Classical Electrodynamics - PKU · The special theory of relativity had its origins in classical electrodynamics . And even after almost 60 years, classical electrodynamics still

by JOHN DAVID JACKSONProfessor of Physics, University of Illinoi s

CLASSICAL

ELECTRODYNAMICS

John Wiley & Sons, Inc., New York • London • Sydney

Classical Electrodynamics

9 10

Copyright (0 1962 by John Wiley & Sons, Inc. All rights reserved.

This book or any part thereof must not

be reproduced in any form without thewritten permission of the publisher .

Printed in the United States of AmericaLibrary of Congress Catalog Card Number : 62-8774

To the memory of my father,

Walter David Jackson

'reface

Classical electromagnetic theory, together with classical and quan-tum mechanics, forms the core of present-day theoretical training for

undergraduate and graduate physicists . A thorough grounding in thesesubjects is a requirement for more advanced or specialized training .

Typically the undergraduate program in electricity and magnetisminvolves two or perhaps three semesters beyond elementary physics, withthe emphasis on the fundamental laws, laboratory verification and elabora-tion of their consequences, circuit analysis, simple wave phenomena, andradiation . The mathematical tools utilized include vector calculus,

ordinary differential equations with constant coefficients, Fourier series,and perhaps Fourier or Laplace transforms, partial differential equations,Legendre polynomials, and Bessel functions .

As a general rule a two-semester course in electromagnetic theory isgiven to beginning graduate students . It is for such a course that my book

is designed . My aim in teaching a graduate course in electromagnetism isat least threefold . The first aim is to present the basic subject matter as acoherent whole, with emphasis on the unity of electric and magneticphenomena, both in their physical basis and in the mode of mathematicaldescription . The second, concurrent aim is to develop and utilize a numberof topics in mathematical physics which are useful in both electromagnetictheory and wave mechanics . These include Green's theorems and Green'sfunctions, orthonormal expansions, spherical harmonics, cylindrical andspherical Bessel functions . A third and perhaps most important pur-pose is the presentation of new material, especially on the interaction of

vii

Viii Preface

relativistic charged particles with electromagnetic fields . In this last areapersonal preferences and prejudices enter strongly . My choice of topics isgoverned by what I feel is important and useful for students interested intheoretical physics, experimental nuclear and high-energy physics, and thatas yet ill-defined field of plasma physics .

The book begins in the traditional manner with electrostatics . The first

six chapters are devoted to the development of Maxwell's theory ofelectromagnetism. Much of the necessary mathematical apparatus is con-structed along the way, especially in Chapters 2 and 3, where boundary-

value problems are discussed thoroughly . The treatment is initially interms of the electric field E and the magnetic induction B, with the derivedmacroscopic quantities, D and H, introduced by suitable averaging overensembles of atoms or molecules . In the discussion of dielectrics, simpleclassical models for atomic polarizability are described, but for magneticmaterials no such attempt is made. Partly this omission was a question ofspace, but truly classical models of magnetic susceptibility are not possible .Furthermore, elucidation of the interesting phenomenon of ferromagnetismneeds almost a book in itself.

The next three chapters (7-9) illustrate various electromagnetic pheno-mena, mostly of a macroscopic sort . Plane waves in different media,including plasmas, as well as dispersion and the propagation of pulses, are

treated in Chapter 7 . The discussion of wave guides and cavities in Chapter8 is developed for systems of arbitrary cross section, and the problems ofattenuation in guides and the Q of a cavity are handled in a very general

way which emphasizes the physical processes involved . The elementarytheory of multipole radiation from a localized source and diffractionoccupy Chapter 9. Since the simple scalar theory of diffraction is coveredin many optics textbooks, as well as undergraduate books on electricity andmagnetism, I have presented an improved, although still approximate,theory of diffraction based on vector rather than scalar Green's theorems .

The subject of magneto hydrodynamics and plasmas receives increasingly

more attention from physicists and astrophysicists . Chapter 1 0 representsa survey of this complex field with an introduction to the main physicalideas involved.

The first nine or ten chapters constitute the basic material of classicalelectricity and magnetism . A graduate student in physics may be expectedto have been exposed to much of this material, perhaps at a somewhatlower level, as an undergraduate . But he obtains a more mature view of it,understands it more deeply, and gains a considerable technical ability inanalytic methods of solution when he studies the subject at the level of this

book. He is then prepared to go on to more advanced topics . The

advanced topics presented here are predominantly those involving the

Preface ix

interaction of charged particles with each other and with electromagneticfields, especially when moving relativistically .

The special theory of relativity had its origins in classical electrodynamics .And even after almost 60 years, classical electrodynamics still impressesand delights as a beautiful example of the covariance of physical laws under

Lorentz transformations . The special theory of relativity is discussed inChapter 11, where all the necessary formal apparatus is developed, variouskinematic consequences are explored, and the covariance of electrodynamics.is established, The next chapter is devoted to relativistic particle kine-matics and dynamics . Although the dynamics of charged particles inelectromagnetic fields can properly be considered electrodynamics, thereader may wonder whether such things as kinematic transformations ofcollision problems can . My reply is that these examples occur naturallyonce one has established the four-vector character of a particle's momentumand energy, that they serve as useful practice in manipulating Lorentz

transformations, and that the end results are valuable and often hard tofind elsewhere.

Chapter 13 on collisions between charged particles emphasizes energyloss and scattering and develops concepts of use in later chapters . Herefor the first time in the book I use semiclassical arguments based on theuncertainty principle to obtain approximate quantum-mechanical ex-pressions for energy loss, etc., from the classical results . This approach, sofruitful in the hands of Niels Bohr and E . J. Williams, allows one to seeclearly how and when quantum-mechanical effects enter to modify classicalconsiderations .

The important subject of emission of radiation by accelerated point

charges is discussed in detail in Chapters 14 and 15 . Relativistic effectsare stressed, and expressions for the frequency and angular dependence ofthe emitted radiation are developed in sufficient generality for all appli-cations . The examples treated range from synchrotron radiation tobremsstrahlung and radiative beta processes . Cherenkov radiation and the

Weizsacker-Williams method of virtual quanta are also discussed . In theatomic and nuclear collision processes semiclassical arguments are again

employed to obtain approximate quantum-mechanical results . I lay con-siderable stress on this point because I feel that it is important for thestudent to see that radiative effects such as bremsstrahlung are almostentirely classical in nature, even though involving small-scale collisions .

A student who meets bremsstrahlung for the first time as an example of acalculation in quantum field theory will not understand its physical basis .

Multipole fields form the subject matter of Chapter 16 . The expansionof scalar and vector fields in spherical waves is developed from first

principles with no restrictions as to the relative dimensions of source and

x Preface

wavelength . Then the properties of electric and magnetic multipole radia-tion fields are considered . Once the connection to the multipole momentsof the source has been made, examples of atomic and nuclear multipole

radiation are discussed, as well as a macroscopic source whose dimensionsare comparable to a wavelength . The scattering of a plane electromagneticwave by a spherical object is treated in some detail in order to illustrate aboundary-value problem with vector spherical .waves .

In the last chapter the difficult problem of radiative reaction is discussed .The treatment is physical, rather than mathematical, with the emphasis ondelimiting the areas where approximate radiative corrections are adequateand on finding where and why existing theories fail . The original Abraham-Lorentz theory of the self-force is presented, as well as more recent classicalconsiderations .

The book ends with an appendix on units and dimensions and a biblio-graphy. In the appendix I have attempted to show the logical stepsinvolved in setting up a system of units, without haranguing the reader asto the obvious virtues of my choice of units . I have provided two tableswhich I hope will be useful, one for converting equations and symbols andthe other for converting a given quantity of something from so manyGaussian units to so many mks units, and vice versa . The bibliographylists books which 1 think the reader may find pertinent and useful forreference or additional study . These books are referred to by author'sname in the reading lists at the end of each chapter .

This book is the outgrowth of a graduate course in classical electro-dynamics which I have taught off and on over the past eleven years, at boththe University of Illinois and McGill University . I wish to thank mycolleagues and students at both institutions for countless helpful remarksand discussions . Special mention must be made of Professor P . R. Wallaceof McGill, who gave me the opportunity and encouragement to teach whatwas then a rather unorthodox course in electromagnetism, and ProfessorsH . W. Wyld and G. Ascoli of Illinois, who have been particularly free withmany helpful suggestions on the treatment of various topics . My thanksare also extended to Dr. A . N . Kaufman for reading and commenting on apreliminary version of the manuscript, and to Mr . G. L . Kane for hiszealous help in preparing the index .

J. D . JACKSON

Urbana, IllinoisJanuary, 1962

Contents

1 .11 .2

1 .3

1 .41 . 51 .6

1 .71 .8

1 .9

1 .101 . 11

chapter 1 . Introduction to Electrostatics

Coulomb's law, 1 .Electric field, 2 .Gauss's law, 4 .Differential form of Gauss's law, 6 .Scalar potential, 7 .Surface distributions of charges and dipoles, 9 .Poisson's and Laplace's equations, 12 .Green's theorem, 14 .Uniqueness theorem, 15 .Formal solution of boundary-value problem, Green's functions, 18 .Electrostatic potential energy, 20 .

1

2 . 12 .22.32 .42 .52 .62 .7

References and suggested reading, 23.Problems, 23.

chapter 2. Boundary-Value Problems in Electrostatics, I

Method of images, 26.Point charge and a grounded conducting sphere, 27 .

Point charge and a charged, insulated, conducting sphere, 31 .Point charge and a conducting sphere at fixed potential, 33 .Conducting sphere in a uniform field, 33 .Method of inversion, 35 .Green's function for a sphere, 40 .

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26

x ii

2.8 Conducting sphere with hemispheres at different potentials, 42 .2.9 Orthogonal functions and expansions, 44 .2.10 Separation of variables in rectangular coordinates, 47 .

References and suggested reading, 50 .Problems, 5 1 .

Contents

chapter 3. Boundary-Va lue Problems in Electrostatics, II 54

3 .1 Laplace's equation in spherical coordinates, 54.3 .2 Legendre polynomials, 56 .3 .3 Boundary-value problems with azimuthal symmetry, 60 .3.4 Spherical harmonics, 64.3.5 Addition theorem for spherical harmonics, 67 .3.6 Cylindrical coordinates, Besse] functions, 69 .3.7 Boundary-value problems in cylindrical coordinates, 75 .3.8 Expansion of Green's functions in spherical coordinates, 77 .3 .9 Use of spherical Green's function expansion, 81 .3 .10 Expansion of Green's functions in cylindrical coordinates, 84 .3 .11 Eigenfunction expansions for Green's functions, 87 .3 .12 Mixed boundary conditions, charged conducting disc, 89 .

References and suggested reading, 93 .Problems, 94 .

4.1

4.2

chapter 4 . Multipoles, E lectrostatics of Macroscopic Media,Die lectrics

Multipole expansion, 98 .Multipole expansion of the energy of a charge distribution in anexternal field, 101 .Macroscopic electrostatics, 103 .Simple dielectrics and boundary conditions, 1 08 .Boundary-value problems with dielectrics, 210 .Molecular polarizability and electric susceptibility, 116 .Models for molecular polarizability, 119.Electrostatic energy in dielectric media, 123 .

98

4 .34 .44 .5

4.64.7

4 . 8

5 . 1

5.25.35.4

5.5

5.6

References and suggested reading, 127 .

Problems, 128 .

chapter 5 . Magnetostatic s

Introduction and definitions, 132 .Blot and Savart law, 133 .Differential equations of magnetostatics, Ampere's law, 137 .Vector potential, 139 .Magnetic induction of a circular loop of current, 141 .Localized current distribution, magnetic moment, 145 .

132

Contents

5.7 Force and torque on localized currents in an external field, 148 .5.8 Macroscopic equations, 150 .5 .9 Boundary conditions, 154 .5.10 Uniformly magnetized sphere, 156.5.11 Magnetized sphere in an external field, permanent magnets, 160 .5 .12 Magnetic shielding, 162 .

References and suggested reading, 164 .Problems, 165.

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chapter 6. Time-Varying Fields, Maxwell's Equations, Con-servation Laws 169

6.1 Faraday's law of induction, 170 .6.2 Energy in the magnetic field, 173 .6 .3 Maxwell's displacement current, Maxwell's equations, 177 .6.4 Vector and scalar potentials, wave equations, 179 .6.5 Gauge transformations, 181 .6 .6 Green's function for the time-dependent wave equation, 1 83 .6.7 Initial-value problem, Kirchhoff's integral representation, 186 .6 .8 Poynting's theorem, 189 .6.9 Conservation laws, 190 .6 .10 Macroscopic equations, 194.

References and suggested reading, 198 .Problems, 198 .

chapter 7. Plane Electromagnetic Waves 202

7.1 Plane waves in a nonconducting medium, 202 .7.2 Linear and circular polarization, 205 .7.3 Superposition of waves, group velocity, 208 .7.4 Propagation of a pulse in a dispersive medium, 212 .7.5 Reflection and refraction, 2 1 6 .7 .6 Polarization by reflection, total internal reflection, 220 .7.7 Waves in a conducting medium, 222 .7.8 Simple model for conductivity, 225 .7.9 Transverse waves in a tenuous plasma, 226 .

References and suggested reading, 231 .Problems, 231 .

chapter 8. Wave Guides and Resonant Cavities 235

8 .1 Fields at the surface of and within a conductor, 236 .8 .2 Cylindrical cavities and wave guides, 240 .8.3 Wave guides, 244 .8 .4 Modes in a rectangular wave guide, 246 .8 .5 Energy flow and attenuation in wave guides, 248 .

X1V Con tents

8.6 Resonant cavities, 252 .8.7 Power losses in a cavity, 255 .8 .8 Dielectric wave guides, 259 .

References and suggested reading, 264 .Problems, 264 .

chapter 9. S imple Radiating Systems and Diffraction

9.1 Fields and radiation of a localized source, 268 .9 .2 Oscillating electric dipole, 271 .9 .3 Magnetic dipole and quadrupole fields, 273 .9 .4 Center-fed linear antenna, 277.9 .5 Kirchhoff's integral for diffraction, 280 .9 .6 Vector equivalents of Kirchhoff's integral, 283 .9.7 Babinet's principle, 288 .9 .8 Diffraction by a circular aperture, 292 .9.9 Diffraction by small apertures, 297 .

9 .10 Scattering by a conducting sphere at short wavelengths, 299 .

References and suggested reading, 3(}4 .Problems, 305 .

chapter 10. Magnetohydrodynam ics and Plasma Physics

10.1 Introduction and definitions, 309.10.2 Magnetohydrodynamic equations, 311 .1 0.3 Magnetic diffusion, viscosity, and pressure, 313 .10.4 Magnetohydrodynamic flow, 316 .10.5 Pinch effect, 3 20 .10.6 Dynamic model of the pinch effect, 322 .10.7 Instabilities, 326.10.8 Magnetohydrodynamic waves, 329 .10.9 High-frequency plasma oscillations, 335 .10.10 Short-wavelength limit, Debye screening distance, 339 .

References and suggested reading, 343.Problems, 343 .

chapter 11. Special Theory of Relativity

11 .111 .211 .311 .411 .511 .611 . 711 .8

Historical background and key experiments, 347 .Postulates of special relativity, Lorentz transformation, 352 .FitzGerald-Lorentz contraction and time dilatation, 357 .Addition of velocities, Doppler shift, 3 60 .Thomas precession, 364 .Proper time and light cone, 369 .Lorentz transformations as orthogonal transformations, 371 .4-vectors and tensors, 374 .

268

309

347

Contents

1 1 .9 Covariance of electrodynamics, 377 .

1 I .10 Transformation of electromagnetic fields, 380 .

11 .11 Covariance of the force equation and the conservation laws, 383 .

References and suggested reading, 386.Problems, 387.

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chapter 12. Relativistic-Particle Kinematics and Dynamics 39 1

12.1 Momentum and energy of a particle, 391 .12 .2 Kinematics of decay of an unstable particle, 394 .

12 .3 Center of momentum transformation, 397 .12 .4 Transformation of momenta from the center of momentum frame

to the laboratory, 400 .12.5 Covariant Lorentz force equation, Lagrangian and Hamiltonian,

404 .12 .6 Relativistic corrections to the Lagrangian for interacting charged

particles, 409 .

12 .7 Motion in a uniform, static, magnetic field, 411 .12.8 Motion in combined uniform, static, electric and magnetic fields,

412 .12.9 Particle drifts in nonuniform magnetic fields, 415.

12.10 Adiabatic invariance of flux through an orbit, 419 .

References and suggested reading, 424 .Problems, 425 .

chapter 13. Collisions be tween Charged Part icles, Energy Loss,and Scattering 429

13.1 Energy transfer in a Coulomb collision, 430 .

13.2 Energy transfer to a harmonically bound charge, 434 .

13 .3 Classical and quantum-mechanical energy loss, 438 .

13 .4 Density effect in collision energy loss, 443 .13 .5 Energy loss in an electronic plasma, 450 .

13.6 Elastic scattering of fast particles by atoms, 451 .

13.7 Mean square angle of scattering, multiple scattering, 456 .

13.8 Electrical conductivity of a plasma, 459 .

References and suggested reading, 462 .

Problems, 462.

chapter 14 . Radiat ion by Moving Charges 464

14.1 Lienard-Wiechert potentials and fields, 464 .14.2 Larmor's radiated power formula and its relativisti c

generalization, 468 .14.3 Angular distribution of radiation, 472 .

14.4 Radiation by an extremely relativistic charged particle, 475 .

X VE Contents

14 .5 General angular and frequency distributions of radiation fro maccelerated charges, 477 .

14.6 Frequency spectrum from relativistic charged particle in an instan-taneously circular orbit, synchrotron radiation, 481 .

14.7 Thomson scattering, 488 .14.$ Scattering by quasi-free charges, 491 .14.9 Cherenkov radiation, 494 .

References and suggested reading, 499.Problems, 500.

chapter 15. Bremsstrahlung , Method of V irtual Quanta, Radia-tive Beta Processes 505

15 .1 Radiation emitted during collisions, 506 .15.2 Bremsstrahlung in nonrelativistic Coulomb collisions, 509 .15.3 Relativistic bremsstrahlung, 513 .15.4 Screening, relativistic radiative energy loss, 516 .15.5 Weizsacker-Williams method of virtual quanta, 520 .15 .6 Bremsstrahlung as the scattering of virtual quanta, 525 .15.7 Radiation emitted during beta decay, 526.15 .8 Radiation emitted in orbital-electron capture, 528 .

References and suggested reading, 533 .Problems, 534.

chapter 16. MuItipole Fields 538

16.1 Scalar spherical waves, 539 .16 .2 Multipole expansion of electromagnetic fields, 543 .16 .3 Properties of multipole fields, energy and angular momentum of

radiation, 546 .16.4 Angular distributions, 550 .16 .5 Sources of multipole radiation, multipole moments, 553 .16.6 Multipole radiation in atoms and nuclei, 557 .16.7 Radiation from a linear, center-fed antenna, 562 .16 .8 Spherical expansion of a vector plane wave, 566 .16 .9 Scattering by a conducting sphere, 569 .16.10 Boundary-value problems with multipole fields, 574 .

References and suggested reading, 574.Problems, 574 .

chapter 17. Radiation Damping, Self-Fields of a Particle,Scattering and Absorption of Radiation by a BoundSystem 578

17 .1 Introductory considerations, 578 .17.2 Radiative reaction force, 581 .

Contents

17.3 Abraham-Lorentz evaluation of the self-force, 584 .17.4 Difficulties with the Abraham-Lorentz model, 589 .17.5 Lorentz transformation properties of the Abraham-Lorentz model ,

Poincare stresses, 590 .17.6 Covariant definitions of self-energy and momentum, 594 .17.7 Integrodifferential equation of motion, including damping, 597 .17.8 Line breadth and level shift of an oscillator, 600 .1 7.9 Scattering and absorption of radiation by an oscillator, 602.

References and suggested reading, 607 .Problems, 608 .

appendix. Units and Dimension s

Bibliography

xvii

611

622

Index 625