Phase jump on reflection, 392, 502, of alternating current, 189 of interference fringes, 384 510 Poynting vector, 231 of interference maxima, 389, Phase shift, 183, 184 Precession, 280 395 Phase structure, 425, 437 Primary light source, 358, 602 of interference rings, 392 Phase velocity, 237, 457 Principal angle of incidence, 498 Order-one and order-two positions in metals, 542 Principal orientation, 155 of a step grating, 422 Phase-contrast method, 437 Principal plane, 335 Oscillations Phasor diagram, 183 of a crystal, 472 autocontrol, 204 Photocell, 297 of a prism, 309 damped, 196 Photodiode, 297 Principal points, 338 electrical, 196 Photometry, 293, 593, 599 Principal ray, 310, 314, 336, 343 energy resonance curve, 209 Photomultiplier, 298 angle of inclination, 314, 336, forced, 205, 207, 520, 548, 566 Piezoelectric crystals, 86 340, 343 resonance frequency, 196 Pinhole camera, 331, 338, 365, 427 Principle of relativity, 132 undamped, 202 Planck’s constant, 277, 582 Prism, 309, 451, 472 Oscillator circuit, electrical, 196, Planck’s radiation formula, 582 direct-vision, 540 214 Plane of incidence, 302 made of ZnO, 539 Oscillator strength, 561 Plasma oscillations, 566 principal plane, 309 Oscillator, electrical, 206 Poisson’s spot, 414 Wollaston, 474 Oscilloscope, 77 Polar molecules, 246, 256, 521, 564 Prism spectrometer, 443 Overcorrection, spherical, 345 Polarizability, 255 Probe coil, 107 Overtones, 218 electric, 523, 552 Prometheus, 585 Oxygen, liquid, in a magnetic field, molecular, 253 Propeller, electric, 58 266 Polarization, 467, 470 Pupils, 340 by birefringence, 469 Purcell’s experiment, 520 P by reflection, 498 Purple, 607 Paper condenser, 61 by scattering, 528 Pyroelectric crystals, 85 Paraelectric materials, 246, 252 electric, of a dielectric material, Pyrometer, 587 Parallax, 601 63, 84, 245, 249 Parallel circuit, 188, 189, 195, 209 magnetic, 262 Q of condensers, 56 of electromagnetic waves, 231, Q-factor, 207, 559 of resistor, coil, and condenser, 233 Quality factor or Q-factor, 207 188 Polarization angle, 498 Quarter-wave plate, 480 of resistors, 17 Polarization foils, 471, 483 Quartz, birefringence, 472, 473, 487 Parallel light beams, 381 Polarized light, 467 Quetelet’s rings, 404 separation by image formation, elliptic, 476, 479 316 interference, 481, 483 R Parallel-bounded beams, 491 Polarized light (partially polarized), Radiance, 373, 579 of light, 300 502 of an emitter, 376 Parallel-plate condenser, 28, 35, 56, Polarizers, 468–470 spectral, of a black body, 581 77, 243 for ultraviolet, 474 Radiant energy, 594 attraction of the plates, 72 for X-rays, 529 Radiant exitance, 375 Paramagnetic materials, 264, 265, Polarizing Radiant flux, 371 274, 280 in the infrared, 498 Radiant intensity, 373, 448 Parsec D 3:08 1016 m, 601 Pole regions direction-independent, 381 Path difference, 383, 478 forces between, 152 of X-rays, 381 Path integral of the electric field, 50, Poles of magnet coils, 90 Radiant intensity (or Intensity), 591 115 Poles of magnets, 154 Radiant power, 296, 297, 401, 578, Path, optical, 304, 397, 444 Polychromatic light, 351 591 Penetration depth Potential, 80 Radiation detectors, 296 mean, of light, 493, 538, 541 Potential difference, 81 Radiation field of a dipole, 228 of X-rays, 544 magnetic, 115 Radiation formula of M. Planck, Permanent magnets, 97, 121, 149 Potentiometer, 18 582 Permeability, 261 Potentiometer circuit, 18 Radiometer, 296, 491 Permeability constant, 106 Rainbow, 419 632 Index

Ram, hydraulic, 176 specific, 16 transition to diffraction, 532 Raster grating, 425 Resistivity, 16 Scherrer, P., 456 Rate of change Resistors Schlieren method, 432 of a magnetic field, 106, 113 in parallel, 17 Schmidt mirror, 352 of an electric field, 123 in series, 17 Secondary light source, 358, 375 Rayleigh scattering, 520, 522 Resolvable angular spacing, 332 Secondary radiation, 522, 533 and compressibility, 525 Resolving power, 331 Sector wheel dependence on the wavelength, of a grating, 447 with radial sectors, 596 522 of a lens, 333 Sector wheel or chopper, 592 extinction constant, 523 of a microscope, 357 Seignette salt crystals, 247 Reactance of a prism, 445 Self-inductance, 173, 199 capacitive, 185 of spectrometers, 448 Series circuit, 187, 195, 209 inductive, 180, 183 Resonance, 186, 188, 207 of a resistor and a coil, 178, 182 Receiver, 371, 519 Resonance curve, 209 of condensers, 56 Redshift, 465 for the energy, 560 of resistor, coil and condenser, Reflection, 302 Resonance frequency, 186, 196 186, 207 avoiding, 502 Resonant transformer, 199 of resistors, 15 elimination of, 401 Resonators Shadow, 413 from a weakly-absorbing properties of optical, 562 Shallow-water lens, 328, 555 material, 501 Retina, 332, 599 Shimmering colors, 611 of linearly-polarized light, 499 Reversing prism, 324 Shine, 611 with strong absorption, 509 Rheostat, 18 Short-circuit current, 168 Reflection coefficient, 232, 603 Ring coil, 91, 262 Shorted rotor, 143 Reflection filters, 401 Ring current, electrodeless, 202 SI D Systeme International Reflection grating, 450 Ring current, molecular, 276 d’Unités, 12 Reflection source, 598, 602 Ring grating, 428 Sine condition, 348, 357, 377 Reflectivity, 232, 497, 501, 509, 542 with only one focal length, 430 Sine-wave grating, 424, 429 metallic, 509, 542 Rochon prism, 474 Skin effect, 199 Refraction, 302, 550, 552 Rotary condenser, 60 Sky blue, 523 by a plane-parallel glass block, Rotating field, 143 color temperature, 589 320 Rotating field, magnetic, 143, 170, Slide projector, 364 reduction to scattering, 545 184 Slip, 143 Table, 552 Rotating-coil ammeter, 7, 11, 145 Soft-iron ammeter, model, 8 Refractive index, 537 Rotating-coil instrument, 8 Solar constant, 376 and gas density, 552 Rotating-field motor, 171 Solar image, diameter of, 315 complex, 508, 509 Rotational frequency of electric Solar surface for X-rays, 551 motors, 167, 171 radiation, 376 imaginary, 568 Rotor, 168 temperature, 583 of mica, 475 in motors, 166 Solid angle, 373 Refractometer, 307 Rowland’s experiment, 94, 129 Space charge, 54 Reis P., 165 Rubber lenses, 352 Spark, 37 Rejection circuit, 189 Ruby glass, extinction spectrum, Spark gap, 197 Relative aperture Nf 570 Sparking, 197 of a lens, 366 Specific resistance, 16 Relativity principle, 103, 132 S Spectral apparatus Relativity, theory of, 103, 133 Saccharimeter (sugar content), 307, after Fabry and Pérot, 398 Relaxation of a conduction current, 487 Spectral lines, 443, 446 179 Saturation magnetization, 266 X-ray, 456 Relaxation time, 60, 178, 208 Savart’s plate, 485 Spectrometer Remanence, 268 Scattered light, 306 after Lummer and Gehrcke, 454 Residual rays, 543 Scattered radiation, 519 and incandescent light, 449 Resistance, 16, 23 Scattering, 300, 492 for X-rays, 455 capacitive, 185 angular dependence, 519 Spectrum electrical, 177 by weakly-absorbing particles, produced by a plane-parallel inductive, 180, 183 530 block, 320 measurement by field decay, 60 from matte surfaces, 532 produced by a plane-parallel negative differential, 204 of X-rays, 526 glass plate, 443 Ohmic, 16, 182 Rayleigh, 520, 522 produced by a prism, 321, 448 significance of, 517 Index 633

Spherical aberration, 344 Thick lenses, 336 measurement using the Spin, 278 Thomson, J. J., 528 astronomical aberration, Spinning top, 98 Three-dimensional viewing of 461 Split-pole motor, 184 two-dimensional images, 370 Versorium, 84 Standard light source, 593 Three-phase current, 171 Vibrations Standing light waves, 403 Tinting and shading triangle, 606 circular, 170 Standing waves, 220, 229 Tints and shades forced, 517, 520, 566 Stark effect, 132 of chromatic colors, 605 Visual sharpness (acuity), 595 Stars Toepler’s schlieren method, 433 Volt balance, 13 deflection of rays, 556 Toggle action, 192 Volt, unit of potential difference, 24, Starter, 168 Toggle oscillations, 205 70 Stefan-Boltzmann law, 582 Tolansky, S., 399 Voltage Strain birefringence, 489 Top, spinning, 278 effective value, 181 Streak focal line, 381 Toriodal coil, 262 examples, 17 Superconductivity, 177, 265, 272 Torque, 81, 82, 147, 166 induced, 106, 113 Superparamagnetism, 282 Torsional waves, 496 Ohmic, 182 Surface, 514 Total reflection, 306, 470, 504 time integral of, 102 Surface tension, reduction by an by electrons, 568 unit of, 70 electric field, 74 critical angle, 306 Voltage divider, 18 Susceptibility Tourmaline crystal, 86 Voltage impulse, 102, 116, 174 dielectric, 245 Transformer, 190 measurement of, 145 magnetic, 262 Transmission coefficient, 609 Voltage or potential difference, 13 Transmitter, 165, 224 Voltage resonance, 187 T Transverse waves, 467, 478 Voltmeter, 14 Tangent condition, 350 Triode, 203 current-carrying or dynamic, 16 Tank circuit, 206 Tungsten lamps, 585 galvanic, 16 Telecentric optical path, 341, 458 Tunnel effect, 504 static, 14 image side, 341 Two-phase AC motor, 171 object side, 367 Tyndall effect, 470 W Telegraph equations, 223 Waitz, J. S., 42 Telegraph of Gauss and Weber, 145 Walkiers de St. Amand, 45 U Telegraphy, wireless, 197 Watt second D 1newtonmeter,24 Ultramicroscope, 569 Telephone as an alternating-current Watt, unit of power, 25 Ultraviolet, 323 generator, 164 Wave center Undercorrection, spherical, 344 Telephoto objective lens, 337 extended, 385 Unipolar induction, 110 Telescope, 407 pointlike, 383, 385 Unit Galilean or Huygens, 360 Wave groups, 237, 383, 449 of electric current, 12 Kepler’s or astronomical, 333, length of, 399 Unit system, 12 356 Wave resistance, 231 Usable wavelength range, 447, 451 magnification, 355, 361 Waveguide, 237 perspective, 370 Wavelength, 322, 537 single-lens, 355 V measurement of, 229 Telescopic optical path, 360 Vacuum tube Waves Temperature measurements with three electrodes (triode), mathematical representation, optical, 586 203 506 Temperature radiation, 577 Van de Graaff generator, 44 Waves, electromagnetic, 211, 230 Temperature, true and black-body, van de Graaff, R. J., 45 energy transport by, 231 587 Variable resistor, 18 polarization of, 231, 233 Terrestrial telescope, 360 Vector diagram, 183 reflection of, 228 Tesla coil, 198 Velocity refraction of, 233 Tesla, unit of the magnetic flux critical (electromagnetism), 140 standing, 220 density, 107 of electromagnetic waves, 223 velocity of, 223, 231 Test charge, 68 of electrons in metals, 141 Waves, electromagnetic, travelling Theory of relativitaty Velocity of light, 140, 231, 457 detection of, 229 special, 130 in vacuum, 130, 223 Wehnelt interrupter, 192, 204 Thermal radiation, 577 measurement by Foucault, 458 White light D natural light, 301 selective, 583 measurement by interference Wien’s displacement law, 583 Thermocouple, 297 experiments, 462 Wilcke, J. C., 30, 32, 38 Thermoelement, 586 measurement by Römer, 457 Wimshurst machine, 19 634 Index

Wind, electric, 58 line spectrum, 456 Z Wire wave, electromagnetic, 222 penetration depth, 544 Zernike, F., 437 Wollaston prism, 474 polarized, 529 Zinc oxide Wood, R.W., 540 refractive index, 537, 551 dispersion of, 539 scattering of, 526 selective radiation, 584 X Zone construction, Fresnel, 413 Xerography, 38 Y Zone plate, 428 X-ray spectrometer, 455 Young, Thomas, 384, 387, 405, 409, Zoom lenses, 352 X-rays, 381, 453, 455 450, 502, 520 absorption of, 537, 541 Young’s interference experiment, diffraction of, 438 388, 406