zotero/storage/UDTGU4MK/.zotero-ft-cache

Classical
ill
Electrodynamics
I
Second Edition
D. JACKSON
J.

Vector Formulas

a • (bxc) = b • (cxa) = c • (axb)

a x (b x c) = (a • c)b-(a • b)c

(axb) • (cxd) = (a • c)(b • d)-(a • d)(b • c)

VxVi// =

V-(Vxa) =

Vx(Vxa) = V(V-a)--V2a

V = + • (i|/a) a • Vi|/ i|/V • a

V = a+ x x (i//a) Viff

<//V x a

V(a • b) = (a • V)b+(b • V)a+ax(Vxb)+bx(Vxa)

V • (axb) = b • (Vxa)-a • (Vxb)

Vx(a xb) = a(V • b)-b(V • a)+(b • V)a-(a • V)b

If x is the coordinate of a point with respect to some origin, with magnitude r=|x|, and n = x/r is a unit radial vector, then
V • x = 3 Vxx =
Vn = -2 Vxn-0
r
(a • V)n = -1 [a-n(a • n)]=ay

Theorems from Vector Calculus

A V In the following </>, and are well-behaved scalar or vector functions, is a
three-dimensional volume with volume element d*x, S is a closed twodimensional surface bounding V, with area element da and unit outward normal n at da.

A A V |

• d 3 x = | • n da

(Divergence theorem)

3
Vijjd

x

=

ifmda

J

J

VxA j*

d3x =

nxA da

J

= 2
(</>V i//+ V<j>

•

Vi|0

d3x

[

<t>n-V*ltda J

(Green's first identity)

= -nda d 2

2

2

(</>V i//-i|/V <f>)

x

j" (<(>Vi(f-i(rV<f>)

J

(Green's theorem)

C In the following S is an open surface and is the contour bounding it, with line
element dl. The normal n to S is defined by the right-hand side rule in relation to the sense of the line integral around C.

A j*

(VxA)

= n da • <j>

• d\

(Stokes's theorem)

j" nxViff da = ^ \\t d\

Classical
Electrodynamics

Digitized by the Internet Archive
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in 2013
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Classical
Electrodynamics
Second Edition
JOHN DAVID JACKSON
Professor of Physics, University of California, Berkeley
JOHN WILEY & SONS, New York • Chichester • Brisbane • Toronto

© & Copyright 1962, 1975, by John Wiley Sons, Inc.
All rights reserved. Published simultaneously in Canada.
Reproduction or translation of any part of this work beyond that permitted by Sections 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc.
Library of Congress Cataloging in Publication Data:

Jackson, John David, 1925Classical electrodynamics.

Bibliography: p. Includes index. 1. Electrodynamics.

I. Title.

QC631.J3 1975 537.6'01 ISBN 0-471-43132-X

75-9962

Printed in the United States of America 10 9 8

To the memory of my father,
Walter David Jackson

Preface
In the thirteen years since the appearance of the first edition, my interest in
classical electromagnetism has waxed and waned, but never fallen to zero. The subject is ever fresh. There are always important new applications and examples.
The present edition reflects two efforts on my part: the refinement and
improvement of material already in the first edition; the addition of new topics
(and the omission of a few).
The major purposes and emphasis are still the same, but there are exten-
A sive changes and additions. major augmentation is the "Introduction and
Survey" at the beginning. Topics such as the present experimental limits on the mass of the photon and the status of linear superposition are treated there. The aim is to provide a survey of those basics that are often assumed to be
well known when one writes down the Maxwell equations and begins to solve specific examples. Other major changes in the first half of the book include a new
treatment of the derivation of the equations of macroscopic electromagnetism from the microscopic description; a discussion of symmetry properties of mechanical and electromagnetic quantities; sections on magnetic monopoles and the quantization condition of Dirac; Stokes's polarization parameters; a unified discussion of the frequency dispersion characteristics of dielectrics, conductors, and plasmas; a discussion of causality and the Kramers-Kronig dispersion relations; a simplified, but still extensive, version of the classic Sommerfeld-
Brillouin problem of the arrival of a signal in a dispersive medium (recently
verified experimentally); an unusual example of a resonant cavity; the normal-
mode expansion of an arbitrary field in a wave guide; and related discussions of
sources in a guide or cavity and the transmission and reflection coefficients of flat obstacles in wave guides.
Chapter 9, on simple radiating systems and diffraction, has been enlarged to include scattering at long wavelengths (the blue sky, for example) and the optical theorem. The sections on scalar and vectorial diffraction have been improved.
vii

viii

Preface

Chapters 11 and 12, on special relativity, have been rewritten almost
completely. The old pseudo-Euclidean metric with x4 = ict has been replaced by g^" (with g 00 = + l, g u =-l, i=l, 2, 3). The change of metric necessitated a complete revision and thus permitted substitution of modern experiments
and concerns about the experimental basis of the special theory for the time-honored aberration of starlight and the Michelson-Morley experiment. Other aspects have been modernized, too. The extensive treatment of relativistic kinematics of the first edition has been relegated to the problems. In its stead is a discussion of the Lagrangian for the electromagnetic fields, the canonical and symmetric stress-energy tensor, and the Proca Lagrangian for massive photons.
Significant alterations in the remaining chapters include a new section on transition radiation, a completely revised (and much more satisfactory) semiclassical treatment of radiation emitted in collisions that stresses momentum transfer
instead of impact parameter, and a better derivation of the coupling of multipole fields to their sources. The collection of formulas and page references to special
functions on the front and back flyleaves is a much requested addition. Of the
278 problems, 117 (more than 40 per cent) are new. The one area that remains almost completely unchanged is the chapter on
magnetohydrodynamics and plasma physics. I regret this. But the book obviously has grown tremendously, and there are available many books devoted exclusively to the subject of plasmas or magnetohydrodynamics.
Of minor note is the change from Maxwell's equations and a Green's function to the Maxwell equations and a Green function. The latter boggles some minds,
but is in conformity with other usage (Bessel function, for example). It is still Green's theorem, however, because that's whose theorem it is.
Work on this edition began in earnest during the first half of 1970 on the
occasion of a sabbatical leave spent at Clare Hall and the Cavendish Laboratory
in Cambridge. I am grateful to the University of California for the leave and indebted to N. F. Mott for welcoming me as a visitor to the Cavendish Laboratory and to R. J. Eden and A. B. Pippard for my appointment as a
Visiting Fellow of Clare Hall. Tangible and intangible evidence at the Cavendish
of Maxwell, Rayleigh and Thomson provided inspiration for my task; the
stimulation of everyday activities there provided necessary diversion.
This new edition has benefited from questions, suggestions, comments and
criticism from many students, colleagues, and strangers. Among those to whom I
owe some specific debt of gratitude are A. M. Bincer, L. S. Brown, R. W. Brown, E. U. Condon, H. H. Denman, S. Deser, A. J. Dragt, V. L. Fitch, M. B. Halpern, A. Hobson, J. P. Hurley, D. L. Judd, L. T. Kerth, E. Marx, M. Nauenberg, A. B. Pippard, A. M. Portis, R. K. Sachs, W. M. Saslow, R. Schleif, V. L. Telegdi, T. Tredon, E. P. Tryon, V. F. Weisskopf, and Dudley Williams. Especially helpful
were D. G. Boulware, R. N. Cahn, Leverett Davis, Jr., K. Gottfried, C. K.
Graham, E. M. Purcell, and E. H. Wichmann. I send my thanks and fraternal greetings to all of these people, to the other readers who have written to me, and

Preface

ix

the countless students who have struggled with the problems (and sometimes written asking for solutions to be dispatched before some deadline!). To my mind, the book is better than ever. May each reader benefit and enjoy!

Berkeley, California, 1974

J. D. Jackson

Preface to the First Edition

Classical electromagnetic theory, together with classical and quantum

mechanics, forms the core of present-day theoretical training for undergraduate
A and graduate physicists. thorough grounding these subjects is a requirement

for more advanced or specialized training. Typically the undergraduate program in electricity and magnetism involves

two or perhaps three semesters beyond elementary physics, with the emphasis

on the fundamental laws, laboratory verification and elaboration of their consequences, circuit analysis, simple wave phenomena, and radiation. The

mathematical tools utilized include vector calculus, ordinary differential equa-

tions 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 is given to
My beginning graduate students. It is for such a course that my book is designed.
aim in teaching a graduate course in electromagnetism is at least threefold. The

first aim is to present the basic subject matter as a coherent whole, with emphasis

on the unity of electric and magnetic phenomena, both in their physical basis and in the mode of mathematical description. The second, concurrent aim is to develop and utilize a number of topics in mathematical physics which are useful

in both electromagnetic theory and wave mechanics. These include Green's

theorems and Green's functions, orthonormal expansions, spherical harmonics,
A cylindrical and spherical Bessel functions. third and perhaps most important
purpose is the presentation of new material, especially on the interaction of

relativistic charged particles with electromagnetic fields. In this last area

My personal preferences and prejudices enter strongly.

choice of topics is

governed by what I feel is important and useful for students interested in

theoretical physics, experimental nuclear and high-energy physics, and that as

yet ill-defined field of plasma physics.

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

xi

xii

Preface to the First Edition

chapters are devoted to the development of Maxwell's theory of electromagnet-
ism. Much of the necessary mathematical apparatus is constructed along the
way, especially in Chapters 2 and 3, where boundary-value problems are
E discussed thoroughly. The treatment is initially in terms of the electric field
D and the magnetic induction B, with the derived macroscopic quantities, and
H, introduced by suitable averaging over ensembles of atoms or molecules. In the discussion of dielectrics, simple classical models for atomic polarizability are described, but for magnetic materials no such attempt is made. Partly this omission was a question of space, but truly classical models of magnetic susceptibility are not possible Furthermore, elucidation of the interesting phenomenon of ferromagnetism needs almost a book in itself.
The next three chapters (7-9) illustrate various electromagnetic phenomena, 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 Chapter 8 is developed for systems of
Q arbitrary cross section, and the problems of attenuation in guides and the of a
cavity are handled in a very general way which emphasizes the physical processes involved. The elementary theory of multipole radiation from a localized source and diffraction occupy Chapter 9. Since the simple scalar theory of diffraction is covered in many optics textbooks, as well as undergraduate books on electricity and magnetism, I have presented an improved, although still approximate, theory of diffraction based on vector rather than scalar Green's theorems.
The subject of magnetohydrodynamics and plasmas receives increasingly more attention from physicists and astrophysicists. Chapter 10 represents a survey of this complex field with an introduction to the main physical ideas
involved.
The first nine or ten chapters constitute the basic material of classical
A electricity and magnetism. graduate student in physics may be expected to
have been exposed to much of this material, perhaps at a somewhat lower level, as an undergraduate. But he obtains a more mature view of it, understands it more deeply, and gains a considerable technical ability in analytic 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 interaction of charged particles with each
other and with electromagnetic fields, especially when moving relativistically. The special theory of relativity had its origins in classical electrodynamics.
And even after almost 60 years, classical electrodynamics still impresses and
delights as a beautiful example of the covariance of physical laws under Lorentz transformations. The special theory of relativity is discussed in Chapter 11, where all the necessary formal apparatus is developed, various kinematic consequences are explored, and the covariance of electrodynamics is established. The next chapter is devoted to relativistic particle kinematics and dynamics. Although the dynamics of charged particles in electromagnetic fields

Preface to the First Edition

xiii

can properly be considered electrodynamics, the reader may wonder whether

My such things as kinematic transformations of collision problems can.

reply is

that these examples occur naturally once one has established the four-vector
character of a particle's momentum and energy, that they serve as useful practice

in manipulating Lorentz transformations, and that the end results are valuable

and often hard to find elsewhere.

Chapter 13 on collisions between charged particles emphasizes energy loss

and scattering and develops concepts of use in later chapters. Here for the first

time in the book I use semiclassical arguments based on the uncertainty principle

to obtain approximate quantum-mechanical expressions for energy loss, etc.,

from the classical results. This approach, so fruitful in the hands of Niels Bohr

and E. J. Williams, allows one to see clearly how and when quantum-mechanical

effects enter to modify classical considerations.

The important subject of emission of radiation by accelerated point charges is

discussed in detail in Chapters 14 and 15. Relativistic effects are stressed, and

expressions for the frequency and angular dependence of the emitted radiation

are developed in sufficient generality for all applications. The examples treated

range from synchrotron radiation to bremsstrahlung and radiative beta proc-

esses. Cherenkov radiation and the Weizsacker-Williams method of virtual quanta

are also discussed. In the atomic and nuclear collision processes semiclassical

arguments are again employed to obtain approximate quantum-mechanical

results. I lay considerable stress on this point because I feel that it is important

for the student to see that radiative effects such as bremsstrahlung are almost
A entirely classical in nature, even though involving small-scale collisions.

student who meets bremsstrahlung for the first time as an example of a

calculation in quantum field theory will not understand its physical basis.

Multipole fields form the subject matter of Chapter 16. The expansion of

scalar and vector fields in spherical waves is developed from first principles with

no restrictions as to the relative dimensions of source and wavelength. Then the

properties of electric and magnetic multipole radiation fields are considered.

Once the connection to the multipole moments of the source has been made,

examples of atomic and nuclear multipole radiation are discussed, as well as a

macroscopic source whose dimensions are comparable to a wavelength. The

scattering of a plane electromagnetic wave by a spherical object is treated in

some detail in order to illustrate a boundary-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 on delimiting

the areas where approximate radiative corrections are adequate and on finding

where and why existing theories fail. The original Abraham-Lorentz theory of

the self-force is presented, as well as more recent classical considerations.

The book ends with an appendix on units and dimensions and a bibliography.

In the appendix I have attempted to show the logical steps involved in setting up

xiv

Preface to the First Edition

a system of units, without haranguing the reader as to the obvious virtues of my
choice of units. I have provided two tables which I hope will be useful, one for converting equations and symbols and the other for converting a given quantity
of something from so many Gaussian units to so many mks units, and vice versa. The bibliography lists books which I think the reader may find pertinent and
useful for reference or additional study. These books are referred to by author's name in the reading lists at the end of each chapter.
This book is the outgrowth of a graduate course in classical electrodynamics which I have taught off and on over the past eleven years, at both the University
of Illinois and McGill University. I wish to thank my colleagues and students at
both institutions for countless helpful remarks and discussions. Special mention
must be made of Professor P. R. Wallace of McGill, who gave me the
opportunity and encouragement to teach what was then a rather unorthodox
course in electromagnetism, and Professors H. W. Wyld and G. Ascoli of Illinois, who have been particularly free with many helpful suggestions on the
treatment of various topics. My thanks are also extended to Dr. A. N. Kaufman
for reading and commenting on a preliminary version of the manuscript, and to Mr. G. L. Kane for his zealous help in preparing the index.

Urbana, Illinois January, 1962

J. D. Jackson

Contents

Introduction and Survey

1.1 Maxwell Equations in Vacuum, Fields, and Sources

2

1.2 The Inverse Square Law or the Mass of the Photon

5

1.3 Linear Superposition

10

1.4 The Maxwell Equations in Macroscopic Media

13

1.5 Boundary Conditions at Interfaces between Different Media

17

1.6 Some Remarks on Idealizations in Electromagnetism

22

References and Suggested Reading

25

Chapter 1. Introduction to Electrostatics

27

1.1 Coulomb's Law

27

1.2 Electric Field

28

1.3 Gauss's Law

30

1.4 Differential Form of Gauss's Law

32

1.5 Another Equation of Electrostatics and the Scalar Potential

33

1.6 Surface Distributions of Charges and Dipoles and Discontinuities in the

Electric Field and Potential

35

1.7 Poisson and Laplace Equations

38

1.8 Green's Theorem

40

1.9 Uniqueness of the Solution with Dirichlet or Neumann Boundary Condi-

tions

42

1.10 Formal Solution of Electrostatic Boundary-Value Problem with Green

Function

43

1.11 Electrostatic Potential Energy and Energy Density, Capacitance

45

References and Suggested Reading

49

Problems

49

Chapter 2. Boundary-Value Problems in

Electrostatics : I

54

2.1 Method of Images

54

2.2 Point Charge in the Presence of a Grounded Conducting Sphere

55

2.3 Point Charge in the Presence of a Charged, Insulated, Conducting Sphere

58

2.4 Point Charge Near a Conducting Sphere at Fixed Potential

60

2.5 Conducting Sphere in a Uniform Electric Field by the Method of Images

60

2.6 Green Function for the Sphere, General Solution for the Potential

62

2.7 Conducting Sphere with Hemispheres at Different Potentials

63

2.8 Orthogonal Functions and Expansions

65

XV

xvi

Contents

2.9 Separation of Variables, Laplace Equation in Rectangular Coordinates

68

A 2.10 Two-dimensional Potential Problem, Summation of a Fourier Series

71

2.1 1 Fields and Charge Densities in Two-dimensional Corners and Along Edges

75

References and Suggested Reading

78

Problems

79

Chapter 3. Boundary-Value Problems in

Electrostatics: II

84

3.1 Laplace Equation in Spherical Coordinates

84

3.2 Legendre Equation and Legendre Polynomials

85

3.3 Boundary-Value Problems with Azimuthal Symmetry

90

3.4 Behavior of Fields in a Conical Hole or near a Sharp Point

94

Y 3.5 Associated Legendre Functions and the Spherical Harmonics lm (0,$)

98

3.6 Addition Theorem for Spherical Harmonics

100

3.7 Laplace Equation in Cylindrical Coordinates, Bessel Functions

102

3.8 Boundary- Value Problems in Cylindrical Coordinates

108

3.9 Expansion of Green Functions in Spherical Coordinates

110

3.10 Solution of Potential Problems with Spherical Green Function Expansion

113

3.11 Expansion of Green Functions in Cylindrical Coordinates

116

3.12 Eigenfunction Expansions for Green Functions

119

3.13 Mixed Boundary Conditions, Conducting Plane with a Circular Hole

121

References and Suggested Reading

127

Problems

128

Chapter 4. Multipoles, Electrostatics of Macroscopic Media, Dielectrics
4.1 Multipole Expansion 4.2 Multipole Expansion of the Energy of a Charge Distribution in an External
Field
4.3 Elementary Treatment of Electrostatics with Ponderable Media 4.4 Boundary- Value Problems with Dielectrics 4.5 Molecular Polarizability and Electric Susceptibility 4.6 Models for the Molecular Polarizability 4.7 Electrostatic Energy in Dielectric Media
References and Suggested Reading Problems

136
136
142 143 147 152 155 158 163 163

Chapter 5. Magnetostatics
5.1 Introduction and Definitions
5.2 Biot and Savart Law

168
168 169

Contents
5.3 The Differential Equations of Magnetostatics and Ampere's Law
5.4 Vector Potential
5.5 Vector Potential and Magnetic Induction for a Circular Current Loop 5.6 Magnetic Fields of a Localized Current Distribution, Magnetic Moment 5.7 Force and Torque on and Energy of a Localized Current Distribution in an
External Magnetic Induction
H 5.8 Macroscopic Equations, Boundary Conditions on B and
5.9 Methods of Solving Boundary-Value Problems in Magnetostatics 5.10 Uniformly Magnetized Sphere 5.11 Magnetized Sphere in an External Field, Permanent Magnets 5.12 Magnetic Shielding, Spherical Shell of Permeable Material in a Uniform
Field
5.13 Effect of a Circular Hole in a Perfectly Conducting Plane with an
Asymptotically Uniform Tangential Magnetic Field on One Side
References and Suggested Reading Problems

xvii
173 175 177 180
184 187 191 194 197
199
201 204 205

Chapter 6. Time-Varying Fields, Maxwell Equations, Conservation Laws
6.1 Faraday's Law of Induction
6.2 Energy in the Magnetic Field 6.3 Maxwell's Displacement Current, Maxwell Equations 6.4 Vector and Scalar Potentials 6.5 Gauge Transformations, Lorentz Gauge, Coulomb Gauge
6.6 Green Functions for the Wave Equation
6.7 Derivation of the Equations of Macroscopic Electromagnetism
6.8 Poynting's Theorem and Conservation of Energy and Momentum for a
System of Charged Particles and Electromagnetic Fields 6.9 Conservation Laws for Macroscopic Media 6.10 Poynting's Theorem for Harmonic Fields, Field Definitions of Impedance
and Admittance 6.11 Transformation Properties of Electromagnetic Fields and Sources under
Rotations, Spatial Reflections, and Time Reversal
6.12 On the Question of Magnetic Monopoles
6.13 Discussion of the Dirac Quantization Condition References and Suggested Reading Problems

209
210 213 217 219 220 223 226
236 240
241
245 251 254 260 261

Chapter 7. Plane Electromagnetic Waves and

Wave Propagation

269

7.1 Plane Waves in a Nonconducting Medium

269

7.2 Linear and Circular Polarization, Stokes Parameters

273

xviii

Contents

7.3 Reflection and Refraction of Electromagnetic Waves at a Plane Interface between Dielectrics
7.4 Polarization by Reflection and Total Internal Reflection 7.5 Frequency Dispersion Characteristics of Dielectrics, Conductors, and
Plasmas 7.6 Simplified Model of Propagation in the Ionosphere and Magnetosphere
7.7 Waves in a Conducting or Dissipative Medium 7.8 Superposition of Waves in One Dimension, Group Velocity
7.9 Illustration of the Spreading of a Pulse as It Propagates in a Dispersive
Medium
D 7.10 Causality in the Connection between and E, Kramers-Kronig Relations
7.11 Arrival of a Signal After Propagation Through a Dispersive Medium
References and Suggested Reading Problems

278 282
284 292 296 299
303 306 313 326 327

Chapter 8. Wave Guides and Resonant Cavities
8.1 Fields at the Surface of and within a Conductor
8.2 Cylindrical Cavities and Wave Guides 8.3 Wave Guides 8.4 Modes in a Rectangular Wave Guide 8.5 Energy Flow and Attenuation in Wave Guides
8.6 Perturbation of Boundary Conditions 8.7 Resonant Cavities
Q 8.8 Power Losses in a Cavity, of a Cavity
8.9 Earth and Ionosphere as a Resonant Cavity, Schumann Resonances 8.10 Dielectric Wave Guides 8.11 Expansion in Normal Modes, Fields Generated by a Localized Source in
Guide 8.12 Reflection and Transmission by Plane Diaphragms, Variational Approxi-
mation 8.13 Impedance of a Flat Strip Parallel to the Electric Field in a Rectangular
Wave Guide
References and Suggested Reading Problems

334
335 339 343 345 346 350 353 356 360 364
369
375
380 384 385

Chapter 9. Simple Radiating Systems, Scattering, and Diffraction
9.1 Fields and Radiation of a Localized Oscillating Source 9.2 Electric Dipole Fields and Radiation 9.3 Magnetic Dipole and Electric Quadrupole Fields 9.4 Center-fed Linear Antenna
9.5 Multipole Expansion for Localized Source or Aperture in Wave Guide

391
391 394 397 401 405

Contents

xix

9.6 Scattering at Long Wavelengths 9.7 Perturbation Theory of Scattering, Rayleigh's Explanation of the Blue Sky,
Scattering by Gases and Liquids 9.8 Scalar Diffraction Theory 9.9 Vector Equivalents of Kirchhoff Integral 9.10 Vectorial Diffraction Theory 9.11 Babinet's Principle of Complementary Screens 9.12 Diffraction by a Circular Aperture, Remarks on Small Apertures 9.13 Scattering in the Short-Wavelength Limit 9.14 Optical Theorem and Related Matters
References and Suggested Reading Problems

411
418 427 432 435 438 441 447 453 459 460

Chapter 10. Magnetohydrodynamics and Plasma
Physics

10.1 10.2 10.3 10.4
10.5 10.6 10.7 10.8 10.9

Introduction and Definitions Magnetohydrodynamic Equations Magnetic Diffusion, Viscosity, and Pressure Magnetohydrodynamic Flow between Boundaries with Crossed Electric
and Magnetic Fields Pinch Effect Instabilities in a Pinched Plasma Column Magnetohydrodynamic Waves Plasma Oscillations Short-wavelength Limit on Plasma Oscillations and the Debye Screening
Distance References and Suggested Reading Problems

469
469 471 472
475 479 482 485 490
494 497 498

Chapter 11. Special Theory of Relativity

11.1 The Situation before 1900, Einstein's Two Postulates 11.2 Some Recent Experiments

11.3 Lorentz Transformations and Basic Kinematic Results of Special Relativity

11.4 11.5 11.6

Addition of Velocities, Four- Velocity
Relativistic Momentum and Energy of a Particle
Mathematical Properties of the Space-Time of Special Relativity

11.7 Matrix Representation of Lorentz Transformations, Infinitesimal

Generators
11.8 Thomas Precession

11.9 Invariance of Electric Charge, Covariance of Electrodynamics

11.10 Transformation of Electromagnetic Fields

11.11 Relativistic Equation of Motion for Spin in Uniform or Slowly Varying External Fields

503
504 507 515 522 525 532
536 541 547 552
556

xx

Contents

11.12 Note on Notation and Units in Relativistic Kinematics References and Suggested Reading Problems

560 561 562

Chapter 12. Dynamics of Relativistic Particles and Electromagnetic Fields
12.1 Lagrangian and Hamiltonian for a Relativistic Charged Particle in External Electromagnetic Fields
12.2 On the Question of Obtaining the Magnetic Field, Magnetic Force, and the
Maxwell Equations from Coulomb's law and Special Relativity 12.3 Motion in a Uniform, Static, Magnetic Field 12.4 Motion in Combined Uniform, Static, Electric and Magnetic Fields 12.5 Particle Drifts in Nonuniform, Static Magnetic Fields 12.6 Adiabatic Invariance of Flux through Orbit of Particle 12.7 Lowest-Order Relativistic Corrections to the Lagrangian for Interacting
Charged Particles, the Darwin Lagrangian 12.8 Lagrangian for the Electromagnetic Field 12.9 Proca Lagrangian, Photon Mass Effects 12.10 Canonical and Symmetric Stress Tensors, Conservation Laws
12.11 Solution of the Wave Equation in Covariant Form, Invariant Green
Functions References and Suggested Reading Problems

571
572
578 581 582 584 588
593 595 597 601
608 612 613

13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8

Chapter 13. Collisions between Charged Particles, Energy Loss, and Scattering
Energy Transfer in a Coulomb Collision Energy Transfer to a Harmonically Bound Charge Classical and Quantum-Mechanical Energy-loss Formulas Density Effect in Collision Energy Loss Cherenkov Radiation Energy Loss in an Electronic Plasma Elastic Scattering of Fast Particles by Atoms Mean Square Angle of Scattering and the Angular Distribution of Multiple
Scattering
References and Suggested Reading Problems

618 619
623 626 632 638 641 643
647 651 651

14.1 14.2

Chapter 14. Radiation by Moving Charges
Lienard-Wiechert Potentials and Fields for a Point Charge Total Power Radiated by an Accelerated Charge: Larmor's Formula and
its Relativistic Generalization

654 654
658

Contents

14.3 14.4
14.5
14.6
14.7 14.8
14.9

Angular Distribution of Radiation Emitted by an Accelerated Charge Radiation Emitted by a Charge in Arbitrary, Extremely Relativistic
Motion Distribution in Frequency and Angle of Energy Radiated by Accelerated
Charges Frequency Spectrum of Radiation Emitted by a Relativistic Charged
Particle in Instantaneously Circular Motion Thomson Scattering of Radiation Scattering of Radiation by Quasi-Free Charges, Coherent and Incoherent
Scattering Transition Radiation
References and Suggested Reading Problems

xxi
662
665
668
672 679
683 685 693 694

15.1 15.2 15.3 15.4 15.5 15.6 15.7

Chapter 15. Bremsstrahlung, Method of Virtual
Quanta, Radiative Beta Processes
Radiation Emitted during Collisions
Bremsstrahlung in Coulomb Collisions Screening Effects, Relativistic Radiative Energy Loss Weizsacker- Williams Method of Virtual Quanta Bremsstrahlung as the Scattering of Virtual Quanta Radiation Emitted During Beta Decay
— Radiation Emitted During Orbital-Electron Capture Disappearance of
Charge and Magnetic Moment
References and Suggested Reading Problems

701
702 708 715 719 724 725
727 733 733

Chapter 16. Multipole Fields 16.1 Basic Spherical Wave Solutions of the Scalar Wave Equation
16.2 Multipole Expansion of the Electromagnetic Fields
16.3 Properties of Multipole Fields, Energy and Angular Momentum of Mul-
tipole Radiation 16.4 Angular Distribution of Multipole Radiation
16.5 Sources of Multipole Radiation, Multipole Moments 16.6 Multipole Radiation in Atomic and Nuclear Systems 16.7 Radiation from a Linear, Center-Fed Antenna 16.8 Spherical Wave Expansion of a Vector Plane Wave 16.9 Scattering of Electromagnetic Waves by a Sphere 16.10 Boundary-Value Problems with Multipole Fields
References and Suggested Reading Problems

739
739 744
747 752 755 758 763 767 769 775 776 776

xxii

Contents

17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8

Chapter 17.

Radiation Damping, Self-Fields of a Particle, Scattering and Absorption of Radiation by a Bound System

Introductory Considerations

Radiative Reaction Force from Conservation of Energy

Abraham-Lorentz Evaluation of the Self-Force

Difficulties with the Abraham-Lorentz Model

Covariant Definitions of Electromagnetic Energy and Momentum

Integrodifferential Equation of Motion, Including Radiation Damping

Line Breadth and Level Shift of an Oscillator

Scattering and Absorption of Radiation by an Oscillator

References and Suggested Reading

Problems

780
780 783 786 790 791 796 798 801 806 807

Appendix on Units and Dimensions
1 Units and Dimensions, Basic Units and Derived Units 2 Electromagnetic Units and Equations
3 Various Systems of Electromagnetic Units
4 Conversion of Equations and Amounts between Gaussian Units and
MKSA Units

811
811 813 816
817

Bibliography

822

Index

828

Classical
Electrodynamics

Introduction and Survey
Although amber and lodestone were known to the ancient Greeks, electrodynamics developed as a quantitative subject in less than a hundred years. Cavendish's remarkable experiments in electrostatics were done from 1771 to 1773. Coulomb's monumental researches began to be published in 1785. This marked the beginning of quantitative research in electricity and magnetism on a worldwide scale. Fifty years later Faraday was studying the effects of timevarying currents and magnetic fields. By 1864 Maxwell had published his famous
paper on a dynamical theory of the electromagnetic field. The story of the development of our understanding of electricity and
magnetism and of light is, of course, much longer and richer than the mention of a few names from one century would indicate. For a detailed account of the
fascinating history, the reader should consult the authoritative volumes by
A Whittaker* briefer account, with emphasis on optical phenomena, appears at
the beginning of Born and Wolf. This book is self-contained in that, though some mathematical background
(vector calculus, differential equations) is assumed, the subject of electrodynamics is developed from its beginnings in electrostatics. Most readers are
not coming to the subject for the first time, however. The purpose of this introduction is therefore not to set the stage for a discussion of Coulomb's law and other basics, but rather to present a review and a survey of classical
electromagnetism. Questions such as the current accuracy of the inverse square law of force (mass of the photon), the limits of validity of the principle of linear superposition, the effects of discreteness of charge and of energy differences are discussed. "Bread and butter" topics such as the boundary conditions for macroscopic fields at surfaces between different media and at conductors are
also treated. The aim is to set classical electromagnetism in context, to indicate
* Italicized surnames are used to denote books that are cited fully in the
Bibliography.
1

2

Classical Electrodynamics

Sect. 1.1

its domain of validity, and to elucidate some of the idealizations that it contains. Some results from later in the book and some nonclassical ideas are used in the
course of the discussion. Certainly a reader beginning electromagnetism for the first time will not follow all the arguments or see their significance. It is intended, however, that for others this introduction will serve as a springboard into the
later parts of the book, beyond Chapter 5, as well as a reminder of how the
subject stands as an experimental science.

1.1 Maxwell Equations in Vacuum, Fields, and Sources
The equations governing electromagnetic phenomena are the Maxwell equations, which for sources in vacuum are
V • E = 47rp

C dt C
(1.1)
VxE + i^ =
C dt
VB =

Implicit in the Maxwell equations is the continuity equation for charge density and current density,

^+V-J =
dt

(v 1.2)
'

This follows from combining the time derivative of the first equation in (1 . 1) with the divergence of the second equation. Also essential for consideration of charged particle motion is the Lorentz force equation,

F=q(E+^xB)

(1.3)

that gives the force acting on a point charge q in the presence of electromagnetic
fields.
These equations have been written in Gaussian units, the system of electromagnetic units used in this book. (Units and dimensions are discussed in the
Appendix.) The Maxwell equations are displayed in the commoner systems of units in Table 2 of the Appendix. Apart from the fields E and B and the sources
p and J, the equations involve a parameter c. This quantity has the dimensions of velocity and is the speed of light in vacuum. It is fundamental to all electromagnetic and relativistic phenomena. Based on our units of length and time, presently defined separately in terms of two different atomic transitions, as

Sect. 1.1

Introduction and Survey

3

discussed in the Appendix, this parameter has the empirical value

c = 299,792,456.2± 1 . 1 meters/second*

This result comes from an experiment using a highly stabilized helium-neon laser in which both the frequency and the wavelength were measured (3.39 fxm methane-stabilized line). In passing we note that the precision here is such that the present definition of the meter is likely to be replaced by one using c and the second. Other evidence [see Section 11.2(c)] indicates that to high accuracy the speed of light in vacuum is independent of frequency from very low frequencies
to at least i^ 1024 Hz (4GeV photons). For most practical purposes we can approximate c — 3xl0 8 m/sec or to be considerably more accurate, c = 2.998 x10 s m/sec.
The electric and magnetic fields E and B in (1.1) were originally introduced by
means of the force equation (1.3). In Coulomb's experiments forces acting between localized distributions of charge were observed. There it is found useful
(see Section 1.2) to introduce the electric field E as the force per unit charge.
Similarly, in Ampere's experiments the mutual forces of current-carrying loops
were studied (see Section 5.2). With the identification of NAqv as a current in a
A N conductor of cross-sectional area with charge carriers per unit volume
moving at velocity v, we see that B in (1.3) is defined in magnitude as a force per unit current. Although E and B thus first appear just as convenient replacements
for forces produced by distributions of charge and current, they have other important aspects. First, their introduction decouples conceptually the sources
from the test bodies experiencing electromagnetic forces. If the fields E and B
from two source distributions are the same at a given point in space, the force
acting on a test charge or current at that point will be the same, regardless of how different the source distributions are. This gives E and B in (1.3) meaning in their own right, independent of the sources. Second, electromagnetic fields can exist in regions of space where there are no sources. They can carry energy, momentum, and angular momentum and so have an existence totally independ-
ent of charges and currents. In fact, though there are recurring attempts to
eliminate explicit reference to the fields in favor of action-at-a-distance descrip-
tions of the interaction of charged particles, the concept of the electromagnetic
field is one of the most fruitful ideas of physics, both classically and quantum
mechanically.
The concept of E and B as ordinary fields is a classical notion. It can be
thought of as the classical limit (limit of large quantum numbers) of a quantum mechanical description in terms of real or virtual photons. In the domain of macroscopic phenomena and even some atomic phenomena the discrete photon
aspect of the electromagnetic field can usually be ignored or at least glossed
over. For example, 1 meter from a 100 watt light bulb, the root mean square
*K. Evenson et al., Phys. Rev. Letters 29, 1346 (1972).

4

Classical Electrodynamics

Sect. 1.1

electric field is of the order of 0.5 volts/cm and there are of the order of 10 15

FM visible photons/cm2 x sec. Similarly, an isotropic

antenna with a power of

100

watts

at

8
10

Hz

produces

a

r.m.s.

electric field

of only

5

microvolts/cm

at

a

distance of 100 kilometers, but this still corresponds to a flux of 10 12

photons/cm2 x sec,

or

about

9
10

photons

in

a

volume

of

1

wavelength

cubed

m (27

3 )

at

that

distance.

Ordinarily

an

apparatus

will

not

be

sensible

to

the

individual photons; the cumulative effect of many photons emitted or absorbed

will appear as a continuous, macroscopically observable response. Then a

completely classical description in terms of the Maxwell equations is permitted

and is appropriate.
How is one to decide a priori when a classical description of the electromag-

netic fields is adequate? Some sophistication is occasionally needed, but the following is usually a sufficient criterion: When the number of photons involved can be taken as large but the momentum carried by an individual photon is small compared to the momentum of the material system, then the response of the

material system can be determined adequately from a classical description of the

FM electromagnetic fields. For example, each 10 8 Hz photon emitted by our

A antenna gives it an impulse of only 2.2xl0~ 34 newton-seconds.

classical

treatment is surely adequate. Again, the scattering of light by a free electron is

governed by the classical Thomson formula (Section 14.7) at low frequencies, but by the laws of the Compton effect as the momentum hoj/c of the incident

photon becomes significant compared to mc. The photoelectric effect is nonclas-

sical for the matter system, since the quasi-free electrons in the metal change

their individual energies by amounts equal to those of the absorbed photons, but

the photoelectric current can be calculated quantum mechanically for the

electrons using a classical description of the electromagnetic fields.

The quantum nature of the electromagnetic fields must, on the other hand, be

taken into account in spontaneous emission of radiation by atoms, or by any

other system where there are no photons present initially and only a small

number of photons present finally. The average behavior may still be describable

in essentially classical terms, basically because of conservation of energy and
momentum. An example is the classical treatment (Section 17.2) of the

cascading of a charged particle down through the orbits of an attractive

potential. At high particle quantum numbers a classical description of particle motion is adequate, and the secular changes in energy and angular momentum

can be calculated classically from the radiation reaction because the energies of

the successive photons emitted are small compared to the kinetic or potential

energy of the orbiting particle.
The sources in (1.1) are p(x, t), the electric charge density, and J(x, t), the

electric current density. In classical electromagnetism they are assumed to be

continuous distributions in x, although we consider from time to time localized

distributions that can be approximated by points. The magnitudes of these point

charges are assumed to be completely arbitrary, but are known to be restricted in

Sect. 1.2

Introduction and Survey

5

reality to discrete values. The basic unit of charge is the magnitude of the charge on the electron,

|qe | = 4.803250(21) x 10" 10 esu

=

1.6021917(70)x

-19
10

coulomb

where the errors in the last two decimal places are shown in parentheses. The

charges on the proton and on all presently known particles or systems of

particles are integral multiples of this basic unit. The experimental accuracy with

which it is known that the multiples are exactly integers is phenomenal (better

than

1

part

in

10

20 ).

The

experiments

are

discussed

in

Section

11.9

where

the

question of the Lorentz invariance of charge is also treated.

The discreteness of electric charge does not need to be considered in most

A macroscopic applications. 1 microfarad capacitor at a potential of 150 volts,

A for example, has a total of 10 15 elementary charges on each electrode. few

A thousand electrons more or less would not be noticed.

current of 1

microampere corresponds to 6.2 x 10 12 elementary charges/second. There are, of

course, some delicate macroscopic or almost macroscopic experiments in which

the discreteness of charge enters. Millikan's famous oil drop experiment is one.
His droplets were typically 10"4 cm in radius and had a few or few tens of

elementary charges on them.

There is a lack of symmetry in the appearance of the source terms in Maxwell

equations (1.1). The first two equations have sources; the second two do not.

This reflects the experimental absence of magnetic charges and currents.

Actually, as is shown in Section 6.12, particles could have magnetic as well as

electric charge. If all particles in nature had the same ratio of magnetic to electric

charge, the fields and sources could be redefined in such a way that the usual

Maxwell equations (1.1) emerge. In this sense it is somewhat a matter of

convention to say that no magnetic charges or currents exist. Throughout most

of this book it is assumed that only electric charges and currents act in the

Maxwell equations, but some consequences of the existence of a particle with a

different magnetic to electric charge ratio, for example, a magnetic monopole,

are described in Chapter 6.

1.2 The Inverse Square Law or the Mass of the Photon
The distance dependence of the electrostatic law of force was shown quantitatively by Cavendish and Coulomb to be an inverse square law. Through Gauss's law and the divergence theorem (see Sections 1.3 and 1.4) this leads to the first of the Maxwell equations (1.1). The original experiments had an accuracy of only a few percent and, furthermore, were at a laboratory length scale. Experiments at higher precision and involving different regimes of size have been performed

6

Classical Electrodynamics

Sect. 1.2

over the years. It is now customary to quote the tests of the inverse square law in

one of two ways:

(a)

Assume

that

the

force

varies

as

2+£
l/r

and

quote

a

value

or

limit

for

e.

(b) Assume that the electrostatic potential has the "Yukawa" form (see

Section

12.9),

_1 _tir re

and

quote

a

value

or

limit

for

fx

or

1
/m" .

Since

m = lJL myC/h, where y is the assumed mass of the photon, the test of the

inverse square law is sometimes phrased in terms of an upper limit on tru,.

m Laboratory experiments usually give e and perhaps fx or 7 ; geomagnetic m experiments give jut or 7 .

The original experiment with concentric spheres by Cavendish* in 1772 gave

an upper limit on e of |e|<0.02. His apparatus is shown in Fig. 1.1. About 100

years later Maxwell performed a very similar experiment at Cambridget and set
an upper limit of |e|<5xl0~ 5 . Two other noteworthy laboratory experiments

based on Gauss's law are those of Plimpton and Lawton,$ which gave |e|<

A 2x

10~ 9 ,

and

the

recent

one

of

Williams,

Faller,

and

Hill.§

schematic drawing

of the apparatus of the latter experiment is shown in Fig. 1.2. Though not a static

experiment (v—4x 106 Hz), the basic idea is almost the same as Cavendish's. He

looked for a charge on the inner sphere after it had been brought into electrical

contact with the charged outer sphere and then disconnected; he found none.

Williams, Faller, and Hill looked for a voltage difference between two concentric

shells when the outer one was subjected to an alternating voltage of ±10 kV with

respect to ground. Their sensitivity was such that a voltage difference of less than

V 10" 12 could have been detected. Their null result, when interpreted by means

of

the

Proca

equations

(Section

12.9),

gives

a

limit

of

e

=

(2.7±3.1)x

10~ 16 .

Measurements of the earth's magnetic field, both on the surface and out from

the surface by satellite observation, permit the best limits to be set on e or
m equivalently the photon mass 7 . The geophysical and also the laboratory
observations are discussed in the reviews by Kobzarev and Okun' and by

Goldhaber and Nieto, listed at the end of this introduction. The surface

measurements of the earth's magnetic field give slightly the best value (see

Problem 12.14), namely,

m7 <4xl0-48 gm
or
/x-^10 10 cm

m For comparison, the electron mass is e = 9.1x 10"28 gm. The laboratory experi-

ment of Williams, Faller, and Hill can be interpreted as setting a limit

m-y<1.6x

-47
10

gm,

only

a

factor

of

4

poorer

than

the

geomagnetic

limit.

* H. Cavendish, Electrical Researches, ed. J. C. Maxwell, Cambridge University Press (1879), pp. 104-113.
t Ibid., see note 19.
$S. J. Plimpton and W. E. Lawton, Phys. Rev. 50, 1066 (1936).
§ E. R. Williams, J. E. Faller, and H. A. Hill, Phys. Rev. Letters 26, 721 (1971).

c
Fig. 1.1 Cavendish's apparatus for establishing the inverse square law of electrostatics.
Top, facsimile of Cavendish's own sketch; bottom, line drawing by a draughtsman. The
inner globe is 12.1 inches in diameter, the hollow pasteboard hemispheres slightly larger.
Both globe and hemispheres were covered with tinfoil "to make them the more perfect conductors of electricity." (Figures reproduced by permission of the Cambridge Univer-
sity Press.)
7

Fig. 1.2 Schematic diagram of the "Cavendish" experiment of Williams, Faller, and

A MHz Hill. The concentric icosahedrons are conducting shells. 4

voltage of 10 kV peak

is applied between shells 5 and 4. Shell 4 and its contiguous shells 2 and 3 are roughly 1.5

8

Sect. 1.2

Introduction and Survey

9

A rough limit on the photon mass can be set quite easily by noting the

existence of very low frequency modes in the earth-ionosphere resonant cavity

(Schumann resonances, discussed in Section 8.9). The double Einstein relation,

hv=

2 rriyC

,

suggests

that

the

photon

mass

must

satisfy

an

inequality,

m-,<hiVc 2 ,

where v is any electromagnetic resonant frequency. The lowest Schumann

m resonance has v —SHz. From this we calculate 7 <6x 10"47 gm, a very small

value only one order of magnitude above the best limit. While this argument has

crude validity, more careful consideration (see Section 12.9 and the references
m R given there) shows that the limit is roughly (R/H) — 10 times larger, —

H— 6400 km being the radius of the earth, and

60 km being the height of the

ionosphere.* In spite of this dilution factor the limit of 10~45 gm set by the mere

existence of Schumann resonances is quite respectable.

The laboratory and geophysical tests show that on length scales of order 1 to 109 cm, the inverse square law holds with extreme precision. At smaller

distances we must turn to less direct evidence often involving additional

assumptions. For example, Rutherford's historical analysis of the scattering of

alpha particles by thin foils substantiates the Coulomb law of force down to distances of the order of 10~ n cm provided the alpha particle and the nucleus

can be treated as classical point charges interacting statically and the charge

cloud of the electrons can be ignored. All of these assumptions can be, and have

been, tested, of course, but only within the framework of the validity of quantum

mechanics, linear superposition (see below), and other (very reasonable) as-

sumptions. At still smaller distances, relativistic quantum mechanics is neces-

sary, and strong interaction effects enter to obscure the questions as well as the

answers. Nevertheless, elastic scattering experiments with positive and negative
electrons at center of mass energies of up to 5 GeV have shown that quantum

electrodynamics (the relativistic theory of point electrons interacting with

We massless photons) holds to distances of the order of 10~ 15 cm.

conclude that

the photon mass can be taken to be zero (the inverse square force law holds)

over the whole classical range of distances and deep into the quantum domain as

well. The inverse square law is known to hold over at least 24 orders of

magnitude in the length scale!

* The basic point is that, to the extent that H/R is negligible, the ELF propagation

TEM is the same as in a parallel plate transmission line in the fundamental

mode. This

propagation is unaffected by a finite photon mass, except through changes in the static

capacitance and inductance per unit length. Explicit photon mass effects occur in order
2
(H/R)/ul .

meters in diameter and contain shell 1 inside. The voltage difference between shells 1 and 2 (if any) appears across the inductor indicated at about 8 o'clock in shell 1. The amplifier and optics system are necessary to extract the voltage information to the outside world. They are equivalent to Cavendish's system of strings that automatically opened the hinged hemispheres and brought up the pith balls to test for charge on the inner sphere.
(Figure reproduced with permission of the authors.)

10

Classical Electrodynamics

Sect. 1.3

1.3 Linear Superposition

The Maxwell equations in vacuum are linear in the fields E and B. This linearity
is exploited so often, for example, with hundreds of different telephone conversations on a single microwave link, that it is taken for granted. There are,
— of course, circumstances where nonlinear effects occur in magnetic materials,
in crystals responding to intense laser beams, even in the devices used to put
those telephone conversations on and off the microwave beam. But here we are concerned with fields in vacuum or the microscopic fields inside atoms and

nuclei.
What evidence do we have to support the idea of linear superposition? At the
macroscopic level, all sorts of experiments test linear superposition at the level
— of 0.1% accuracy groups of charges and currents produce electric and mag-
netic forces calculable by linear superposition, transformers perform as expected,
— standing waves are observed on transmission lines the reader can make a list.
In optics, slit systems show diffraction patterns; X-ray diffraction tells us about
crystal structure; white light is refracted by a prism into the colors of the rainbow and recombined into white light again. At the macroscopic and even at the
atomic level, linear superposition is remarkably valid. It is in the subatomic domain that departures from linear superposition can be
legitimately sought. As charged particles approach each other very closely, electric field strengths become enormous. If we think of a charged particle as a localized distribution of charge, we see that its electromagnetic energy grows larger and larger as the charge is localized more and more. It is natural, in order to avoid infinite self-energies of point particles, to speculate that some sort of saturation occurs, that fields strengths have some upper bound. Such classical nonlinear theories have been studied in the past. One well-known example is the theory of Born and Infeld.* The vacuum is given electric and magnetic

permeabilities,

H r e=

F i+ (B2 - E2)

(l4)

where b is a maximum field strength. Equation (1.4) is actually a simplified form
proposed earlier by Born alone. It suffices to illustrate the general idea. Fields
are obviously modified at short distances; all electromagnetic energies are finite.
But such theories suffer from arbitrariness in the manner of how the nonlinearity occurs and also from grave problems with a transition to a quantum theory. Furthermore, there is no evidence of this kind of classical nonlinearity. The quantum mechanics of many-electron atoms is described to high precision by normal quantum theory with the interactions between nucleus and electrons and
between electrons and electrons given by a linear superposition of pairwise potentials (or retarded relativistic interactions for fine effects). Field strengths of

* M. Born and L. Infeld, Proc. Roy. Soc. A144, 425 (1934). See M. Born, Atomic Physics, Blackie, London, Appendix VI, for an elementary discussion.

Sect. 1.3

Introduction and Survey

11

k4

Fig. 1.3 The scattering of light by light. Schematic diagram of the process by which photon-photon scattering occurs.
the order of 109-10 15 volts/cm exist at the orbits of electrons in atoms, while the electric field at the edge of a heavy nucleus is of the order of 10 19 volts/cm. Energy level differences in light atoms like helium, calculated on the basis of linear superposition of electromagnetic interactions, are in agreement with
experiment to accuracies that approach 1 part in 106 . And Coulomb energies of
heavy nuclei are consistent with linear superposition of electromagnetic effects. It is possible, of course, that for field strengths greater than 10 19 volts/cm
nonlinear effects could occur. One place to look for such effects is in superheavy nuclei (Z>110), both in the atomic energy levels and in the nuclear Coulomb energy.* At the present time there is no evidence for any classical nonlinear behavior of vacuum fields at short distances.
There is a quantum-mechanical nonlinearity of electromagnetic fields that arises because the uncertainty principle permits the momentary creation of an electron-positron pair by two photons and the subsequent disappearance of the pair with the emission of two different photons, as indicated schematically in Fig. 1.3. This process is called the scattering of light by light. t The two incident plane waves eikl '*~k°1< and e*2 ~ x i0i2t do not merely add coherently, as expected with linear superposition, but interact and (with small probability) transform into two different plane waves with wave vectors k3 and k4 . This nonlinear feature of quantum electrodynamics can be expressed, at least for slowly varying fields, in
* An investigation of the effect of a Born-Infeld type of nonlinearity on the atomic energy levels in superheavy elements has been made by J. Rafelski, W. Greiner, and L. P.
Fulcher, Nuovo Cimento 13B, 135 (1973).
t When two of the photons in Fig. 1.3 are virtual photons representing interaction to second order with a static nuclear Coulomb field, the process is known as Delbriick
scattering. See Section 15.8 of J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons, Addison-Wesley, Reading, Mass. (1955).

12

Classical Electrodynamics

Sect. 1.3

terms of electric and magnetic permeability tensors of the vacuum:

where

X E Di =

€ ik k ,

k

X Bi = j^ikHk
k

= + - + 7B B eik

5ik

g4ft

A-

4

7 [2(E 2

2 )fi» k

]+ t k

•••

-£ 7£ £ = + u,k

Sik

4J

eh
47
71?n c

[2(B 2

+ 2
)§ik

]+- i k

••

m Here e and are the charge and mass of the electron. These results were first

We obtained by Euler and Kockel in 1935.*

observe that in the classical limit

(h—>0), these nonlinear effects go to zero. Comparison with the classical

Born-Infeld expression (1.4) shows that for small nonlinearities, the quantum-

mechanical field strength

plays a role analogous to the Born-Infeld parameter b. Here r = e 2/mc 2 — 2.8 x 10~ 13 cm is the classical electron radius and e/r 2 = 1.8x 10 18 volts/cm is the electric field at the surface of such a classical electron. Two comments in passing:

(a) the eik and /mik in (1.5) are approximations that fail for field strengths

approaching bq or when the fields vary too rapidly in space or time (h/mc setting

the

critical

scale

of

length

and

2
fi/mc

of

time);

(b)

the

chance

numerical

coincidence of bq and e/2r 2 is suggestive, but probably not significant since bq

involves Planck's constant h.

In analogy with the polarization P = (D-E)/47T, we speak of the field-

dependent terms in (1.5) as vacuum polarization effects. In addition to the

scattering of light by light or Delbriick scattering, vacuum polarization causes

very small shifts in atomic energy levels. The dominant contribution involves a

virtual electron-positron pair, just as in Fig. 1.3, but with only two photon lines

instead of four. If the photons are real, the process contributes to the mass of the

photon and is decreed to vanish. If the photons are virtual, however, as in the

electromagnetic interaction between a nucleus and an orbiting electron, or

indeed for any externally applied field, the creation and annihilation of a virtual

electron-positron pair from time to time causes observable effects. The first

effect is a reduction in the observed charge of the nucleus from its value in the

absence of interaction. This renormalization of the bare charge can be under-

stood in simple electrostatic terms. The electron of the pair is attracted and the

positron repelled by the positive charge of the nucleus. This vacuum polarization

effect causes the nuclear charge to be screened and to appear less in magnitude

than before. Since charged particles are always surrounded by this cloud of

virtual electron-positron pairs, their observed charges must be interpreted as

*H. Euler and B. Kockel, Naturwiss. 23, 246 (1935).

Sect. 1.4

Introduction and Survey

13

their renormalized charges. Beyond the unobservable renormalization of charge, the vacuum polarization induces a charge density within distances of the order of h/2mc or less that causes the electrostatic potential energy between two charges to be greater in magnitude than the Coulomb potential energy. This causes a very small shift in atomic energy levels in the direction of increased binding. The lowest order added potential is proportional to aqext , where a = e 2/hc= 1/137 and qext is the charge producing the external field. It is thus linear in the external field and produces a small linear modification of the Maxwell equations. It is nonlinear in the sense that the strength of the effect depends on the fine structure constant times the external field and so involves the third power of charge in the added potential. Higher order effects, such as Fig. 1.3 with three of the photons corresponding to the third power of the external field, give totally nonlinear vacuum polarization effects.
In electronic atoms the vacuum polarization effects are a small part of the total radiative correction, but are still observable. In mu-mesic atoms, the effects are relatively larger because the atomic orbits are mainly inside the region where the potential is modified. Then vacuum polarization effects are important in their own right.
The final conclusion about linear superposition of fields in vacuum is that in the classical domain of sizes and attainable field strengths there is abundant evidence for the validity of linear superposition and no evidence against it. In the atomic and subatomic domain there are small quantum-mechanical nonlinear effects whose origins are in the coupling between charged particles and the electromagnetic field. They modify the interactions between charged particles and cause interactions between electromagnetic fields even if physical particles
are absent.

1.4 The Maxwell Equations in Macroscopic Media

So far we have considered electromagnetic fields and sources in vacuum. The

Maxwell equations (1.1) for the electric and magnetic fields E and B can be

thought of as equations giving the fields everywhere in space, provided all the

sources p and J are specified. For a small number of definite sources, determina-

tion of the fields is a tractable problem, but for macroscopic aggregates of matter

the solution of the equations is almost impossible. There are two aspects here.
One is that the number of individual sources, the charged particles in every atom

and nucleus, is prohibitively large. The other aspect is that for macroscopic

observations the detailed behavior of the fields, with their drastic variations in

space over atomic distances, is not relevant. What is relevant is the average of a

field or a source over a volume large compared to the volume occupied by a

We single atom or molecule.

call such averaged quantities the macroscopic fields

and macroscopic sources. It is shown in detail in Section 6.7 that the macroscopic

14

Classical Electrodynamics

Maxwell equations are

V-D = 4ttp '

Sect. 1.4

c dt c (1.6)
VxE+±^ =
c dt
VB =
where E and B are the averaged E and B of the microscopic or vacuum Maxwell
D equations (1.1). The two new field quantities and H, usually called the electric
displacement and magnetic field (B is then called the magnetic induction), have
components given by

V

, a*

/

Hu

=

B c<

-4t7(Mq

+-

•

•)

The quantities P, M, Q'^, and similar higher order objects, represent the macroscopically averaged electric dipole, magnetic dipole, and electric quadrupole, and higher moment densities of the material medium in the presence of applied fields. Similarly, the charge and current densities p and J are macroscopic averages of the "free" charge and current densities in the medium. The bound charges and currents appear in the equations via P, M, and Qap.
The macroscopic Maxwell equations (1.6) are a set of eight equations involving the components of the four fields E, B, D, and H. The four homogeneous
equations can be solved formally by expressing E and B in terms of the scalar
potential and the vector potential A, but the inhomogeneous equations cannot
D H be solved until the derived fields and are known in terms of E and B. These
connections, which are implicit in (1.7), are known as constitutive relations,

D = D[E, B] H = H[E, B]

(1.8)

In addition, for conducting media there is the generalized Ohm's law,

J = J[E,B]

(1.8')

The square brackets are intended to signify that the connections are not necessarily simple and may depend on past history (hysteresis), may be
nonlinear, etc.
In most materials the electric quadrupole and higher terms in (1.7) are
M completely negligible. Only the electric and magnetic polarizations P and are
significant. This does not mean, however, that the constitutive relations are then simple. There is a tremendous diversity in the electric and magnetic properties of

Sect. 1.4

Introduction and Survey

15

matter, especially in crystalline solids, with ferroelectric and ferromagnetic
M materials having nonzero P or in the absence of applied fields, as well as more
ordinary dielectric, diamagnetic, and paramagnetic substances. The study of these properties is one of the provinces of solid-state physics. In this book we touch only very briefly and superficially on some more elementary aspects. Solid-state books such as Kittel should be consulted for a more systematic and
extensive treatment of the electromagnetic properties of bulk matter.
In substances other than ferroelectrics or ferromagnets, for weak enough
fields the presence of an applied electric or magnetic field induces an electric or
We magnetic polarization proportional to the magnitude of the applied field.
then say that the response of the medium is linear and write the Cartesian
D H components of and in the form,*

The tensors €aP and iA'af} are called the electric permittivity or dielectric tensor and the inverse magnetic permeability tensor. They summarize the linear response of the medium and are dependent on the molecular and perhaps crystalline structure of the material, as well as bulk properties like density and temperature. For simple materials the linear response is often isotropic in space.
D Then €a3 and /ll^ are diagonal with all three elements equal, and = eE,
H=fx'B.

To be generally correct Eqs. (1.9) should be understood as holding for the Fourier
transforms in space and time of the field quantities. This is because the basic linear
D H connection between and E (or and B) can be nonlocal. Thus

D Z = a (x, Jdvjdt'e^x', t')E,(x-x', t-t')

(1.10)

where €qP (x', t') may be localized around x' = 0, = t' 0, but is nonvanishing for some range
D E away from the origin. If we introduce the Fourier transforms a (k, to), p (k, to), and
€a0 (k, a>) through
Oe-— /(k, a>) = Jd3 xjdt/(x,

Eq. (1.9) can be written in terms of the Fourier transforms as

D = £ a (k, to)

ea3 (k, <o)Ep (k, to)

(1. 11)

A H similar equation can be written B a (k, to) in terms of 3 (k, co). The permeability tensors
are therefore functions of frequency and wave vector in general. For visible light or

£ * Precedent would require writing Ba = M-apHp, but this reverses the natural roles H of B as the basic magnetic field and as the derived quantity. In Chapter 5 we revert to
the traditional usage.

16

Classical Electrodynamics

Sect. 1.4

electromagnetic radiation of longer wavelength it is often permissible to neglect the nonlocality in space. Then ea3 and u-^ are functions only of frequency. This is the situation discussed in Chapter 7 where a simplified treatment of the high frequency properties of matter is given and the consequences of causality explored. For conductors and
superconductors long-range effects can be important. For example, when the electronic collisional mean free path in a conductor becomes large compared to the skin depth, a spatially local form of Ohm's law is no longer adequate. Then the dependence on wave vector also enters. In the understanding of a number of properties of solids the concept of a dielectric constant as a function of wave vector and frequency is fruitful. Some exemplary references are given in the suggested reading at the end of this introduction.

For orientation we mention that at low frequencies (i^106 Hz) where all

charges, regardless of their inertia, respond to applied fields, solids have
— dielectric constants typically in the range of €<*<* 2-20, with larger values not
uncommon. Systems with permanent molecular dipole moments can have much

larger and temperature sensitive dielectric constants. Distilled water, for ex-
ample, has a static dielectric constant of e = 88 at 0°C and e = 56 at 100°C. At

optical frequencies only the electrons can respond significantly. The dielectric
~ constants are in the range, eaa 1.7-10, with eaa — 2-3 for most solids. Water has

€=1.77-1.80 over the visible range, essentially independent of temperature

from to 100°C.

The type of response of materials to an applied magnetic field depends on the

properties of the individual atoms or molecules and also on their interactions.

Diamagnetic substances consist of atoms or molecules with no net angular

momentum. The response to an applied magnetic field is the creation of

circulating atomic currents that produce a very small bulk magnetization

opposing the applied field. With the definition of /utLp in (1.9) and the form of

(1.7), this means /m«a >l. Bismuth, the most diamagnetic substance known, has

— (/Xaa— 1)

1.8x

10~ 4 .

Thus

diamagnetism

is

a

very

small

effect.

If

the

basic

atomic

unit of the material has a net angular momentum from unpaired electrons, the

substance is paramagnetic. The magnetic moment of the odd electron is aligned

parallel to the applied field. Hence jLtaa <l. Typical values are in the range (1 - |Wa«)— 10 -2-10~ 5 at room temperature, but decreasing at higher temperatures

because of the randomizing effect of thermal excitations.

Ferromagnetic materials are paramagnetic but, because of interactions be-

tween atoms, show drastically different behavior. Below the Curie temperature

(1040°K for Fe, 630°K for Ni), ferromagnetic substances show spontaneous

magnetization, that is, all the magnetic moments in a microscopically large

region called a domain are aligned. The application of an external field tends to

cause the domains to change and the moments in different domains to line up

together, leading to the saturation of the bulk magnetization. Removal of the

field leaves a considerable fraction of the moments still aligned, giving a

permanent

magnetization

that

can

be

as

large

as

B r

=

47rMr

^

10 4

gauss.

For data on the dielectric and magnetic properties of materials the reader can

Sect. 1.5

Introduction and Survey

17

consult some of the basic physics handbooks* from which he or she will be led to more specific and detailed compilations.
Materials that show a linear response to weak fields eventually show nonlinear behavior at high enough field strengths as the electronic or ionic oscillators are driven to large amplitudes. The linear relations (1.9) are modified to, for
example,

D X XeS a = eSEp +

E E + r (3 7

-••

(1.12)

For static fields the consequences are not particularly dramatic, but for time-
A varying fields it is another matter. large amplitude wave of two frequencies coi
and (i)2 generates waves in the medium with frequencies 0, 2a>i, 2o>2 , + o>i co2 , coi-o)2 , as well as the original co, and w2 . From cubic and higher nonlinear terms
an even richer spectrum of frequencies can be generated. With the development of lasers, nonlinear behavior of this sort has become a research area of its own, called nonlinear optics, and also a laboratory tool. At present, lasers are capable of generating light pulses with peak electric fields approaching 10 10 or even 10 11 volts/cm. The static electric field experienced by the electron in its orbit in a
hydrogen atom is e/a 2 — 5xl0 9 volts/cm. Such laser fields are thus seen to be
capable of driving atomic oscillators well into their nonlinear regime, capable
indeed of destroying the sample under study! References to some of the literature of this specialized field are given in the suggested reading at the end of this introduction. The reader of this book will have to be content with basically linear phenomena.

1.5 Boundary Conditions at Interfaces between Different Media

The Maxwell equations (1.6) are differential equations applying locally at each point in space-time (x, t). By means of the divergence theorem and Stokes's
V theorem, they can be cast in integral form. Let be a finite volume in space, S
the closed surface (or surfaces) bounding it, da an element of area on the surface, and n a unit normal to the surface at da pointing outward from the enclosed volume. Then the divergence theorem applied to the first and last equations of (1.6) yields the integral statements

(j> D-nda = 47r| pd 3 x

(1.13)

(j>B-nda =

(1.14)

* Handbook of Chemistry and Physics, ed. R. C. Weast, Chemical Rubber Publishing House, Cleveland, Ohio.
American Institute of Physics Handbook, ed. D. E. Gray, McGraw Hill, New York, 3rd
edition (1972), Sects. 5.d and 5.f.

18

Classical Electrodynamics

Sect. 1.5

D The first relation is just Gauss's law that the total flux of out through the
surface is proportional to the charge contained inside. The second is the
magnetic analog, with no net flux of B through a closed surface because of
the nonexistence of magnetic charges.
C Similarly, let be a closed contour in space, S' an open surface spanning the
contour, d\ a line element on the contour, da an element of area on S', and n' a unit normal at da pointing in the direction given by the right-hand rule from the sense of integration around the contour. Then applying Stokes's theorem to the middle two equations in (1.6) gives the integral statements

Equation (1.15) is the Ampere-Maxwell law of magnetic fields and (1.16) is
Faraday's law of electromagnetic induction. These familiar integral equivalents of the Maxwell equations can be used
directly to deduce the relationship of various normal and tangential components of the fields on either side of a surface between different media, perhaps with a
surface charge or current density at the interface. An appropriate geometrical arrangement is shown in Fig. 1.4. An infinitesimal Gaussian pillbox straddles the
boundary surface between two media with different electromagnetic properties

Fig. 1,4 Schematic diagram of boundary surface (heavy line) between different media.
The boundary region is assumed to carry idealized surface charge and current densities cr
V and K. The volume is a small pillbox, half in one medium and half in the other, with the normal n to its top pointing from medium 1 into medium 2. The rectangular contour C is
partly in one medium and partly in the other and is oriented with its plane perpendicular
to the surface so that its normal t is tangent to the surface.

Sect. 1.5

Introduction and Survey

19

C Similarly, the infinitesimal contour has its long arms on either side of the

boundary and is oriented so that the normal to its spanning surface is tangent to

We the interface.

first apply the integral statements (1.13) and (1.14) to the

volume of the pillbox. In the limit of a very shallow pillbox, the side surface does

not contribute to the integrals on the left in (1.13) and (1.14). Only the top and

bottom contribute. If the top and bottom are parallel, tangent to the surface, and

of area Aa, then the left-hand integral in (1.13) is

(j> D-nda = (D2 -Di)-nAa

and similarly for (1.14). If the charge density p is singular at the interface so as to produce an idealized surface charge density cr, then the integral on the right in
(1.13) is

47r| pd3 x = 47rcrAa

D Thus the normal components of and B on either side of the boundary surface

are related according to

(02-00-0 = 4770-

(1.17)

(B2 -Bi)-n =

(1.18)

In words, we say that the normal component of B is continuous and the
D discontinuity of the normal component of at any point is equal to 477 times the
surface charge density at that point.
In an analogous manner the infinitesimal Stokesian loop can be used to
determine the discontinuities of the tangential components of E and H. If the
C short arms of the contour in Fig. 1.4 are of negligible length and the long arms
are each parallel to the surface and of length A/, then the left-hand integral of
(1.16) is
E-dI=(tXn)-(E2 -Ei)AJ
<J>

and similarly for the left-hand side of (1.15). The right-hand side of (1.16) vanishes because dB/dt is finite at the surface and the area of the loop is zero as the length of the short sides goes to zero. The right-hand side of (1.15) does not
K vanish, however, if there is an idealized surface current density flowing exactly
on the boundary surface. In such circumstances the integral on the right of (1.15)

im. ^ r \^ s+

tda= K . tAl

Js lC

C dt J

C

The second term in the integral vanishes by the same argument that was just
H given. The tangential components of E and on either side of the boundary are

20

Classical Electrodynamics

\ \

Sect. 1.5

Fig. 1.5 Moving boundary between two media. The pillbox and loop are as in Fig. 1.4 and are stationary in the laboratory. The dashed lines show the interface a moment before and after the instant shown.

therefore related by

nx(E2 -E!) =

(1.19)

nx(H2 -Hi) =

(1.20)

K In Eq. (1.20) it is understood that the surface current has only components
parallel to the surface at every point. The tangential component of E across an
H interface is continuous, while the tangential component of is discontinuous by
an amount whose magnitude is equal to 47r/c times the magnitude of the surface current density and whose direction is parallel to Kxn.
The discontinuity equations (I.17)-(1.20) are useful in solving the Maxwell
equations in different regions and then connecting the solutions to obtain the
fields throughout all space.

The discontinuity formulas presented above hold in the common circumstance that the interface between the two media is fixed as a function of time. In some applications it may be useful to have the discontinuities for a moving boundary.* The results for a boundary surface moving with velocity v = cJJ can be obtained in essentially the same way as previously, provided a little care is taken. The moving boundary surface between the two
media is shown schematically in Fig. 1.5, along with the infinitesimal Gaussian pillbox and Stokesian loop. The pillbox and loop are fixed in the laboratory. The boundary surface sweeps past them with velocity v. If we now consider the derivation of the discontinuity
D formulas (1.17) and (1.18) for and B, we see that the same arguments starting from
*P. D. Noerdlinger, Am. J. Physics 39, 191 (1971).

Sect. 1.5

Introduction and Survey

21

(1.13) and (1.14) are valid without change, provided cr is interpreted as the surface charge
density on the moving surface as observed in the laboratory. Therefore the discontinuity
D formulas for and B, (1.17) and (1.18), hold without modification for a moving interface. H The discontinuity formulas (1.19) and (1.20) for E and are modified, however. This
comes about because the time derivative terms on the right-hand sides of (1.15) and (1.16)
C no longer vanish. The sweeping of the interface past the stationary loop gives a
contribution. To determine its value, consider the surface integral of the time derivative
of D/c over the open surface identical in shape to C, but moving with the interface at
C velocity v and instantaneously coincident with in Fig. 1.5. The integral is

I = \- ^Wt),t)-tda c
We have indicated the implicit time dependence of the coordinate x to emphasize that the
integration is over a moving surface. In the limit that the area of the open surface vanishes
C as the short arms of the rectangular loop become vanishingly small, the integral I
vanishes. (From the viewpoint of special relativity, an observer in an inertial frame moving with velocity v sees the interface at rest, and observes Lorentz-transformed fields that are not singular at the interface.) The integral I can, however, be related to the integral appearing in (1.15) through the convective derivative expansion:

= I=J^(x(0, 0-tda

= j^'tda+J[(p.V)D].tda

Using a vector identity the second term can be transformed and the required integral becomes
±25 . t da = J[Vx(pxD)-pV • D] • t da
J

The first term on the right can be transformed by Stokes's theorem into a loop integral

We and the second can be expressed in terms of the charge density p.

therefore have as

C the application of (1.15) to the loop in Fig. 1.5 the expression

[H-pxD]-dI = ^j"[J-pv]-tda
<j>

By the same steps as above (1.19) and (1.20), we obtain from this relation the discontinuity
formula,

-H -D ^ t • {nx[H2

-px(D

1

2

= 1 )]}

1 (K-crv) • t

where all quantities are evaluated in the laboratory frame. Some vector manipulation and
use of (1.17) leads to

-H K t • [nx(H2

)+n
x

•

P(D2 -D0]=

•t

(1.21)

A completely similar derivation from (1.16) yields

t

•

[nx(E2 -E0-n

•

-B P(B 2

1 )]

=

(1.22)

as the discontinuity formula for the tangential components of E (and B). Equations (1.21)
and (1.22) represent the generalizations of (1.19) and (1.20) to the circumstance of a moving interface between two media.

2.2

Classical Electrodynamics

Sect. 1.6

D— H— In the simplified situation where

E and

B in both media (or these hold in one

medium and the other medium is an excellent conductor with all fields essentially zero

inside), the relation involving the surface current simplifies considerably. Equation (1.22)

can be written (without approximation) as

-E (E2 = 1 )tan -(n • P)nx(B2 -B,)

D^E Then with H-^B and

in (1.21), Eq. (1.23) can be substituted to give

(1.23)

[l-(n • p) 2]nx(B2 -B 1 )=^I K

(1.24)

The motion of the interface between the media introduces only an overall multiplicative

factor

into

(1.20),

a

correction

of

relative

order

22
v /c .

1.6 Some Remarks on Idealizations in Electromagnetism
In the previous section we made use of the idea of surface distributions of charge
and current. These are obviously mathematical idealizations that do not exist in the physical world. There are other abstractions that occur throughout elec-
tromagnetism. In electrostatics, for example, we speak of holding objects at a fixed potential with respect to some zero of potential usually called "ground." The relation of such idealizations to the real world is perhaps worthy of a little discussion, even though to the experienced hand most will seem obvious.
First we consider the question of maintaining some conducting object at a fixed electrostatic potential with respect to some reference value. Implicit is the idea that the means does not significantly disturb the desired configuration of charges and fields. To maintain an object at fixed potential it is necessary, at least
from time to time, to have a conducting path or its equivalent from the object to a source of charge far away ("at infinity") so that as other charged or uncharged objects are brought in the vicinity charge can flow to or from the object, always maintaining its potential at the desired value. Although more sophisticated
means are possible, metallic wires are commonly used to make the conducting path. Intuitively we expect small wires to be less perturbing than large ones. The
reason is as follows. "Since the quantity of electricity on any given portion of a wire at a given potential diminishes indefinitely when the diameter of the wire is indefinitely diminished, the distribution of electricity on bodies of considerable dimensions will not be sensibly affected by the introduction of very fine metallic wires into the field, such as are used to form electrical connexions between these bodies and the earth, an electrical machine, or an electrometer."* The electric field in the immediate neighborhood of the thin wire is very large, of course. However, at distances away of the order of the size of the "bodies of
considerable dimensions" the effects can be made small. An important historical
A * J. C. Maxwell, Treatise on Electricity and Magnetism, Dover, New York, 1954
reprint of the 3rd edition (1891), Vol. 1, p. 96.

Sect. 1.6

Introduction and Survey

23

illustration of Maxwell's words is given by the work of Henry Cavendish 200

years ago. By experiments done in a converted stable of his father's house, using

Ley den jars as his sources of charge, thin wires as conductors, and suspending

the objects in the room, Cavendish measured the amounts of charge on

cylinders, discs, etc., held at fixed potential and compared them to the charge on

a sphere (the same sphere shown in Fig. 1.1) at the same potential. His values of

capacitance, so measured, are accurate to a few per cent. For example, he found

the ratio of the capacitance of a sphere to that of a thin circular disc of the same

radius was 1.57. The theoretical value is it/2.

There is a practical limit to the use of finer and finer wires. The charge per unit

length decreases only logarithmically (as the reciprocal of In (d/a), where a is the

mean radius of the wire and d is a typical distance of the wire from some

conducting surface). To minimize the perturbation of the system below some

level, it is necessary to resort to other means to maintain potentials, comparison

methods using beams of charged particles intermittently, for example.

When a conducting object is said to be grounded, it is assumed to be connected

by a very fine conducting filament to a remote reservoir of charge that serves as

the common zero of potential. Objects held at fixed potentials are similarly

connected to one side of a voltage source, such as a battery, the other side of

which is connected to the common "ground." Then, when initially electrified

objects are moved relative to one another in such a way that their distributions

of electricity are altered, but their potentials remain fixed, the appropriate

amounts of charge flow from or to the remote reservoir, assumed to have an

inexhaustible supply. The idea of grounding something is a well-defined concept

in electrostatics where time is not a factor, but for oscillating fields the finite

speed of propagation blurs the concept. In other words, stray inductive and

capacitive effects can enter significantly. Great care is then necessary to ensure a

"good ground."

Another idealization in macroscopic electromagnetism is the idea of a surface

charge density or a surface current density. The physical reality is that the charge

or current is confined to the immediate neighborhood of the surface. If this

region has thickness small compared to the length scale of interest, we may

approximate the reality by the idealization of a region of infinitesimal thickness
and speak of a surface distribution. Two different limits need to be distinguished.

One is the limit in which the "surface" distribution is confined to a region near

An the surface that is macroscopically small, but microscopically large.

example is

the penetration of time-varying fields into a very good, but not perfect,

conductor, described in Section 8.1. It is found that the fields are confined to a

thickness 5, called the skin depth, and that for high enough frequencies and good

enough conductivities 8 can be macroscopically very small. It is then appropriate

to integrate the current density J over the direction perpendicular to the surface
K to obtain an effective surface current density eff .
The other limit is truly microscopic and is set by quantum-mechanical effects in

24

Classical Electrodynamics

Sect. 1.6

Fig. 1.6 Distribution of excess charge at the surface of a conductor and of the normal
component of the electric field. The ions of the solid are confined to x<0 and are
approximated by a constant continuous charge distribution through which the electrons move. The bulk of the excess charge is confined to within ±2 angstroms of the "surface."

the atomic structure of materials. Consider, for instance, the distribution of

excess charge of a conducting body in electrostatics. It is well known that this

We charge lies entirely on the surface of a conductor.

then speak of a surface

charge density cr. There is no electric field inside the conductor, but there is, in

accord with (1.17), a normal component of electric field just outside the surface.

At the microscopic level the charge is not exactly at the surface and the field does

not change discontinuously. The most elementary considerations would indicate

that the transition region is a few atomic diameters in extent. The ions in a metal

can be thought of as relatively immobile and localized to 1 angstrom or better;

the lighter electrons are less constrained. The results of model calculations* are

shown in Fig. 1.6. They come from a solution of the quantum-mechanical

many-electron problem in which the ions of the conductor are approximated by
a continuous constant charge density for x<0. The electron density = (rs 5) is roughly appropriate to copper and the heavier alkali metals. The excess

* N. D. Lang and W. Kohn, Phys. Rev. Bl, 4555 (1970); B3, 1215 (1971); V. E. Kenner, R. E. Allen, and W. M. Saslow, Phys. Letters 38A, 255 (1972).

Refs. I

Introduction and Survey

25

electronic charge is seen to be confined to a region within ±2 angstroms of the "surface" of the ionic distribution. The electric field rises smoothly over this
region to its value of 4ircr "outside" the conductor. For macroscopic situations
where 10~ 7 cm is a negligible distance, we can idealize the charge density and E electric field behavior as p(x) = cr8(x) and = n (x) 47rcr0(x), corresponding to a truly surface density and a step-function jump of the field.
We see that the theoretical treatment of classical electromagnetism involves
several idealizations, some of them technical and some physical. The subject of
electrostatics, discussed in the first chapters of the book, developed as an experimental science of macroscopic electrical phenomena, as did virtually all
other aspects of electromagnetism. The extension of these macroscopic laws, even for charges and currents in vacuum, to the microscopic domain was for the most part an unjustified extrapolation. Earlier in this introduction we have discussed some of the limits to this extrapolation. The point to be made here is the following. With hindsight we know that many aspects of the laws of classical electromagnetism apply well into the atomic domain provided the sources are treated quantum mechanically, that the averaging of electromagnetic quantities over volumes containing large numbers of molecules so smooths the rapid
fluctuations that static applied fields induce static average responses in matter,
that excess charge is on the surface of a conductor in a macroscopic sense. Thus Coulomb's and Ampere's macroscopic observations and our mathematical abstractions from them have a wider applicability than might be supposed by a supercautious physicist. The absence for air of significant electric or magnetic
susceptibility certainly simplified matters!

REFERENCES AND SUGGESTED READING
The history of electricity and magnetism is in large measure the history of science itself.
We have already cited
Whittaker's two volumes, the first covering the period up to 1900, as well as the shorter account emphasizing
optics in
Born and Wolf. Another readable account, with perceptive discussion of the original experiments, is
N. Feather, Electricity and Matter, University Press, Edinburgh (1968). The experimental tests of the inverse square nature of Coulomb's law or, in modern
language, the mass of the photon, are reviewed by
I. Yu. Kobzarev and L. B. Okun', Uspekhi Fiz. Nauk 95, 131 (1968) [transl., Sov.
Phys. Uspekhi 11, 338 (1968).] and
A. S. Goldhaber and M. M. Nieto, Rev. Mod. Phys. 43, 277 (1971). Suggested reading on the topic of the macroscopic Maxwell equations and their
derivation from the microscopic equations can be found at the end of Chapter 6. The basic physics of dielectrics, ferroelectrics, and magnetic materials can be found in

26

Classical Electrodynamics

Refs. I

numerous books on solid state physics, for example, Beam,
Kittel,
Wert and Thomson, Wooten. The first of these is aimed at electrical engineers and stresses practical topics like semiconductors. The last one is mainly on optical properties. The need for spatial nonlocality in treating the surface impedance of metals (the anomalous skin effect) is discussed in several places by A. B. Pippard, Advances in Electronics and Electron Physics, Vol. VI, ed. L.
Marton, Academic, New York (1954), pp. 1-45; Reports on Progress in Physics,
Vol. XXIII, pp. 176-266 (1960); The Dynamics of Conduction Electrons,
Gordon and Breach, New York (1965).
The concept of a wave- vector and frequency-dependent dielectric constant e(k, co) is developed by Kittel, Advanced Topic D.
D. Pines, Elementary Excitations in Solids, W. A. Benjamin, New York (1963),
Chapters 3 and 4.
F. Stern, Solid State Physics, Vol. 15, eds. F. Seitz and D. Turnbull, Academic, New
York, pp. 299-408. The rapidly growing field of nonlinear optics is beginning to have a book literature of its
own, but much of it is still in the research or review journals or in summer school
proceedings. An introduction can be found in
J. A. Giordmaine, Physics Today 22 (1) 38 (1969),
N. Bloembergen, Am. J. Phys. 35, 989 (1967), G. C. Baldwin, Introduction to Nonlinear Optics, Plenum, New York (1969). Somewhat more advanced discussions arc available in S. A. Akhmanov and R. V. Khokhlov, Usp. Fiz. Nauk. 88, 439 (1966); 95, 231
(1968), [trans., Sov. Phys. Uspekhi, 9, 210 (1966); 11, 394 (1968)],
N. Bloembergen, Nonlinear Optics, W. A. Benjamin, New York (1965),
Quantum Optics, Proc. Int. School of Physics "Enrico Fermi," Varenna, Course
XLII, 1967, ed. R. J. Glauber, Academic, New York (1969), articles by J.
Ducuing, Y. R. Shen, J. A. Giordmaine, P. S. Pershan, Progress in Optics, Vol. V, ed. E. Wolf, North-Holland, Amsterdam
(1966), pp. 83-144.

1
Introduction to
Electrostatics

We — begin our discussion of electrodynamics with the subject of electrostatics
phenomena involving time-independent distributions of charge and fields. For

most readers this material is in the nature of a review. In this chapter especially

We we do not elaborate significantly.

introduce concepts and definitions that are

important for later discussion and present some essential mathematical ap-

paratus. In subsequent chapters the mathematical techniques are developed and

applied.
One point of physics should be mentioned. Historically, electrostatics developed as a science of macroscopic phenomena. As indicated at the end of the

Introduction, such idealizations as point charges or electric fields at a point must

be viewed as mathematical constructs that permit a description of the
phenomena at the macroscopic level, but that may fail to have meaning

microscopically.

1.1 Coulomb's Law
All of electrostatics stems from the quantitative statement of Coulomb's law concerning the force acting between charged bodies at rest with respect to each other. Coulomb, in an impressive series of experiments, showed experimentally that the force between two small charged bodies separated in air a distance large compared to their dimensions
(1) varied directly as the magnitude of each charge, (2) varied inversely as the square of the distance between them, (3) was directed along the line joining the charges, (4) was attractive if the bodies were oppositely charged and repulsive if the
bodies had the same type of charge. Furthermore it was shown experimentally that the total force produced on one small charged body by a number of the other small charged bodies placed
27

28

Classical Electrodynamics

Sect. 1.2

around it was the vector sum of the individual two-body forces of Coulomb.

Strictly speaking, Coulomb's conclusions apply to charges in vacuum or in media

We of negligible susceptibility.

defer consideration of charges in dielectrics to

Chapter 4.

1.2 Electric Field

Although the thing that eventually gets measured is a force, it is useful to introduce a concept one step removed from the forces, the concept of an electric field due to some array of charged bodies. At the moment, the electric field can be defined as the force per unit charge acting at a given point. It is a vector
function of position, denoted by E. One must be careful in its definition,
however. It is not necessarily the force that one would observe by placing one unit of charge on a pith ball and placing it in position. The reason is that one unit
of charge may be so large that its presence alters appreciably the field
configuration of the array. Consequently one must use a limiting process whereby the ratio of the force on the small test body to the charge on it is measured for smaller and smaller amounts of charge.* Experimentally, this ratio and the direction of the force will become constant as the amount of test charge
is made smaller and smaller. These limiting values of magnitude and direction
define the magnitude and direction of the electric field E at the point in question. In symbols we may write

F=qE

(1.1)

E where F is the force, the electric field, and q the charge. In this equation it is
assumed that the charge q is located at a point, and the force and the electric field are evaulated at that point.
Coulomb's law can be written down similarly. If F is the force on a point
charge qi, located at Xi, due to another point charge q2 , located at x2 , then Coulomb's law is

F=kq!q2

(1.2)

Note that qi and q2 are algebraic quantities which can be positive or negative. The constant of proportionality k depends on the system of units used.
The electric field at the point x due to a point charge qi at the point Xi can be
obtained directly:

E(x) = kq!

(1.3)

* The discreteness of electric charge (see Section 1.1) means that this mathematical limit is impossible to realize physically. This is an example of a mathematical
idealization in macroscopic electrostatics.

Sect. 1.2

Introduction to Electrostatics

2S

Fig. 1.1

as indicated in Fig. 1.1. The constant k is determined by the unit of charge

chosen. In electrostatic units (esu), unit charge is chosen as that charge which

exerts a force of one dyne on an equal charge located one centimeter away.

Thus, with cgs units, k = 1 and the unit of charge is called the "stat-coulomb." In

MKSA the

system, k = (4ir€ y\ where e (= 8.854 x 10~ 12 farad/meter) is the

We permittivity of free space.

will use esu.*

The experimentally observed linear superposition of forces due to many

charges means that we may write the electric field at x due to a system of point

charges qh located at x<, i= 1, 2, . . . , n, as the vector sum:

If the charges are so small and so numerous that they can be described by a charge density p(x') [if Aq is the charge in a small volume Ax Ay Az at the point x', then Aq = p(x') Ax Ay Az], the sum is replaced by an integral:

E(x) = Jp(x')^^<f*'

(L5)

where d 3 x'=dx' dy' dz' is a three-dimensional volume element at x'.
At this point it is worth while to introduce the Dirac delta function. In one dimension, the delta function, written S(x-a), is a mathematically improper function having the
properties:
(1) 8(x-a) = for x*a, and (2) j" 8(x-a) dx = 1 if the region of integration includes x = a, and is zero otherwise. The delta function can be given, an intuitive, but nonrigorous, meaning as the limit of a
peaked curve such as a Gaussian which becomes narrower and narrower, but higher and higher, in such a way that the area under the curve is always constant. L. Schwartz's theory of distributions is a comprehensive rigorous mathematical approach to delta functions and their manipulations.!
From the definitions above it is evident that, for an arbitrary function /(x), (3) J/(x)8(x-a)dx = /(a). The integral of f(x) times the derivative of a delta function is simply understood if the delta function is thought of as a well-behaved, but sharply peaked, function. Thus the

* The question of units is discussed in detail in the Appendix.
A t useful, rigorous account of the Dirac delta function is given by Lighthill. See
also Dennery and Krzywicki, Sect. III. 13. (Full references for items cited in the text or
footnotes by italicized author only will be found in the Bibliography.)

30

Classical Electrodynamics

Sect. 1.3

definition is
(4) U(x)8'(x-a)dx = -f(a)
where a prime denotes differentiation with respect to the argument. If the delta function has as argument a function /(x) of the independent variable x, it
can be transformed according to the rule,

I r^ = (5) 8(/(x)) — w 1

rS(x-x,)
(x

dx

|

I

where f(x) is assumed to have only simple zeros, located at = x X;. In more than one dimension, we merely take products of delta functions in each
dimension. In three dimensions, for example, with Cartesian coordinates,

-X -X -X (6)

S(x-X) = S(x 1

1 ) 8(x2

2) 8(x3

3)

is a function which vanishes everywhere except at x = X, and is such that

(7)( v

8(x-X)^x

=

{;

V X A if contains x = V X A if does not not contain x =

Note that a delta function has the dimensions of an inverse volume in whatever number of
dimensions the space has.
A discrete set of point charges can be described with a charge density by means of delta
functions. For example,

Iq p(x) =

S(x-x

i

i)

i=l

(1.6)

represents

a

distribution

of

n

point

charges

qh

located

at

the

points

x. ;

Substitution

of

this

charge density (1.6) into (1.5) and integration, using the properties of the delta function,

yields the discrete sum (1.4).

1.3 Gauss's Law

The integral (1.5) is not the most suitable form for the evaluation of electric

fields. There is another integral result, called Gauss's law, which is sometimes
more useful and which furthermore leads to a differential equation for E(x). To

obtain Gauss's law we first consider a point charge q and a closed surface S, as shown in Fig. 1.2. Let r be the distance from the charge to a point on the surface,

n be the outwardly directed unit normal to the surface at that point, da be an
element of surface area. If the electric field E at the point on the surface due to

the charge q makes an angle 6 with the unit normal, then the normal component
of E times the area element is:

E * n da = q °°2 ^ da

(1.7)

Since E is directed along the line from the surface element to the charge q, cos 6 da=r2 dft, where dCl is the element of solid angle subtended by da at the

position of the charge. Therefore
En da = qdQ,

(1.8)

32

Classical Electrodynamics

Sect. 1.4

If we now integrate the normal component of E over the whole surface, it is easy

to see that

E <j>

• n da=l^ 7TC*

Js

lO

if q lies inside S if q lies outside S

. ,
^ ''

This result is Gauss's law for a single point charge. For a discrete set of charges, it is immediately apparent that

(j> E-nda = 47rXqi

(1.10)

where the sum is over only those charges inside the surface S. For a continuous
charge density p(x), Gauss's law becomes:

E = <j>

• n da

J"
47r

p(x) d 3 x

(1.11)

where V is the volume enclosed by S.
Equation (1.11) is one of the basic equations of electrostatics. Note that it depends upon
(1) the inverse square law for the force between charges, (2) the central nature of the force,
(3) the linear superposition of the effects of different charges.
Clearly, then, Gauss's law holds for Newtonian gravitational force fields, with matter density replacing charge density.
It is interesting to note that, even before the experiments of Cavendish and Coulomb, Priestley, taking up an observation of Franklin that charge seemed to reside on the outside, but not the inside, of a metal cup, reasoned by analogy with Newton's law of universal gravitation that the electrostatic force must obey an inverse square law with distance. The present status of the inverse square law
is discussed in Section 1.2.

1.4 Differential Form of Gauss's Law

Gauss's law can be thought of as being an integral formulation of the law of

We electrostatics.

can obtain a differential form (i.e., a differential equation) by

using the divergence theorem. The divergence theorem states that for any
V well-behaved vector field A(x) defined within a volume surrounded by the

closed surface S the relation

A-nda^J

A V •

d3x

<j>

A holds between the volume integral of the divergence of and the surface
integral of the outwardly directed normal component of A. The equation in fact
can be used as the definition of the divergence (see Stratton, p. 4).

Sect. 1.5

Introduction to Electrostatics

33

To apply the divergence theorem we consider the integral relation expressed
in Gauss's theorem:
£ E • n da = 47rj p(x) d3 x

Now the divergence theorem allows us to write this as:

(V-E-47rp)d3 x =
J

(1.12)

We for an arbitrary volume V.

can, in the usual way, put the integrand equal to

zero to obtain

V-E=4ttp

(1.13)

which is the differential form of Gauss's law of electrostatics. This equation can itself be used to solve problems in electrostatics. However, it is often simpler to deal with scalar rather than vector functions of position, and then to derive the vector quantities at the end if necessary (see below).

1.5 Another Equation of Electrostatics and the Scalar Potential

The single equation (1.13) is not enough to specify completely the three components of the electric field E(x). Perhaps some readers know that a vector field can be specified almost* completely if its divergence and curl are given
everywhere in space. Thus we look for an equation specifying curl E as a
function of position. Such an equation, namely,

VxE =

(1.14)

follows directly from our generalized Coulomb's law (1.5):

E(x) = Jp(x')j^dV

The vector factor in the integrand, viewed as a function of x, is the negative
gradient of the scalar l/|x-x'|:

Since the gradient operation involves x, but not the integration variable x', it can
be taken outside the integral sign. Then the field can be written

E(x)=-v[j£^jdV

(1.15)

Up *

to the gradient of a scalar function that satisfies the Laplace equation. See

Section 1.9 on uniqueness.

34

Classical Electrodynamics

Sect. 1.5

Since the curl of the gradient of any well-behaved scalar function of position

vanishes (VxVi// = 0, for all

(1.14) follows immediately from (1.15).

VxE Note that

= depends on the central nature of the force between

charges, and on the fact that the force is a function of relative distances only, but

does not depend on the inverse square nature.

In (1.15) the electric field (a vector) is derived from a scalar by the gradient

operation. Since one function of position is easier to deal with than three, it is

worth while concentrating on the scalar function and giving it a name. Conse-

quently we define the scalar potential <£(x) by the equation:

E = -VcJ>

(1.16)

Then (1.15) shows that the scalar potential is given in terms of the charge density by

*(x) = lbS| dV

(L17)

where the integration is over all charges in the universe, and <E> is arbitrary only

to the extent that a constant can be added to the right side of (1.17).

The scalar potential has a physical interpretation when we consider the work

done on a test charge q in transporting it from one point (A) to another point (B) in the presence of an electric field E(x), as shown in Fig. 1.3. The force acting on

the charge at any point is

F=qE

A so that the work done in moving the charge from

B to

is

B
W=-j F-dl=-qj" E-dl

(1.18)

The minus sign appears because we are calculating the work done on the charge against the action of the field. With definition (1.16) the work can be written

W= q^V® = = - *

• d\ q

d<D

q(<D B

A <*> )

J"

(1.19)

which shows that q<& can be interpreted as the potential energy of the test charge
in the electrostatic field.

Fig. 1.3

Sect. 1.6

Introduction to Electrostatics

35

From (1.18) and (1.19) it can be seen that the line integral of the electric field
between two points is independent of the path and is the negative of the potential difference between the points:

B
£ E-dl=-(4>B -4>A)

(1.20)

This follows directly, of course, from definition (1.16). If the path is closed, the
line integral is zero,

E-dl=0

(1.21)

a result that can also be obtained directly from Coulomb's law. Then application of Stokes's theorem [if A(x) is a well-behaved vector field, S is an arbitrary open
C surface, and is the closed curve bounding S,

A-dl=J (VxA)-nda
<j>
C where d\ is a line element of C, n is the normal to S, and the path is traversed Vx in a right-hand screw sense relative to n] leads immediately back to E = 0.

1.6 Surface Distributions of Charges and Dipoles and Discontinuities in the Electric Field and Potential
One of the common problems in electrostatics is the determination of electric
field or potential due to a given surface distribution of charges. Gauss's law (1.11) allows us to write down a partial result directly. If a surface S, with a unit normal n directed from side 1 to side 2 of the surface, has a surface-charge density of o-(x) (measured in statcoulombs per square centimeter) and electric
fields Ei and E2 on either side of the surface, as shown in Fig. 1.4, then Gauss's

Fig. 1.4 Discontinuity in the normal component of electric field across a surface layer of
charge.

36

Classical Electrodynamics

Sect. 1.6

law tells us immediately that

(E2 -Ei) -n = 47ro-

(1.22)

This does not determine Ei and E 2 unless there are no other sources of field and the geometry and form a are especially simple. All that (1.22) says is that there is
a discontinuity of Attct in the normal component of electric field in crossing a surface with a surface-charge density o\ the crossing being made in the direction
of n.
The tangential component of electric field can be shown to be continuous
across a boundary surface by using (1.21) for the line integral of E around a
closed path. It is only necessary to take a rectangular path with negligible ends and one side on either side of the boundary.
An expression for the potential (and hence the field, by differentiation) at any
point in space (not just at the surface) can be obtained from (1.17) by replacing
p d3 x by a da :

(1.23)

For volume or surface distributions of charge the potential is everywhere continuous, even within the charge distribution. This can be shown from (1.23)
or from the fact that E is bounded, even though discontinuous across a surface
distribution of charge. With point or line charges, or dipole layers, the potential is no longer continuous, as will be seen immediately.
Another problem of interest is the potential due to a dipole-layer distribution
A on a surface S. dipole layer can be imagined as being formed by letting the
surface S have a surface-charge density o-(x) on it, and another surface S', lying close to S, have an equal and opposite surface-charge density on it at neighboring points, as shown in Fig. 1.5. The dipole-layer distribution of strength D(x) is formed by letting S' approach infinitesimally close to S while the surface- charge density cr(x) becomes infinite in such a manner that the product of cr(x) and the local separation d(x) of S and S' approaches the limit D(x):
lim o-(x) d(x) = D(x)
d(x)->-0
S

s
S'
Fig. 1.5 Limiting process involved in creating a dipole layer.

Sect. 1.6

Introduction to Electrostatics

37

Fig. 1.6 Dipole- layer geometry.
The direction of the dipole moment of the layer is normal to the surface S and in
the direction going from negative to positive charge.
To find the potential due to a dipole layer we can consider a single dipole and then superpose a surface density of them, or we can obtain the same result by
performing mathematically the limiting process described in words above on the surface-density expression (1.23). The first way is perhaps simpler, but the
second gives useful practice in vector calculus. Consequently we proceed with the limiting process. With n, the unit normal to the surface S, directed away from S', as shown in Fig. 1.6, the potential due to the two close surfaces is

4>(x)=[
Js

j^rda'-f

x-x'

JS '

°$2
,x-xx''++nndd|

For

small

d

we

can

expand

_1
|x-x'+nd| .

Consider

the

general

expression

-1
|x+a| ,

where

|a|«|x|.

We

write

a

Taylor

series

expansion

in

three

dimensions:

1
|x+a| x

d^O In this way we find that as

the potential becomes

(1.24)

In passing we note that the integrand in Eq. (1.24) is the potential of a point
dipole with dipole moment p=n D da' . The potential at x caused by a dipole pat
x' is

<D(x) <D(x)

P-(x-x')
x—

(1-25)

We Equation (1.24) has a simple geometrical interpretation.

note that

_,/ 1 \ , ,

cos 6 da'

,~

where dO is the element of solid angle subtended at the observation point by the
area element da', as indicated in Fig. 1.7. Note that dCl has a positive sign if 6 is an acute angle, i.e., when the observation point views the "inner" side of the

38

Classical Electrodynamics

S

Sect. 1.7

D Fig. 1.7 The potential at P due to the dipole layer on the area element da' is just the D negative product of and the solid angle element dtl subtended by da' at P.
dipole layer. The potential can be written:

(1.26)

For a constant surface-dipole-moment density D, the potential is just the
product of the moment and the solid angle subtended at the observation point by
the surface, regardless of its shape.
There is a discontinuity in potential in crossing a double layer. This can be seen by letting the observation point come infinitesimally close to the double
layer. The double layer is now imagined to consist of two parts, one being a small disc directly under the observation point. The disc is sufficiently small that it is
sensibly flat and has constant surface-dipole-moment density D. Evidently the total potential can be obtained by linear superposition of the potential of the disc
and that of the remainder. From (1.26) it is clear that the potential of the disc
alone has a discontinuity of 4ttD in crossing from the inner to the outer side,
being — 2rrD on the inner side and +2itD on the outer. The potential of the
remainder alone, with its hole where the disc fits in, is continuous across the plane of the hole. Consequently the total potential jump in crossing the surface

is:

4>2-4>i = 4ttD

(1.27)

This result is analogous to (1.22) for the discontinuity of electric field in crossing a surface-charge density. Equation (1.27) can be interpreted "physically" as a potential drop occurring "inside" the dipole layer, and can be calculated as the product of the field between the two layers of surface charge times the separation before the limit is taken.

1.7 Poisson and Laplace Equations

In Sections 1.4 and 1.5 it was shown that the behavior of an electrostatic field can be described by the two differential equations:

V • E = 4irp

(1.13)

Sect. 1.7

Introduction to Electrostatics

39

and

VxE =

(1.14)

the latter equation being equivalent to the statement that E is the gradient of a
scalar function, the scalar potential 4>:

E = -V4)

(1.16)

Equations (1.13) and (1.16) can be combined into one partial differential
equation for the single function 4>(x):

V2 4>

=

-4ttp

(1.28)

This equation is called the Poisson equation. In regions of space where there is no charge density, the scalar potential satisfies the Laplace equation:

V2 <D

=

(1.29)

We already have a solution for the scalar potential in expression (1.17):

•»-JjSf^

(L17)

To verify directly that this does indeed satisfy the Poisson equation (1.28) we operate with the Laplacian on both sides. Because it turns out that the resulting integrand is singular, we invoke a limiting procedure. Define the "^-potential"
<*>a(x) by

J V(x-x') 2 +a +<2 2:

The actual potential (1.17) is then the limit of the "a-potential" as<z ->0. Taking
the Laplacian of the "^-potential" gives
V^(x)=}p(x^( =L=)^'
7

=-M(^] dV

(l30)

where r= x-x' . The square-bracketed expression is the negative Laplacian of \ \ l/Vr 2 + a 2 . It is well-behaved everywhere for nonvanishing a, but as a tends to
zero it becomes infinite at r=0 and vanishes for r^O. It has a volume integral
equal to 47r for arbitrary a. For the purposes of integration divide space into two
regions by a sphere of fixed radius R centered on x. Choose R such thatp(x') changes little over the interior of the sphere, and imagine a much smaller than R
and tending towards zero. If p(x') is such that (1.17) exists, the contribution to
the integral from the exterior of the sphere will vanish like a 2 as**—^-0. We thus
need consider only the contribution from inside the sphere. With a Taylor series
expansion of the well-behaved p(x') around x'=x, one finds

f V^.OO = -4n

(x)-^p 3fl2
rp

+ • •• lr2 dr+ 0(a>)

+a 2
Jo (r

)L

o

J

40

Classical Electrodynamics

Direct integration yields

Sect. 1.8

V2

4> fl

(x)

=

-4ttp(x)

(l

+

0(a 2 lR 2 ))

+

0(a\

2 \og a) V 2p + ...

a^O, In the limit

we obtain the Poisson equation (1.28).

The singular nature of the Laplacian of 1/r can be exhibited formally in terms

of

a

Dirac

delta

function.

Since

V = 2 (l/r)

for r^O and its volume integral is

-4tt,

we

can

write

the

formal

equation,

V2 (l/r)

=

-4ir

8(x)

or,

more

generally,

(1.31)

1.8 Green's Theorem
If electrostatic problems always involved localized discrete or continuous distributions of charge with no boundary surfaces, the general solution (1.17) would be the most convenient and straightforward solution to any problem. There would be no need of the Poisson or Laplace equation. In actual fact, of course, many, if not most, of the problems of electrostatics involve finite regions of space, with or without charge inside, and with prescribed boundary conditions
on the bounding surfaces. These boundary conditions may be simulated by an
appropriate distribution of charges outside the region of interest (perhaps at
infinity), but (1.17) becomes inconvenient as a means of calculating the potential, except in simple cases (e.g., method of images).
To handle the boundary conditions it is necessary to develop some new
mathematical tools, namely, the identities or theorems due to George Green (1824). These follow as simple applications of the divergence theorem. The divergence theorem:

A V applies to any well-behaved vector field defined in the volume bounded by

the closed surface S. Let A=</> Vi//, where <f> and ijj are arbitrary scalar fields.
Now

and

V = V • (<f> Vifr)

2
i//+ V<j) • Vi|/
<f>

(1.32)

dip
(1.33)
dn

where d/dn is the normal derivative at the surface S (directed outwards from
inside the volume V). When (1.32) and (1.33) are substituted into the divergence

Sect. 1.8

Introduction to Electrostatics

41

theorem, there results Green's first identity:

f

= 2

3

((/>V i//+V4>- Vi//)d x <j>

<t>^da

Jv

Js dn

(1.34)

If we write down (1.34) again with <#> and i// interchanged, and then subtract it from (1.34), the V<j> • Vi// terms cancel, and we obtain Green's second identity or
Green's theorem:

(*V*-*^)d'x = j[[*|*-*U]da
fv

(1.35)

The Poisson differential equation for the potential can be converted into an
= integral equation if we choose a particular \p, namely l/R l/|x— x'|, where x is
the observation point and x' is the integration variable. Further, we put =
the scalar potential, and make use of V2 = -47rp. From (1.31) we know that
V2(1/R) = -4tt8(x-x'), so that (1.35) becomes

L

*<-{ [*£(*Hi?]

If the point x lies within the volume V, we obtain:

If x lies outside the surface S, the left-hand side of (1.36) is zero.* [Note that this is consistent with the interpretation of the surface integral as being the potential
D due to a surface-charge density cr= (l/47r)(d<I>/dn') and a dipole layer =
-(1/4tt)<&. The discontinuities in electric field and potential (1.22) and (1.27) across the surface then lead to zero field and zero potential outside the volume
v.]
Two remarks are in order about result (1.36). First, if the surface S goes to infinity and the electric field on S falls off faster than R~\ then the surface
integral vanishes and (1.36) reduces to the familiar result (1.17). Second, for a charge-free volume the potential anywhere inside the volume (a solution of the Laplace equation) is expressed in (1.36) in terms of the potential and its normal derivative only on the surface of the volume. This rather surprising result is not a solution to a boundary-value problem, but only an integral statement, since the arbitrary specification of both 4> and d<&/dn (Cauchy boundary conditions) is an overspecification of the problem. This will be discussed in detail in the next sections, where techniques yielding solutions for appropriate boundary conditions will be developed using Green's theorem (1.35).
*The reader may complain that (1.36) has been obtained in an illegal fashion since l/|x-x'| is not well-behaved inside the volume V. Rigor can be restored by using a
limiting process, as in the previous section, or by excluding a small sphere around the offending point, x=x'. The result is still (1.36).

42

Classical Electrodynamics

1.9 Uniqueness of the Solution with Dirichlet or
Neumann Boundary Conditions

Sect. 1.9

The question arises as to what are the boundary conditions appropriate for the

Poission (or Laplace) equation in order that a unique and well-behaved (i.e.,

physically reasonable) solution exist inside the bounded region. Physical experi-

ence leads us to believe that specification of the potential on a closed surface

(e.g., a system of conductors held at different potentials) defines a unique

potential problem. This is called a Dirichlet problem, or Dirichlet boundary

conditions. Similarly it is plausible that specification of the electric field (normal

derivative of the potential) everywhere on the surface (corresponding to a given

surface-charge density) also defines a unique problem. Specification of the
normal derivative is known as the Neumann boundary condition. We now

proceed to prove these expectations by means of Green's first identity (1.34).

We want to show the uniqueness of the solution of the Poisson equation,

V V 2 <£

=

— 477p,

inside

a

volume

subject to either Dirichlet or Neumann

We boundary conditions on the closed bounding surface S.

suppose, to the

contrary, that there exist two solutions <I>i and <J> 2 satisfying the same boundary

conditions. Let

U = <!>2-<S>i

(1.37)

U Then V2 = inside V, and 17=0 or dU/dn = on S for Dirichlet and Neumann
boundary conditions, respectively. From Green's first identity (1.34), with 4>= ifj=U, we find

(1.38)

With the specified properties of 17, this reduces (for both types of boundary
conditions) to:

VU U which implies

= 0. Consequently, inside V,

is constant. For Dirichlet

U boundary conditions, = on S so that, inside V, 4>i=<J>2 and the solution is

unique. Similarly, for Neumann boundary conditions, the solution is unique,

apart from an unimportant arbitrary additive constant.

From the right-hand side of (1.38) it is evident that there is also a unique

solution to a problem with mixed boundary conditions (i.e., Dirichlet over part

of the surface S, and Neumann over the remaining part).

It should be clear that a solution to the Poisson equation with both and

d<t>/dn specified arbitrarily on a closed boundary (Cauchy boundary conditions)
does not exist, since there are unique solutions for Dirichlet and Neumann

conditions separately and these will in general not be consistent. This can be

verified with (1.36). With arbitrary values of 3> and d<t>/dn inserted on the

Sect. 1.10

Introduction to Electrostatics

43

right-hand side, it can be shown that the values of 4>(x) and VO(x) as x approaches the surface are in general inconsistent with the assumed boundary values. The question of whether Cauchy boundary conditions on an open surface define a unique electrostatic problem requires more discussion than is warranted here. The reader may refer to Morse and Feshbach, Section 6.2, pp. 692-706, or to Sommerfeld, Partial Differential Equations in Physics, Chapter II, for a detailed discussion of these questions. The conclusion is that electrostatic
problems are specified only by Dirichlet or Neumann boundary conditions on a closed surface (part or all of which may be at infinity, of course).

1.10 Formal Solution of Electrostatic Boundary-Value Problem with Green Function
V The solution of the Poisson or Laplace equation in a finite volume with either
Dirichlet or Neumann boundary conditions on the bounding surface S can be
obtained by means of Green's theorem (1.35) and so-called "Green functions."
— — In obtaining result (1.36) not a solution we chose the function i// to be
l/|x-x'|, it being the potential of a unit point charge, satisfying the equation:

(^) 2
V'

= -4tt8(x-x')

(1.31)

The function l/|x-x'| is only one of a class of functions depending on the variables x and x', and called Green functions, which satisfy (1.31). In general,

where

V' 2 G(x, x') = -4tt8(x-x') G(x,x') = j^+F(x,x')

(1.39) (1.40)

with the function F satisfying the Laplace equation inside the volume V:

2
V' F(x,x')

=

(1.41)

In facing the problem of satisfying the prescribed boundary conditions on <I> or
d<£/dn, we can find the key by considering result (1.36). As has been pointed out
already, this is not a solution satisfying the correct type of boundary conditions because both and d<E>/dn appear in the surface integral. It is at best an integral relation for <£. With the generalized concept of a Green function and its
additional freedom [via the function F(x, x')], there arises the possibility that we can use Green's theorem with i//= G(x, x') and choose F(x, x') to eliminate one or the other of the two surface integrals, obtaining a result which involves only Dirichlet or Neumann boundary conditions. Of course, if the necessary G(x, x') depended in detail on the exact form of the boundary conditions, the method would have little generality. As will be seen immediately, this is not required,
and G(x, x') satisfies rather simple boundary conditions on S.

44

Classical Electrodynamics

Sect. 1.10

With Green's theorem (1.35), = i(r= G(x, x'), and the specified properties
G of (1.39), it is simple to obtain the generalization of (1.36):

d>(x) = p(x')G(x,x')dV+^^ [G(x,x')g-<D(x')^^]da' (1.42)
Jv

G The freedom available in the definition of (1.40) means that we can make the
surface integral depend only on the chosen type of boundary conditions. Thus, for Dirichlet boundary conditions we demand:

GD (x,x') =

for x' on S

(1.43)

Then the first term in the surface integral in (1.42) vanishes and the solution is

x'~ ^ *(x)= p(x')GD (x, x') d3 Jv

j> <fc(x')

da'

(1.44)

For Neumann boundary conditions we must be more careful. The obvious
choice of boundary condition on G(x, x') seems to be

—da—Gnr (x, x') =

for x' on S

since that makes the second term in the surface integral in (1.42) vanish, as desired. But an application of Gauss's theorem to (1.39) shows that

iS' da' = -^

G Consequently the simplest allowable boundary condition on N is

~= (x, *')

for x! on S

an

b

(1.45)

where S is the total area of the boundary surface. Then the solution is

G <D(x) = <<D)s+ f p(x')GN (x,x') dV+-^-(j>

Jv

477 Js

^on7

N da'

(1.46)

where (<J>)S is the average value of the potential over the whole surface. The customary Neumann problem is the so-called "exterior problem" in which the
V volume is bounded by two surfaces, one closed and finite, the other at infinity.
Then the surface area S is infinite; the boundary condition (1.45) becomes
homogeneous; the average value (<J>)S vanishes.
We note that the Green functions satisfy simple boundary conditions (1.43) or
(1.45) which do not depend on the detailed form of the Dirichlet (or Neumann)
boundary values. Even so, it is often rather involved (if not impossible) to
We determine G(x, x') because of its dependence on the shape of the surface S.
will encounter such problems in Chapter 2 and 3.
The mathematical symmetry property G(x, = x') G(x', x) can be proved for the

Sect. 1.11

Introduction to Electrostatics

45

Green functions satisfying the Dirichlet boundary condition (1.43) by means of Green's theorem with </>=G(x, y) and i//=G(x', y), where y is the integration variable. Since the Green function, as a function of one of its variables, is a potential due to a unit point charge, the symmetry merely represents the physical
interchangeability of the source and the observation points. For Neumann
boundary conditions the symmetry is not automatic, but can be imposed as a
separate requirement.
As a final, important remark we note the physical meaning of F(x, x'). It is a
V solution of the Laplace equation inside and so represents the potential of a
system of charges external to the volume V. It can be thought of as the potential due to an external distribution of charges so chosen as to satisfy the homogeneous boundary conditions of zero potential (or zero normal derivative) on the
surface S when combined with the potential of a point charge at the source point x'. Since the potential at a point x on the surface due to the point charge depends
on the position of the source point, the external distribution of charge F(x, x')
must also depend on the "parameter" x\ From this point of view, we see that the method of images (to be discussed in Chapter 2) is a physical equivalent of the
determination of the appropriate F(x, x') to satisfy the boundary conditions (1.43) or (1.45). For the Dirichlet problem with conductors, F(x, x') can also be interpreted as the potential due to the surface-charge distribution induced on the conductors by the presence of a point charge at the source point x'.

1.11 Electrostatic Potential Energy and Energy Density, Capacitance

In Section 1.5 it was shown that the product of the scalar potential and the charge of a point object could be interpreted as potential energy. More precisely, if a point charge q{ is brought from infinity to a point x< in a region of localized electric fields described by the scalar potential (which vanishes at infinity), the work done on the charge (and hence its potential energy) is given by

W^qM*)

(1.47)

The potential <I> can be viewed as produced by an array of (n-1) charges q,(j= 1,2,..., n— 1) at positions x,. Then

* (Xi)= "^i^i j=l |Xj Xj|

(L48)

so that the potential energy of the charge q; is

Wi = qi ZrjLi /-i |xi— x,|

(1-49)

The total potential energy of all the charges due to all the forces acting between

46

Classical Electrodynamics

them is:

Sect. 1.11

= i l j<i |Xj X, I

A as can be seen most easily by adding each charge in succession.

more

symmetric form can be written by summing over i and / unrestricted, and then

dividing by 2:

It is understood that = i j terms (infinite "self-energy" terms) are omitted in the
double sum. For a continuous charge distribution [or, in general, using the Dirac delta
functions (1.6)] the potential energy takes the form:
W

Another expression, equivalent to (1.52), can be obtained by noting that one of the integrals in (1.52) is just the scalar potential (1.17). Therefore

W=|Jp(x)<I>(x)<fx

(1.53)

Equations (1.51), (1.52), and (1.53) express the electrostatic potential energy in terms of the positions of the charges and so emphasize the interactions
between charges via Coulomb forces. An alternative, and very fruitful, approach
is to emphasize the electric field and to interpret the energy as being stored in the
electric field surrounding the charges. To obtain this latter form, we make use of
the Poisson equation to eliminate the charge density from (1.53):

I

2
J(DV <D

Integration by parts leads to the result:

W=± ± 2 J|V<D|

d3x

=

2
J|E|

d3x

(1.54)

where the integration is over all space. In (1.54) all explicit reference to charges has gone, and the energy is expressed as an integral of the square of the electric field over all space. This leads naturally to the identification of the integrand as an energy density w:

w = ^|E| 2

(1.55)

This expression for energy density is intuitively reasonable, since regions of high fields "must" contain considerable energy.

Sect. 1.11

Introduction to Electrostatics

47

Fig. 1.8
There is perhaps one puzzling thing about (1.55). The energy density is positive definite. Consequently its volume integral is necessarily nonnegative. This seems to contradict our impression from (1.51) that the potential energy of two charges of opposite sign is negative. The reason for this apparent contradiction is that (1.54) and (1.55) contain "self-energy" contributions to the energy density, whereas the double sum in (1.51) does not. To illustrate this, consider two point charges qi and q2 located at x x and x2 , as in Fig. 1.8. The electric field at the point P with coordinate x is
term gives the proper result for the interaction potential energy we integrate
over all space: (1.57)
A change of integration variable to p = (x-Xi)/|xi-x2 yields |
(1.58)
where n is a unit vector in the direction (xi~x2). Using the fact that (p+n)/
-V |p+n| 3 = p (l/|p+n|), the dimensionless integral can easily be shown to have the
value 47r, so that the interaction energy reduces to the expected value. Forces acting between charged bodies can be obtained by calculating the
change in the total electrostatic energy of the system under small virtual displacements. Examples of this are discussed in the problems. Care must be taken to exhibit the energy in a form showing clearly those factors which vary with a change in configuration and those which are kept constant.
As a simple illustration we calculate the force per unit area on the surface of a
conductor with a surface-charge density cr(x). In the immediate neighborhood of

48

Classical Electrodynamics

the surface the energy density is

Sect. 1.11

w = ^-|E| 2 = 27ra2

(1.59)

If we now imagine a small outward displacement Ax of an elemental area A a of the conducting surface, the electrostatic energy decreases by an amount which is the product of energy density w and the excluded volume Ax Aa:

AW=-27Ta2 AaAx

(1.60)

This

means

that

there

is

an

outward

force

per

unit

area

equal

to

= 2
2ttgt

w

at

the

surface of the conductor. This result is normally derived by taking the product of

the surface-charge density and the electric field, with care taken to eliminate the

electric field due to the element of surface-charge density itself.

For a system of n conductors, each with potential Vi and total charge Q*
= (i 1, 2, . . . , n) in otherwise empty space, the electrostatic potential energy can
be expressed in terms of the potentials alone and certain geometrical quantities

called coefficients of capacity. For a given configuration of the conductors, the

linear functional dependence of the potential on the charge density implies that

the potential of the ith conductor can be written as

Q Vi=Ipy

(i=l,2,...,n)

where the piy depend on the geometry of the conductors. These n equations can be inverted to yield the charge on the ith conductor in terms of all the potentials:

Q tc = i

ii Vi

0=1,2,. ..,n)

(1.61)

C C The coefficients u are called capacities or capacitances while the ih i^j, are
called coefficients of induction. The capacitance of a conductor is therefore the total charge on the conductor when it is maintained at unit potential, all other conductors being held at zero potential. Sometimes the capacitance of a system of conductors is also defined. For example, the capacitance of two conductors carrying equal and opposite charges in the presence of other grounded conductors is defined as the ratio of the charge on one conductor to the potential difference between them. The equations (1.61) can be used to express this capacitance in terms of the coefficients Q.
The potential energy (1.53) for the system of conductors is

W=i I QV.-4 t t QV.V,

Z ,= 1

Z i=i = , i

(1.62)

The expression of the energy in terms of the potentials Vi and the G„ or in terms
of the charges Qi and the coefficients pih permits the application of variational methods to obtain approximate values of capacitances. It can be shown (see

Prob. 1

Introduction to Electrostatics

49

Problems 1.17 and 1.18) that there are variational principles giving upper and lower bounds on Gi. The principles permit estimation with known error of the capacitances of relatively involved configurations of conductors. High-speed computational techniques permit the use of elaborate trial functions involving several parameters. It must be remarked, however, that the need for a Green function satisfying Dirichlet boundary conditions in the lower bound makes the error estimate nontrivial. Further consideration of this technique for calculating capacitances is left to the problems at the end of this and subsequent chapters.

REFERENCES AND SUGGESTED READING
On the mathematical side, the subject of delta functions is treated simply but rigorously
by
Lighthill,
Dennery and Kryzwicki. For a discussion of different types of partial differential equations and the appropriate
boundary conditions for each type, see Morse and Feshbach, Chapter 6, Sommerfeld, Partial Differential Equations in Physics, Chapter II, Courant and Hilbert, Vol. II, Chapters III-VI.
The general theory of Green functions is treated in detail by Friedman, Chapter 3, Morse and Feshbach, Chapter 7.
The general theory of electrostatics is discussed extensively in many of the older books. Notable, in spite of some old-fashioned notation, are Maxwell, Vol. 1, Chapters II and IV,
Jeans, Chapters II, VI, VII, Kellogg.
Of more recent books, mention may be made of the treatment of the general theory
by Stratton, Chapter III, and parts of Chapter II. Readers interested in variational methods applied to electromagnetic problems can
consult
Cairo and Kahan, Collin, Chapter 4, and Polya and Szego for elegant and powerful mathematical techniques.
PROBLEMS
1.1 Use Gauss's theorem (and Eq. (1.21) if necessary) to prove the following:
(a) Any excess charge placed on a conductor must lie entirely on its surface. (A
conductor by definition contains charges capable of moving freely under the action of
applied electric fields.)
A (b) closed, hollow conductor shields its interior from fields due to charges outside,
but does not shield its exterior from the fields due to charges placed inside it.

50

Classical Electrodynamics

Prob. 1

(c) The electric field at the surface of a conductor is normal to the surface and has a magnitude 4tt<j, where a is the charge density per unit area on the surface.
1.2 The Dirac delta function in three dimensions can be taken as the improper limit as
a—() of the Gaussian function

exp[-^(x • D(a;x,y,z) = (277)- 3/2 a- 3

+ + 2

2

2

y

2 )]

Consider a general orthogonal coordinate system specified by the surfaces, u =
W constant, inconstant, w = constant, with length elements du/ U, dv/ V, dw/ in the
three perpendicular directions. Show that
UVW 8(x-x') = 5(u-u') S(u-u') S(w-w') •
by considering the limit of the above Gaussian. Note that as a—»0 only the
infinitesimal length element need be used for the distance between the points in the
exponent.

1.3 Using Dirac delta functions in the appropriate coordinates, express the following charge distributions as three-dimensional charge densities p(x).
Q (a) In spherical coordinates, a charge uniformly distributed over a spherical shell of
radius R. (b) In cylindrical coordinates, a charge A per unit length uniformly distributed over a
cylindrical surface of radius b.
Q (c) In cylindrical coordinates, a charge spread uniformly over a flat circular disc of
negligible thickness and radius R. (d) The same as (c), but using spherical coordinates.

1.4 Each of three charged spheres of radius a, one conducting, one having a uniform charge density within its volume, and one having a spherically symmetric charge density which varies radially as r" (n>-3), has a total charge Q. Use Gauss's theorem to obtain the electric fields both inside and outside each sphere. Sketch the behavior of the fields as a function of radius for the first two spheres, and for the third with n = -2, +2.

1.5 The time-average potential of a neutral hydrogen atom is given by

where q is the magnitude of the electronic charge, and a' 1 = a /2, cio being the Bohr
radius. Find the distribution of charge (both continuous and discrete) which will give this potential and interpret your result physically.

A 1.6

simple capacitor is a device formed by two insulated conductors adjacent to each

other. If equal and opposite charges are placed on the conductors, there will be a

certain difference of potential between them. The ratio of the magnitude of the charge

on one conductor to the magnitude of the potential difference is called the capacitance

(in electrostatic units it is measured in centimeters). Using Gauss's law, calculate the

capacitance of

(a) two large, flat, conducting sheets of area A, separated by a small distance d;

(b) two concentric conducting spheres with radii a, b (b>a);

(c) two concentric conducting cylinders of length L, large compared to their radii a, b

(b>a).

(d) What is the inner diameter of the outer conductor in an air-filled coaxial cable

Prob. 1

Introduction to Electrostatics

51

mm whose center conductor is a cylindrical wire of diameter 1

and whose capacitance

is 0.5 micromicrofarad/cm? 0.05 micromicrofarad/cm?

1.7 Two long, cylindrical conductors of radii a, and a2 are parallel and separated by a distance d which is large compared with either radius. Show that the capacitance per
unit length is given approximately by

where a is the geometrical mean of the two radii. Approximately what gauge wire (state diameter in millimeters) would be necessary
to make a two- wire transmission line with a capacitance of 0.1 fi/xf/cm if the separation of the wires was 0.5 cm? 1.5 cm? 5.0 cm?

1.8 (a) For the three capacitor geometries in Problem 1.6 calculate the total electro-

Q static energy and express it alternatively in terms of the equal and opposite charges

-Q and

placed on the conductors and the potential difference between them.

(b) Sketch the energy density of the electrostatic field in each case as a function of the

appropriate linear coordinate.

1.9 Calculate the attractive force between conductors in the parallel plate capacitor (Problem 1.6a) and the parallel cylinder capacitor (Problem 1.7) for (a) fixed charges on each conductor; (b) fixed potential difference between conductors.

1.10 Prove the mean value theorem: For change-free space the value of the electrostatic
potential at any point is equal to the average of the potential over the surface of any sphere centered on that point.

1.11 Use Gauss's theorem to prove that at the surface of a curved charged conductor the normal derivative of the electric field is given by

R where Ri and 2 are the principal radii of curvature of the surface.
1.12 Prove Green's reciprocation theorem: If 3> is the potential due to a volume-charge
V density p within a volume and a surface-charge density cr on the conducting surface
S bounding the volume V, while <£' is the potential due to another charge distribution p' and a', then

A 1.13 Two infinite grounded parallel conducting planes are separated by a distance d.
point charge q is placed between the planes. Use the reciprocation theorem of Green to prove that the total induced charge on one of the planes is equal to (-q) times the fractional perpendicular distance of the point charge from the other plane. (Hint: Choose as your comparison electrostatic problem with the same surfaces one whose charge densities and potential are known and simple.)

A 1.14

volume V is bounded by a surface S consisting of several separate surfaces

^ (conductors) Si, one perhaps at infinity, each held at potential V . Let (x) be a (
well-behaved function in V and on 5, with a value equal to V, on each surface Si, but

otherwise arbitrary for the present. Define the energylike quantity, otherwise arbitr-

52

Classical Electrodynamics

ary for the present. Define the energylike quantity,

Prob. 1

Prove the following theorem:
W[^], which is nonnegative by definition, is stationary and an absolute minimum if
^ V and only if satisfies the Laplace equation inside and takes on the specified values
V; on the surfaces S . (
1.15 Prove Thomson's theorem: If a number of surfaces are fixed in position and a given total charge is placed on each surface, then the electrostatic energy in the region bounded by the surfaces is an absolute minimum when the charges are placed so that every surface is an equipotential, as happens when they are conductors.
1.16 Prove the following theorem: If a number of conducting surfaces are fixed in position with a given total charge on each, the introduction of an uncharged, insulated conductor into the region bounded by the surfaces lowers the electrostatic energy.
1.17 Consider a configuration of conductors as in problem 1.14 with one conductor held at unit potential and all the other conductors at zero potential. (a) Show that the capacitance of the one conductor is given by

where <I>(x) is the solution for the potential.
C (b) Use the theorem of Problem 1 . 14 to show that the true capacitance is always less
than or equal to the quantity
^ where is any trial function satisfying the boundary conditions on the conductors.
This is a variational principle for the capacitance that yields an upper bound. 1.18 Consider the configuration of conductors of Problem 1.17, with all conductors
except Si held at zero potential.
V (a) Show that the potential 4>(x) anywhere in the volume and on any of the surfaces
S, can be written
where o-i(x') is the surface charge density on Si and G(x, x') is the Green function potential for a point charge in the presence of all the surfaces that are held at zero potential (but with Si absent). Show also that the electrostatic energy is
where the integrals are only over the surface Si. (b) Show that the variational expression

Prob. 1

Introduction to Electrostatics

53

with an arbitrary integrable function cr(x) defined on Si, is stationary for small

variations of cr away from cr,. Use Thomson's theorem to prove that the reciprocal of

C_1 [cr]

gives

a

lower

bound

to

the

true

capacitance

of

the

conductor

S,.

1.19 For the cylindrical capacitor of Problem 1.6(c), evaluate the variational upper
^ bound of Problem 1.17(b) with the naive trial function, ,(p) = (b- p)/(b- a) . Compare

the variational result with the exact result for b/a= 1.5, 2, 3. Explain the trend of your

results in terms of the functional form of

An improved trial function is treated by

Collin, pp. 151-152.

1.20 In estimating the capacitance of a given configuration of conductors, comparison

with known capacitances is often helpful. Consider two configurations of n conductors

in which the (n-1) conductors held at zero potential are the same, but the one

conductor whose capacitance we wish to know is different. In particular, let the

conductor in one configuration have a closed surface Si and in the other configuration

have surface S[, with S[ totally inside Si.

(a) Use the theorem of Problem 1.14 and the variational principle of Problem 1.17 to
C prove that the capacitance of the conductor with surface S[ is less than or equal to
C the capacitance of the conductor with surface Si that encloses SJ.

(b) Set upper and lower limits for the capacitance of a conducting cube of side a.

C— Compare your limits and also their average with the numerical value,

0.655a.

2
Boundary-Value Problems
in Electrostatics: I
Many problems in electrostatics involve boundary surfaces on which either the potential or the surface-charge density is specified. The formal solution of such
problems was presented in Section 1.10, using the method of Green functions. In practical situations (or even rather idealized approximations to practical situations) the discovery of the correct Green function is sometimes easy and sometimes not. Consequently a number of approaches to electrostatic boundaryvalue problems have been developed, some of which are only remotely connected to the Green function method. In this chapter we will examine two of these special techniques: (1) the method of images, which is closely related to the use of Green functions; (2) expansion in orthogonal functions, an approach directly through the differential equation and rather remote from the direct
A construction of a Green function. major omission is the use of complex-
variable techniques, including conformal mapping, for the treatment of two-
dimensional problems. The topic is important, but lack of space and the existence of self-contained discussions elsewhere accounts for its absence. The interested reader may consult the references cited at the end of the chapter.
2.1 Method of Images
The method of images concerns itself with the problem of one or more point charges in the presence of boundary surfaces, for example, conductors either grounded or held at fixed potentials. Under favorable conditions it is possible to infer from the geometry of the situation that a small number of suitably placed
charges of appropriate magnitudes, external to the region of interest, can simulate the required boundary conditions. These charges are called image charges, and the replacement of the actual problem with boundaries by an enlarged region with image charges but not boundaries is called the method of
images. The image charges must be external to the volume of interest, since their
54

Sect. 2.2

Boundary-Value Problems in Electrostatics: I

55

Fig. 2.1 Solution by method of images. The original potential problem is on the left, the equivalent-image problem on the right.
potentials must be solutions of the Laplace equation inside the volume; the "particular integral" (i.e., solution of the Poisson equation) is provided by the
sum of the potentials of the charges inside the volume.
A simple example is a point charge located in front of an infinite plane
conductor at zero potential, as shown in Fig. 2.1. It is clear that this is equivalent to the problem of the original charge and an equal and opposite charge located at the mirror-image point behind the plane defined by the position of the
conductor.

2.2 Point Charge in the Presence of a Grounded Conducting Sphere

As an illustration of the method of images we consider the problem illustrated in

Fig. 2.2 of a point charge q located at y relative to the origin around which is

We centered a grounded conducting sphere of radius a.

seek the potential <$(x)

such that = = 4>(|x| a) 0. By symmetry it is evident that the image charge q'

Fig. 2.2 Conducting sphere of radius a, with charge q and image charge q'.

56

Classical Electrodynamics

Sect. 2.2

(assuming that only one image is needed) will lie on the ray from the origin to the
charge q. If we consider the charge q outside the sphere, the image position y' will lie inside the sphere. The potential due to the charges q and q[ is:

r r ^ + ^(x) =

fl -
1

fl

(2.1)

We now must try to choose q' and |y'| such that this potential vanishes at |x| = a. If
n is a unit vector in the direction x, and n' a unit vector in the direction y, then

—3
<S>(x)=i
\xn—

yn\H-h,|xn—^Vy tnt|

(2.2)

If x is factored out of the first term and y' out of the second, the potential at x = a
becomes:

3>(x=a)=

q

i

..

+

q

— i n,

a:

.

;n

y

From the form of (2.3) it will be seen that the choices:

(2.3)

a y" ay'

make <£(x = a) = 0, for all possible values of n«n'. Hence the magnitude and
position of the image charge are

q,'=-a-% y, =a-2

A\ (2.4)

We note that, as the charge q is brought closer to the sphere, the image charge
grows in magnitude and moves out from the center of the sphere. When q is just
outside the surface of the sphere, the image charge is equal and opposite in magnitude and lies just beneath the surface.
Now that the image charge has been found, we can return to the original
problem of a charge q outside a grounded conducting sphere and consider various effects. The actual charge density induced on the surface of the sphere can be calculated from the normal derivative of <I> at the surface:

J_cKP I cr=
4tt dx x=a

q /a\

\y

Aira

— 1H 2- 2 -cos 7

y

y

where 7 is the angle between x and y. This charge density in units of -q/4ira2 is shown plotted in Fig. 2.3 as a function of 7 for two values of y/a. The
concentration of charge in the direction of the point charge q is evident,
especially for y/a = 2. It is easy to show by direct integration that the total

Sect. 2.2

Boundary-Value Problems in Electrostatics: I

57

Fig. 2.3 Surface-charge density cr induced on the grounded sphere of radius a due to the presence of a point charge q located a distance y away from the center of the sphere, cr is plotted in units of -q/4ira 2 as function of the angular position 7 away from the radius to
the charge for y = 2a, 4a.
induced charge on the sphere is equal to the magnitude of the image charge, as it
must according to Gauss's law.
The force acting on the charge q can be calculated in different ways. One (the easiest) way is to write down immediately the force between the charge q and the image charge q'. The distance between them is y - y' = y(l - a 2 /y 2). Hence
the attractive force, according to Coulomb's law, is:

For large separations the force is an inverse cube law, but close to the sphere it is proportional to the inverse square of the distance away from the surface of the

sphere.

The alternative method for obtaining the force is to calculate the total force

acting on the surface of the sphere. The force on each element of area da is

2
27Tcr

da,

where

cr

is

given

by

(2.5),

as

indicated

in

Fig.

2.4.

But

from

symmetry

it

is clear that only the component parallel to the radius vector from the center of

58

Classical Electrodynamics

dF = 27ra 2 da

Sect. 2.3

Fig. 2.4
the sphere to q contributes to the total force. Hence the total force acting on the sphere (equal and opposite to the force acting on q) is given by the integral:
(2.7)
Integration immediately yields (2.6).
The whole discussion has been based on the understanding that the point
charge q is outside the sphere. Actually, the results apply equally for the charge q
inside the sphere. The only change necessary is in the surface-charge density (2.5), where the normal derivative out of the conductor is now radially inwards, implying a change in sign. The reader may transcribe all the formulas, remem-
< bering that now y a. The angular distributions of surface charge are similar to
those of Fig. 2.3, but the total induced surface charge is evidently equal to -q, independent of y.

2.3 Point Charge in the Presence of a Charged, Insulated, Conducting Sphere

In the previous section we considered the problem of a point charge q near a

grounded sphere and saw that a surface-charge density was induced on the
sphere. This charge was of total amount q' — —aq/y, and was distributed over the

surface in such a way as to be in equilibrium under all forces acting.

If we wish to consider the problem of an insulated conducting sphere with
Q total charge in the presence of a point charge q, we can build up the solution
for the potential by linear superposition. In an operational sense, we can imagine

that we start with the grounded conducting sphere (with its charge q' distributed

We over its surface).

then disconnect the ground wire and add to the sphere an

amount of charge (Q-q'). This brings the total charge on the sphere up to Q. To

find the potential we merely note that the added charge (Q-q') will distribute

itself uniformly over the surface, since the electrostatic forces due to the point

charge q are already balanced by the charge q'. Hence the potential due to the
added charge (Q— q') will be the same as if a point charge of that magnitude were

at the origin, at least for points outside the sphere.

Sect. 2.3

Boundary-Value Problems in Electrostatics: I

59

The potential is the superposition of (2.1) and the potential of a point charge (Q-q') at the origin:

L <D(x) =

,-

r|x-y|

SSL a

Q + ^q

(2.8)

The force acting on the charge q can be written down directly from Coulomb's
law. It is directed along the radius vector to q and has the magnitude:

U
yi

y(f-a2f J

(2.9)

In the limit of y »a, the force reduces to the usual Coulomb's law for two small charged bodies. But close to the sphere the force is modified because of the induced charge distribution on the surface of the sphere. Figure 2.5 shows the force as a function of distance for various ratios of Q/q. The force is expressed in

Q/q = 3

Fy 2 1 q2

1

11

/ /

/

1

i

1

3

4 y/a

5
5

1

1

1

1

|

-1 -1
-2

-3

-4

-5
Fig. 2.5 The force on a point charge q due to an insulated, conducting sphere of radius a carrying a total charge Q. Positive values mean a repulsion, negative an attraction. The asymptotic dependence of the force has been divided out. Fy 2/q 2 is plotted versus y/a for Q/q = -l, 0, 1, 3. Regardless of the value of Q, the force is always attractive at close
distances because of the induced surface charge.

60

Classical Electrodynamics

Sect. 2.5

units

of

22
q /y ;

positive

(negative)

values

correspond

to

a

repulsion

(attraction).

If

the sphere is charged oppositely to q, or is uncharged, the force is attractive at all

Q distances. Even if the charge is the same sign as q, however, the force becomes

attractive at very close distances. In the limit of 0»q, the point of zero force

— (unstab le equilibrium point) is very close to the sphere, namely, at y

a(l + 2>/q/Q). Note that the asymptotic value of the force is attained as soon as

the charge q is more than a few radii away from the sphere. This example exhibits a general property which explains why an excess of

charge on the surface does not immediately leave the surface because of mutual

repulsion of the individual charges. As soon as an element of charge is removed

from the surface, the image force tends to attract it back. If sufficient work is

done, of course, charge can be removed from the surface to infinity. The work

function of a metal is in large part just the work done against the attractive image

force in order to remove an electron from the surface.

2.4 Point Charge Near a Conducting Sphere at Fixed Potential
Another problem which can be discussed easily is that of a point charge near a conducting sphere held at a fixed potential V. The potential is the same as for the charged sphere, except that the charge (Q-q') at the center is replaced by a
charge (Va). This can be seen from (2.8), since at = |x| a the first two terms
V cancel and the last term will be equal to as required. Thus the potential is

The force on the charge q due to the sphere at fixed potential is
For corresponding values of Va/q and Q/q this force is very similar to that of the charged sphere, shown in Fig. 2.5, although the approach to the asymptotic
value (Vaq/y 2) is more gradual. For Va »q, the unstable equilibrium point has the equivalent location y — a(l+Wq/Va).
2.5 Conducting Sphere in a Uniform Electric Field by Method of Images
As a final example of the method of images we consider a conducting sphere of
E A radius a in a uniform electric field . uniform field can be thought of as being
produced by appropriate positive and negative charges at infinity. For example,
if there are two charges ±Q, located at positions z = =FR, as shown in Fig. 2.6a,

Sect. 2.5

Boundary-Value Problems in Electrostatics: I

61

+Q z= -R

S / aQ R ~ R JC \ V

2

R \ a\

-Q z=R

(b)
Fig. 2.6 Conducting sphere in a uniform electric field by the method of images.

then in a region near the origin whose dimensions are very small compared to R

E — there is an approximately constant electric field

2Q/R 2 parallel to the z axis.

Q— In the limit as R,

with Q/R 2 constant, this approximation becomes exact.

If now a conducting sphere of radius a is placed at the origin, the potential will

±Q be that due to the charges

at

and their images =FQa/R at z = IFa 2/R:

+R 2
(r

2 +2rRcos

1/2
6)

Q

+R 2

2

(r

-2rRcos

m
0)

aQ

aQ
1/2 (2.12)

where 3> has been expressed in terms of the spherical coordinates of the

R observation point. In the first two terms is much larger than r by assumption.

R Hence we can expand the radicals after factoring out

2
. Similarly, in the third

and

fourth

terms,

we

can

factor

out

2 r

and

then

expand.

The

result

is:

<D=[-f rcosO+fr^4ccose]

• •

R— where the omitted terms vanish in the limit

In that limit 2Q/R 2 becomes

the applied uniform field, so that the potential is

4> = -E (r-^-) cose

(2.14)

62

Classical Electrodynamics

Sect. 2.6

E The first term (~E z) is, of course, just the potential of a uniform field which

could have been written down directly instead of the first two terms in (2.12).

The second is the potential due to the induced surface charge density or,

equivalently, the image charges. Note that the image charges form a dipole of

strength

D=Qa/Rx2a2/R = E

a3 .

The

induced

surface-charge

density

is

1 a4> I
<r=-T-ir\

3
=T-Eocos6

4tt dr = a \ r 4tt

(2.15)

We note that the surface integral of this charge density vanishes, so that there is
no difference between a grounded and an insulated sphere.

2.6 Green Function for the Sphere, General Solution for the Potential
In preceding sections the problem of a conducting sphere in the presence of a
point charge has been discussed by the method of images. As was mentioned in
Section 1.10, the potential due to a unit charge and its image (or images), chosen to satisfy homogeneous boundary conditions, is just the Green function (1.43 or
1.45) appropriate for Dirichlet or Neumann boundary conditions. In G(x, x') the
variable x' refers to the location P' of the unit charge, while the variable x is the
point P at which the potential is being evaluated. These coordinates and the sphere are shown in Fig. 2.7. For Dirichlet boundary conditions on the
sphere of radius a the potential due to a unit charge and its image is given by
(2.1) with q = 1 and relations (2.4). Transforming variables appropriately, we
z

Fig. 2.7

Sect. 2.7

Boundary-Value Problems in Electrostatics: I

63

obtain the Green function:
rV g <»'*'> = ,i

Vti

(216)

In terms of spherical coordinates this can be written:

^ — ,2^, V G(x,x')

=

(x 2 +x'2 -2xx',c_o„s u7/) 12/2 -

7-2-72

(x

2

x'\
\-a

2

-2-xx

,

cos

TT72
71

(2.17)

where 7 is the angle between x and x\ The symmetry in the variables x and x' is
G obvious in the form (2.17), as is the condition that = if either x or x' is on the
surface of the sphere.
For solution (1.44) of the Poisson equation we need not only G, but also dG/dri. Remembering that n' is the unit normal outwards from the volume of interest, i.e., inwards along x' toward the origin, we have

dG I

-a 2

2

(x

)

dn'U~ a(x2 +a2

-2axcos

3/2
7)

n1

(

}

[Note that this is essentially the induced surface-charge density (2.5).] Hence the solution of the Laplace equation outside a sphere with the potential specified on
its surface is, according to (1.44),

*'
fe«

*>

(^J-tfL

y

r

dn '

(219)

where dCl' is the element of solid angle at the point (a, 0', <f>') and cos 7 =
cos cos 0'+sin sin 0' cos (</>-</>')• For the interior problem, the normal deriva-

tive is radially outwards, so that the sign of dG/dri is opposite to (2.18). This is

equivalent

to

replacing

the

factor

-a 2

2

(x

)

by

-x 2

2

(a

)

in

(2.19).

For

a

problem

with a charge distribution, we must add to (2.19) the appropriate integral in

(1.44), with the Green function (2.17).

2.7 Conducting Sphere with Hemispheres at Different Potentials
As an example of the solution (2.19) for the potential outside a sphere with prescribed values of potential on its surface, we consider the conducting sphere of radius a made up of two hemispheres separated by a small insulating ring. The
hemispheres are kept at different potentials. It will suffice to consider the
potentials as ±V, since arbitrary potentials can be handled by superposition of the solution for a sphere at fixed potential over its whole surface. The insulating ring lies in the z = plane, as shown in Fig. 2.8, with the upper (lower) hemisphere at potential +V(-V).

64

Classical Electrodynamics

z

Sect. 2.7

y
-v
x Fig. 2.8

From (2.19) the solution for 4>(x, 0, <f>) is given by the integral:

<D(x, 0,

=f
47r Jo

l J 2 ^ (X2 " a

I Jfo

d(cos 0')"
Jf-°i

d(cos 0') FT J( a +3C

— 92ax cosg 7)

(2.20)

By a suitable change of variables in the second integral ($'—»7r— 0', (/>'—»<(>'+ 77),
this can be cast in the form:

"^ f'Wf = <D(x, 0, <)>)

Va( * 2

^cos 0')[(a 2 +x2 -2ax cos 7)" 3/2

477

Jo

Jo

-(a 2 +x 2 +2axcos7)- 3/2]

(2.21)

Because of the complicated dependence of cos 7 on the angles (0', <#>') and (0, <f>), equation (2.21) cannot in general be integrated in closed form.
As a special case we consider the potential on the positive z axis. Then cos 7 = cos 0' since = 0. The integration is elementary, and the potential can be shown to be

<j>( 2)=v[i-i^^l
L zvz +a J

(2.22)

V At z = a, this reduces to = <I> as required, while at large distances it goes

— asymptotically as <E>

Va 2

2

3

/2z .

In the absence of a closed expression for the integrals in (2.21), we can expand

the denominator in power series and integrate term by term. Factoring out

+x 2

2

(a

)

from

each denominator,

we

obtain

& Zf/+aY F 0, =

2

y d</>

d(C ° S

')[(1_2a cos T)"3/2 -(l + 2a cos y

3/2 ]

jo

(2.23)

We where

a

=

+x 2

2

ax/(a

).

observe that in the expansion of the radicals only odd

powers of a cos 7 will appear:

[(l-2a cos 7)- 3/2 -(l + 2a cos = 7)" 3/2] 6a cos 7+35a 3 cos3 7+. . • (2.24)

Sect. 2.8

Boundary-Value Problems in Electrostatics: I

65

It is now necessary to integrate odd powers of cos y over d<f>' d(cos 0'):

rvr d(cOS 0') COS 7 = 7T cos

Jo

Jo

dp ^ d(cos

0')

3
cos

7

=

cos 0(3 - cos2 0)

J

J

(2.25)

If (2.24) and (2.25) are inserted into (2.23), the potential becomes

^(R TO7 J ^^ aV ^(I/>(x,0,m*\) = 3VaVx3(^x2^-jac2) o\ s0[l1+x 35

„(3-cos2

...

1

e)+-..J

„ -~
(2.26)

We note that only odd powers of cos appear, as required by the symmetry of

the

problem.

If

the

expansion

parameter

is

22
(a /x ),

rather

than

a2 ,

the

series

takes on the form:

^^[cos = <D(x, 0, <f>)

+ (§

3
cos

0-1

cos

0)

- • •]

(2.27)

For large values of x/a this expansion converges rapidly and so is a useful representation for the potential. Even for x/a =5, the second term in the series is only of the order of 2 per cent. It is easily verified that, for cos 0=1, expression (2.27) agrees with the expansion of (2.22) for the potential on the axis. [The particular choice of angular factors in (2.27) is dictated by the definitions of the
Legendre polynomials. The two factors are, in fact, Pi (cos 0) and P3 (cos 0), and
We the expansion of the potential is one in Legendre polynomials of odd order.
shall establish this in a systematic fashion in Section 3.3.]

2.8 Orthogonal Functions and Expansions

The representation of solutions of potential problems (or any mathematical
physics problem) by expansions in orthogonal functions forms a powerful
technique that can be used in a large class of problems. The particular orthogonal set chosen depends on the symmetries or near symmetries involved. To recall the general properties of orthogonal functions and expansions in terms of them, we consider an interval (a, b) in a variable £ with a set of real or
complex functions l/n (£), n= 1, 2, . . . , square integrable and orthogonal on the
interval (a, b). The orthogonality condition on the functions (/„(£) is expressed by

Ut(t)Um (Z)dt = 0,
J

m*n

(2.28)

We If n = m, the integral is nonzero.

assume that the functions are normalized

so that the integral is unity. Then the functions are said to be orthonormal, and

they satisfy

fV^) 1/^)^=8™

(2.29)

66

Classical Electrodynamics

Sect. 2.8

An arbitrary function /(£), square integrable on the interval (a, b), can be
expanded in a series of the orthonormal functions U„(£). If the number of terms
in the series is finite (say N),

f(&**t<lnUn(& . n= 1

(2.30)

then we can ask for the "best" choice of coefficients a„ so that we get the "best" representation of the function /(£). If "best" is defined as minimizing the mean
M square error N :

Mn= f I f{&- 1 "nUn (Z) \(%

Ja \

n=l

|

(2.31)

it is easy to show that the coefficients are given by

b
an=J U!(€)f(©de a

(2.32)

where the orthonormality condition (2.29) has been used. This is the standard

result for the coefficients in an orthonormal function expansion.

N If the number of terms

in series (2.30) is taken larger and larger, we

intuitively expect that our series representation of /(£) is "better" and "better." Our intuition will be correct provided the set of orthonormal functions is

complete, completeness being defined by the requirement that there exist a finite

N N>N M number such that for

the mean square error N can be made smaller

than any arbitrarily small positive quantity. Then the series representation

n= 1
with an given by (2.32) is said to converge in the mean to /(£). Physicists generally
leave the difficult job of proving completeness of a given set of functions to the mathematicians. All orthonormal sets of functions normally occurring in mathematical physics have been proved to be complete.
Series (2.33) can be rewritten with the explicit form (2.32) for the coefficients
a„:

«f)=f{i^(?)Un(&}m de

(2.34)

Since this represents any function /(£) on the interval (a, b), it is clear that the
sum of bilinear terms Ut(ff)Un(& must exist only in the neighborhood of = £
In fact, it must be true that

Iu!(?)U»(€) = 8(r-{)
n=l

(2.35)

This is the so-called completeness or closure relation. It is analogous to the

Sect. 2.8

Boundary-Value Problems in Electrostatics: I

67

orthonormality condition (2.29), except that the roles of the continuous variable £ and the discrete index n have been interchanged.
The most famous orthogonal functions are the sines and cosines, an expansion in terms of them being a Fourier series. If the interval in x is (-a/2, a/2), the
orthonormal functions are

l^> [2 . (2irmx\
Va S,n

[2 (2imxx\
Va COS l^-)

m m where

is an integer and for = the cosine function is 1/Va. The series

equivalent to (2.33) is customarily written in the form:

where

/(x) = Uo+| [Aw cos (^P)+Bm sin (^p)]

(2.36)

_

2 a/2

(

, . (2irmx\ .

(2.37)

If the interval spanned by the orthonormal set has more than one dimension,
formulas (2.28)-(2.33) have obvious generalizations. Suppose that the space is two dimensional, and that the variable £ ranges over the interval (a, b) while the variable -n has the interval (c, d). The orthonormal functions in each dimension
V are U„(£) and m(r)). Then the expansion of an arbitrary function /(£, -n) is

L Z U = f(fc t|)

anm n (£)Vm (r])

(2.38)

where

jNf ^ anm =

dT,U?(€) (T,)/(& T])

(2 - 39)

U If the interval (a, b) becomes infinite, the set of orthogonal functions n(&
may become a continuum of functions, rather than a denumerable set. Then the
Kronecker delta symbol in (2.29) becomes a Dirac delta function. An important

example is the Fourier integral. Start with the orthonormal set of complex

exponentials,

U ™ = -^e m (x)

i(2 x/a)

va

(2.40)

m = 0,

±1,

±2, . .

,

.

on

the

interval

(-a/2, a/2),

with

the

expansion:

where

t A ™ f(*)=4=

i(2
me

x/a)

va m =-=c

a/2
f(x') dx'

(2.41) (2.42)

68

Classical Electrodynamics

Sect. 2.9

Then let the interval become infinite (a-»o°), at the same time transforming

2irm

[ dm = -— f dk >

m

J-oo

Z7T J oc

(2.43)

A(k)

The resulting expansion, equivalent to (2.41), is the Fourier integral,

where

/(x)= vfef Mk)eikxdk

A(k) =

7W ik
e"

dx

^=J°°

(2.44) (2.45)

The orthogonality condition is

P ^-

i(k - k)x e

dx

=

8(k-k')

Z7T J-oo

while the completeness relation is

(2.46)

-M = dk ik(x~x) e

8(x-x')

Z7T J-oo

(2.47)

We These last integrals serve as convenient representations of a delta function.
note in (2.44)-(2.47) the complete equivalence of the two continuous variables x and k.

2.9 Separation of Variables, Laplace Equation in Rectangular Coordinates

The partial differential equations of mathematical physics are often solved

conveniently by a method called separation of variables. In the process, one often generates orthogonal sets of functions which are useful in their own right. Equations involving the three-dimensional Laplacian operator are known to be

separable in eleven different coordinate systems (see Morse and Feshbach, pp.

We — 509, 655).

will discuss only three of these in any detail rectangular,

— spherical, and cylindrical and will begin with the simplest, rectangular coordi-

nates.

The Laplace equation in rectangular coordinates is

Sect. 2.9

Boundary-Value Problems in Electrostatics: I

69

A solution of this partial differential equation can be found in terms of three
ordinary differential equations, all of the same form, by the assumption that the potential can be represented by a product of three functions, one for each
coordinate:

d>(x,y,z) = X(x)Y(y)Z(z)

(2.49)

_L<q ^ £Z_ Substitution into (2.48) and division of the result by (2.49) yields

X(x)

dx 2

+

1
Y(y)

dy 2

1
Z(z) dz'

(2.50)

where total derivatives have replaced partial derivatives, since each term

involves a function of one variable only. If (2.50) is to hold for arbitrary values of the independent coordinates, each of the three terms must be separately

constant:

X 1

d

2

a 2=~OL

2

X dx

where

l^=-fl2
Ydy 2 * ZIcdfzZ72=72

a2

+

2
p

=

2
7

(2.51)

If

we

arbitrarily

choose

a2

and

2 |3

to

be

positive,

then

the

solutions

of

the

three

ordinary differential equations (2.51) are exp(±iax), exp(±i/3y),

exp

(±Va 2

2
+|3 z).

The

potential

(2.49)

can

thus

be

built

up

from

the

product

solutions:

= e**V *to
<S> c

,/=J * IIz

(2.52)

At this stage a and |3 are completely arbitrary. Consequently (2.52), by linear
superposition, represents a very large class of solutions to the Laplace equation.
To determine a and |3 it is necessary to impose specific boundary conditions on the potential. As an example, consider a rectangular box, located as shown in
Fig. 2.9, with dimensions (a, b, c) in the (x, y, z) directions. All surfaces of the
box are kept at zero potential, except the surface = z c, which is at a potential
V(x, y). It is required to find the potential everywhere inside the box. Starting
with the requirement that = <I> for x = 0, y = 0, z = 0, it is easy to see that the
required forms of X, Y, Z are

X=sin ax Y=sin (3y

Z = sinh (Va 2 +02 z)

70

Classical Electrodynamics

f = V(x,y)

$ = o-

f= =6

Sect. 2.9

v# =

Fig. 2.9 Hollow, rectangular box with five sides at zero potential, while the sixth (z = c)
has the specified potential <J>= V(x, y).

In order that 4> = at x = a and y = b, it is necessary that aa = nir and |3b = mix.
With the definitions,

= M77
Oin

T 0~ =

(2.54)

2+ 2
b

O we can write the partial potential nm , satisfying all the boundary conditions
except one,

3> nm = sin (anx) sin (0m y) sinh (ynm z)

(2.55)

The potential can be expanded in terms of these <£> nm with initially arbitrary coefficients (to be chosen to satisfy the final boundary condition):

£ A = <£(x, y, z)

«m sin («**) sin (Pmy) sinh (7nmZ)

n,m=l

There remains only the boundary condition <$= V(x, y) at z = c:

(2.56)

X A V(x, y) =

«m sin (a„x) sin (|3m y) sinh (y„m c)

(2.57)

This is just a double Fourier series for the function V(x, y). Consequently the
A coefficients nm are given by:

ab sinh (ynm c) j^dxj^dy V(x, y) sin (anx) sin (/^y)

(2.58)

Sect. 2.10

Boundary-Value Problems in Electrostatics: I

71

If the rectangular box has potentials different from zero on all six sides, the required solution for the potential inside the box can be obtained by a linear superposition of six solutions, one for each side, equivalent to (2.56) and (2.58). The problem of the solution of the Poisson equation, that is, the potential inside the box with a charge distribution inside, as well as prescribed boundary conditions on the surface, requires the construction of the appropriate Green function, according to (1.43) and (1.44). Discussion of this topic will be deferred
until we have treated the Laplace equation in spherical and cylindrical coordinates. For the moment, we merely note that solution (2.56) and (2.58) is equivalent to the surface integral in the Green function solution (1.44).

2.10 A Two-Dimensional Potential Problem, Summation
of a Fourier Series

We now consider briefly the solution by separation of variables of the two-
dimensional Laplace equation in Cartesian coordinates. By two-dimensional problems we mean those in which the potential can be assumed to be
independent of one of the coordinates, say, z. This is usually only an approxima-
tion, but may hold true to high accuracy, as in a long uniform transmission line.
If the potential is independent of z, the basic solutions of the previous section reduce to the products
e

where a is any real or complex constant. The imposition of boundary conditions

on the potential will determine what values of a are permitted and the form of

the linear superposition of different solutions required.
A simple problem that can be used to demonstrate the separation of variables

technique and also establish connection with the use of complex variables is

indicated in Fig. 2.10. The potential in the region, 0<x<a, y^O, is desired,

V subject to the boundary conditions that 3> = at x = and x = a, while <!>= at
y = 0for0<x<a and <£—»0 for large y. Inspection of the basic solutions shows
that a is real and that, in order to have the potential vanish at x = and x = a for

all y and as y—

the

proper

linear

combinations

are

ay
e~

sin

(ax)

with

a

=

mr/a.

The linear combination of solutions satisfying the boundary conditions on three

of the four boundary surfaces is thus

Z A y) =

* ex P (-niry/a) sin (mrx/a)

n=l

(2.59)

V The coefficients A„ are determined by the requirement that 3> = for y = 0,
0<x<a. As discussed in Section 2.8, the Fourier coefficients are

(2.60)

72

Classical Electrodynamics

= <t>

*=

$ = o-

Sect. 2.10

y/a = 0.5

= ;v/a 0.1
3.

Fig. 2.10 Two-dimensional potential problem.

With 4>(x, 0)= V, one finds

A= 4Vfl 7rnl0

for n odd for n even

The potential <I>(x, y) is therefore determined to be
— h ®(x>y) = 4V v —1 exp (-mry/a) sin (mrx/a) 77 noddH

(2.61)

For small values of y many terms in the series are necessary to give an accurate
approximation, but for y>alir it is evident that only the first few terms are
appreciable. The potential rapidly approaches its asymptotic form given by the
first term,

—4V <£(x, y)—>

exp (-Try/a) sin (-rrx/a)

(2.62)

Paranthetically, we remark that this general behavior is characteristic of all
boundary-value problems of this type, independently of whether <I>(x, 0) is a constant or not, provided the first term in the series is nonvanishing. The