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Classical Field Theory CLASSICAL FIELD THEORY ELECTROMAGNETISM AND GRAVITATION Francis E. Low Wiley-VCH Verlag GmbH & Co. KGaA This Page Intentionally Left Blank CLASSICAL FIELD THEORY This Page Intentionally Left Blank CLASSICAL FIELD THEORY ELECTROMAGNETISM AND GRAVITATION Francis E. Low Wiley-VCH Verlag GmbH & Co. KGaA All books published by Wiley-VCH are carefully produced. Neverthele\s, author\, editors, and publisher do not warrant the information contained in these books, including this book, to be free oferrors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other item may inadvertently be inaccurate. Library of Congress Card No.: Applied for British Library Cataloging-in-Publication Data: A catalogue record for this book is available from the British Library Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliogralie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. 0 1997 by John Wiley & Sons, Inc. 0 2004 WILEY-VCN Verlag GmbH & Co. KGaA, Weinheim Cover Design Edward Smith Design. Inc Cover Photographs Albert Einstein (above). W. F. Meggers collection, AIP Einilio Scgri: Visual Archives; James Clerk Maxwell (below), AIP Eniilio Segre Visual Archives All rights reserved (including those oftranslation into other languages). No part of this book may be reproduced in any form - nor transmitted or translated into machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Printed in the Federal Republic of Germany Printed on acid-free paper Printing Strauss (imbl I, Miirlenbach Bookbinding Litges & Dopf Buchbinderei Gmbll, Heppenheim ISBN-13: 978-0-47 1-5955 1-9 ISBN-10: 0-471-59551-9 Contents Preface ix 1. Electrostatics 1 1.1. Coulomb’s Law, 1 1.2. Multipoles and Multipole Fields, 9 1.3. Energy and Stress in the Electrostatic Field, 12 1.4. Electrostatics in the Presence of Conductors: Solving for Electrostatic Configurations, 16 1.5. Systems of Conductors, 20 1.6. Electrostatic Fields in Matter, 24 1.7. Energy in a Dielectric Medium, 32 Problems, 36 2. Steady Currents and Magnetostatics 47 2.1. Steady Currents, 47 2.2. Magnetic Fields, 51) 2.3. Magnetic Multipoles, 56 2.4. Magnetic Fields in Matter, 61 2.5. Motional Electromotive Force and Electromagnetic Induction, 66 2.6. Magnetic Energy and Force, 69 2.7. Diamagnetism, 73 Problems, 77 3. Time-Dependent Fields and Currents 81 3.1. Maxwell’s Equations, 81 3.2. Electromagnetic Fields in Matter, 84 3.3. Momentum and Energy, 91 3.4. Polarizability and Absorption by Atomic Systems, 95 3.5. Free Fields in Isotropic Materials, 102 3.6. Reflection and Refraction, 109 3.7. Propagation in Anisotropic Media, 114 vi Contents 3.8. Helicity and Angular Momentum, 118 Problems, 123 4. Radiation by Prescribed Sources 134 4.1. Vector and Scalar Potentials, 134 4.2. Green’s Functions for the Radiation Equation, 137 4.3. Radiation from a Fixed Frequency Source, 140 4.4. Radiation by a Slowly Moving Point Particle, 144 4.5. Electric and Magnetic Dipole and Electric Quadrupole Radiation, 146 4.6. Fields of a Point Charge Moving at Constant High Velocity v: Equivalent Photons, 150 4.7. A Point Charge Moving with Arbitrary Velocity Less Than c: The LiCnard-Wiechert Potentials, 156 4.8. Low-Frequency Bremsstrahlung, 159 4.9. Lidnard-Wiechert Fields, 165 4.10. Cerenkov Radiation, 170 Problems, 176 5. Scattering 181 5.1. Scalar Field, 181 5.2. Green’s Function for Massive Scalar Field, 188 5.3. Formulation of the Scattering Problem, 191 5.4. The Optical Theorem, 194 5.5. Digression on Radial Wave Functions, 198 5.6. Partial Waves and Phase Shifts, 203 5.7. Electromagnetic Field Scattering, 208 5.8. The Optical Theorem for Light, 210 5.9. Perturbation Theory of Scattering, 211 5.10. Vector Multipoles, 217 5.11. Energy and Angular Momentum, 227 5.12. Multipole Scattering by a Dielectric, 230 Problems, 240 6. Invariance and Special Relativity 245 6.1. Invariance, 245 6.2. The Lorentz Transformation, 248 6.3. Lorentz Tensors, 257 6.4. Tensor Fields: Covariant Electrodynamics, 260 6.5. Equations of Motion for a Point Charge in an Electromagnetic Field, 269 6.6. Relativistic Conservation Laws, 271 Problems, 277 Contents vii 7. Lagrangian Field Theory 281 7.1. Review of Lagrangians in Mechanics, 281 7.2. Relativistic Lagrangian for Particles in a Field, 284 7.3. Lagrangian for Fields, 290 7.4. Interacting Fields and Particles, 298 7.5. Vector Fields, 304 7.6. General Covariance, 313 7.7. Local Transformation to a Pseudo-Euclidean System, 323 7.8. Alternative Construction of a Covariantly Conserved, Symmetric Stress-Energy Tensor, 326 Problems, 331 8. Gravity 338 8.1. The Nature of the Gravitational Field, 338 8.2. The Tensor Field, 341 8.3. Lagrangian for the Gravitational Field, 345 8.4. Particles in a Gravitational Field, 349 8.5. Interaction of the Gravitational Field, 356 8.6. Curvature, 367 8.7. The Einstein Field Equations and the Precession of Orbits, 370 8.8. Gravitational Radiation, 376 Problems, 384 Appendix A. Vectors and Tensors 391 A.l. Unit Vectors and Orthogonal Transformations, 391 A.2. Transformation of Vector Components, 394 A.3. Tensors, 396 A.4. Pseudotensors, 398 AS. Vector and Tensor Fields, 399 A.6. Summary of Rules of Three-Dimensional Vector Algebra and Analysis, 401 Problems, 402 Appendix B. Spherical Harmonics and Orthogonal Polynomials 406 B. 1. Legendre Polynomials, 406 B.2. Spherical Harmonics, 410 B.3. Completeness of the Yf,m,415 Problems, 418 Index 421 This Page Intentionally Left Blank Preface It is hard to fit a graduate course on electromagnetic theory into one semester. On the other hand, it is hard to stretch it to two semesters. This text is based on a two-semester MIT ccurse designed to solve the problem by a compromise: Allow approximately one and a half semesters for electromagnetic theory, including scattering theory, special relativity and Lagrangian field theory, and add approximately one-half semester on gravitation. It is assumed throughout that the reader has a physics background that includes an intermediate-level knowledge of electromagnetic pheno- mena and their theoretical description. This permits the text to be very theory-centered, starting in Chapters 1 and 2 with the simplest experi- mental facts (Coulomb’s law, the law of Biot and Savart, Faraday’s law) and proceeding to the corresponding differential equations; theoretical constructs, such as energy, momentum, and stress; and some applications, such as fields in matter, fields in the presence of conductors, and forces on matter. In Chapter 3, Maxwell’s equations are obtained by introducing the displacement current, thus making the modified form of Ampkre’s law consistent for fields in the presence of time-dependent charge and current densities. The remainder of Chapters 3-5 applies Maxwell’s equations to wave propagation, radiation, and scattering. In Chapter 6, special relativity is introduced. It is also assumed here that the reader comes with prior knowledge of the historic and experi- mental background of the subject. The major thrust of the chapter is to translate the physics of relativistic invariance into the language of four- dimensional tensors. This prepares the way for Chapter 7, in which we study Lagrangian methods of formulating Lorentz-covariant theories of interacting particles and fields. The treatment of gravitation is intended as an introduction to the subject. It is not a substitute for a full-length study of general relativity, such as might be based on Wcinberg’s book.’ Paralleling the treatment ‘Steven Weinberg, Gravitation and Coymology, New York: John Wiley & Sons. 1972. ix X Preface of electromagnetism in earlier chapters, we start from Newton’s law of gravitation. Together with the requirements of Lorentz covariance and the very precise proportionality of inertial and gravitational mass, this law requires that the gravitational potential consist of a second-rank (or higher) tensor. In complete analogy with the earlier treatment of the vector (electro- magnetic) field, following Schwinger,2 we develop a theory of the free tensor field. Just as Maxwell’s equations required that the vector field be coupled to a conserved vector source (the electric current density), the tensor field equations require that their tensor source be conserved. The only available candidate for such a tensor source is the stress-energy tensor, which in the weak field approximation we take as the stress-energy tensor of all particles and fields other than the gravitational field. This leads to a linear theory of gravitation that incorporates all the standard tests of general relativity (red shift, light deflection, Lense-Thirring effect, gravitational radiation) except for the precession of planetary orbits, whose calculation requires nonlinear corrections to the gravitational po- te nti a]. In order to remedy the weak field approximation, we note that the linear equations are not only approximate, but inconsistent. The reason is that the stress-energy tensor of the sources alone is not conserved, since the sources exchange energy and momentum with the gravitational field. The remedy is to recognize that the linear equations are, in fact, consistent in a coordinate system that eliminates the gravitational field, that is, one that brings the tensor g,, locally to Minkowskian form. The consistent equations in an arbitrary coordinate system can then be written down immediately-they are Einstein’s equations. The basic requirement is that the gravitational potential transform like a tensor under general coordinate transformations. Our approach to gravitation is not historical. However, it parallels the way electromagnetism developed: experiment + equations without the displacement current; consistency plus the displacement current + Maxwell’s equations.
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