MAXWELL’S EQUATIONS AND THE PRINCIPLES OF ELECTROMAGNETISM “fm” — 2007/12/27 — 12:39 — pagei—#1 LICENSE, DISCLAIMER OF LIABILITY, AND LIMITED WARRANTY By purchasing or using this book (the “Work”), you agree that this license grants permission to use the contents contained herein, but does not give you the right of ownership to any of the textual content in the book or ownership to any of the information or products contained in it. This license does not permit use of the Work on the Internet or on a network (of any kind) without the written consent of the Publisher. Use of any third party code contained herein is limited to and subject to licensing terms for the respective products, and permission must be obtained from the Publisher or the owner of the source code in order to reproduce or network any portion of the textual material (in any media) that is contained in the Work. 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The use of “implied warranty” and certain “exclusions” vary from state to state, and might not apply to the purchaser of this product. “fm” — 2007/12/27 — 12:39 — page ii — #2 MAXWELL’S EQUATIONS AND THE PRINCIPLES OF ELECTROMAGNETISM RICHARD FITZPATRICK, PH.D. University of Texas at Austin Infinity Science Press LLC Hingham, Massachusetts New Delhi “fm” — 2007/12/27 — 12:39 — page iii — #3 Copyright 2008 by Infinity Science Press LLC All rights reserved. This publication, portions of it, or any accompanying software may not be reproduced in any way, stored in a retrieval system of any type, or transmitted by any means or media, electronic or mechanical, including, but not limited to, photocopy, recording, Internet postings or scanning, without prior permission in writing from the publisher. Publisher: David Pallai INFINITY SCIENCE PRESS LLC 11 Leavitt Street Hingham, MA 02043 Tel. 877-266-5796 (toll free) Fax 781-740-1677 info@infinitysciencepress.com www.infinitysciencepress.com This book is printed on acid-free paper. Richard Fitzpatrick. Maxwell’s Equations and the Principles of Electromagnetism. ISBN: 978-1-934015-20-9 The publisher recognizes and respects all marks used by companies, manufacturers, and developers as a means to distinguish their products. All brand names and product names mentioned in this book are trademarks or service marks of their respective companies. Any omission or misuse (of any kind) of service marks or trademarks, etc. is not an attempt to infringe on the property of others. Library of Congress Cataloging-in-Publication Data Fitzpatrick, Richard. Maxwell’s equations and the principles of electromagnetism / Richard Fitzpatrick. p. cm. Includes bibliographical references and index. ISBN-13: 978-1-934015-20-9 (hardcover with cd-rom : alk. paper) 1. Maxwell equations. 2. Electromagnetic theory. I. Title. QC670.F545 2008 530.14 1–dc22 2007050220 Printed in the United States of America 08091054321 Our titles are available for adoption, license or bulk purchase by institutions, corporations, etc. For additional information, please contact the Customer Service Dept. at 877-266-5796 (toll free). Requests for replacement of a defective CD-ROM must be accompanied by the original disc, your mailing address, telephone number, date of purchase and purchase price. Please state the nature of the problem, and send the information to Infinity Science Press, 11 Leavitt Street, Hingham, MA 02043. The sole obligation of Infinity Science Press to the purchaser is to replace the disc, based on defective materials or faulty workmanship, but not based on the operation or functionality of the product. “fm” — 2007/12/27 — 12:39 — page iv — #4 For Faith “fm” — 2007/12/27 — 12:39 — pagev—#5 “fm” — 2007/12/27 — 12:39 — page vi — #6 CONTENTS Chapter 1. Introduction 1 Chapter 2. Vectors and Vector Fields 5 2.1 Introduction 5 2.2 Vector Algebra 5 2.3 Vector Areas 8 2.4 The Scalar Product 10 2.5 The Vector Product 12 2.6 Rotation 15 2.7 The Scalar Triple Product 17 2.8 The Vector Triple Product 18 2.9 Vector Calculus 19 2.10 Line Integrals 20 2.11 Vector Line Integrals 23 2.12 Surface Integrals 24 2.13 Vector Surface Integrals 26 2.14 Volume Integrals 27 2.15 Gradient 28 2.16 Divergence 32 2.17 The Laplacian 36 2.18 Curl 38 2.19 Polar Coordinates 43 2.20 Exercises 45 Chapter 3. Time-Independent Maxwell Equations 49 3.1 Introduction 49 3.2 Coulomb’s Law 49 3.3 The Electric Scalar Potential 54 3.4 Gauss’ Law 57 3.5 Poisson’s Equation 66 3.6 Ampère’s Experiments 68 vii “fm” — 2007/12/27 — 12:39 — page vii — #7 viii CONTENTS 3.7 The Lorentz Force 71 3.8 Ampère’s Law 75 3.9 Magnetic Monopoles? 76 3.10 Ampère’s Circuital Law 79 3.11 Helmholtz’s Theorem 86 3.12 The Magnetic Vector Potential 91 3.13 The Biot-Savart Law 95 3.14 Electrostatics aND Magnetostatics 97 3.15 Exercises 101 Chapter 4. Time-Dependent Maxwell Equations 107 4.1 Introduction 107 4.2 Faraday’s Law 107 4.3 Electric Scalar Potential? 112 4.4 Gauge Transformations 113 4.5 The Displacement Current 116 4.6 Potential Formulation 123 4.7 Electromagnetic Waves 124 4.8 Green’s Functions 132 4.9 Retarded Potentials 136 4.10 Advanced Potentials? 142 4.11 Retarded Fields 145 4.12 Maxwell’s Equations 149 4.13 Exercises 151 Chapter 5. Electrostatic Calculations 157 5.1 Introduction 157 5.2 Electrostatic Energy 157 5.3 Ohm’s Law 163 5.4 Conductors 165 5.5 Boundary Conditions on the Electric Field 171 5.6 Capacitors 172 5.7 Poisson’s Equation 178 5.8 The Uniqueness Theorem 179 5.9 One-Dimensional Solutions of Poisson’s Equation 184 5.10 The Method of Images 186 5.11 Complex Analysis 195 5.12 Separation of Variables 202 5.13 Exercises 209 “fm” — 2007/12/27 — 12:39 — page viii — #8 CONTENTS ix Chapter 6. Dielectric and Magnetic Media 215 6.1 Introduction 215 6.2 Polarization 215 6.3 Electric Susceptibility and Permittivity 217 6.4 Boundary Conditions For E and D 218 6.5 Boundary Value Problems with Dielectrics 220 6.6 Energy Density within a Dielectric Medium 226 6.7 Force Density within a Dielectric Medium 228 6.8 The Clausius-Mossotti Relation 230 6.9 Dielectric Liquids in Electrostatic Fields 233 6.10 Polarization Current 237 6.11 Magnetization 237 6.12 Magnetic Susceptibility and Permeability 240 6.13 Ferromagnetism 241 6.14 Boundary Conditions for B and H 243 6.15 Boundary Value Problems with Ferromagnets 244 6.16 Magnetic Energy 249 6.17 Exercises 251 Chapter 7. Magnetic Induction 255 7.1 Introduction 255 7.2 Inductance 255 7.3 Self-Inductance 257 7.4 Mutual Inductance 261 7.5 Magnetic Energy 264 7.6 Alternating Current Circuits 270 7.7 Transmission Lines 274 7.8 Exercises 280 Chapter 8. Electromagnetic Energy and Momentum 283 8.1 Introduction 283 8.2 Energy Conservation 283 8.3 Electromagnetic Momentum 287 8.4 Momentum Conservation 291 8.5 Angular Momentum Conservation 294 8.6 Exercises 297 Chapter 9. Electromagnetic Radiation 299 9.1 Introduction 299 9.2 The Hertzian Dipole 299 “fm” — 2007/12/27 — 12:39 — page ix — #9 x CONTENTS 9.3 Electric Dipole Radiation 306 9.4 Thompson Scattering 307 9.5 Rayleigh Scattering 310 9.6 Propagation in a Dielectric Medium 312 9.7 Dielectric Constant of a Gaseous Medium 313 9.8 Dispersion Relation of a Plasma 314 9.9 Faraday Rotation 318 9.10 Propagation in a Conductor 322 9.11 Dispersion Relation of a Collisional Plasma 324 9.12 Normal Reflection at a Dielectric Boundary 326 9.13 Oblique Reflection at a Dielectric Boundary 330 9.14 Total Internal Reflection 336 9.15 Optical Coatings 339 9.16 Reflection at a Metallic Boundary 342 9.17 Wave-Guides 343 9.18 Exercises 348 Chapter 10.Relativity and Electromagnetism 351 10.1 Introduction 351 10.2 The Relativity Principle 351 10.3 The Lorentz Transformation 352 10.4 Transformation of Velocities 357 10.5 Tensors 359 10.6 Physical Significance of Tensors 364 10.7 Space-Time 365 10.8 Proper Time 370 10.9 4-Velocity and 4-Acceleration 371 10.10 The Current Density 4-Vector 372 10.11 The Potential 4-Vector 374 10.12 Gauge Invariance 374 10.13 Retarded Potentials 375 10.14 Tensors and Pseudo-Tensors 377 10.15 The Electromagnetic Field Tensor 381 10.16 The Dual Electromagnetic Field Tensor 384 10.17 Transformation of Fields 386 10.18 Potential Due to a Moving Charge 387 10.19 Field Due to a Moving Charge 388 10.20 Relativistic Particle Dynamics 391 10.21 Force on a Moving Charge 393 10.22 The Electromagnetic Energy Tensor 394 10.23 Accelerated Charges 397 “fm” — 2007/12/27 — 12:39 — pagex—#10 CONTENTS xi 10.24 The Larmor Formula 402 10.25 Radiation Losses 406 10.26 Angular Distribution of Radiation 407 10.27 Synchrotron Radiation 409 10.28 Exercises 412 Appendix A.