Classical Electrodynamics - Ctrl + F [3 ed.]

Citation preview

‘vecmr 1* ermiilas a (b>-l CO -i('0 >-i('0 CD ted in Chapter 7. The discussion o """c3wave guides and cavities in Chapter 8 is developed for systems of arbitrary cross section, and the problems of attenuation in guides and the Q of j-A -

r-l~

xi

X11

Preface to the First Edition

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 electricitv and magnetism, I have presented an improved, although still approxK

::§Q3 te, t :5" (D CD ry of diffi1 no ('5 C" ‘:5 CT‘Q"sed on vector rather than scalar Green’s theorems. The subject of magnetohydrodynamics and plasmas receives increasingly more attention from physicists and astrophyS (0 Ln Chapte(>-1 CD >-:(0 "C3 ""t (0 U0 (0 C: OD Q0 02C. l-1 vey of this complex field with an introduction to the main physical ideas involved. Q-+p-.1-

L:-A n

1-A -

h-I 1

s J

l'U1b§U11 '(111Ll .l_/(1lJ1flUU 1..'4L1L[C1L1L)11b

_zj_1.j_.l

\oE>o'-1

1.10

l—* l-—* l—* l-—* l\) I-—*

1.13

24

co

.|I>.

Green’s Theorem 35 Uniqueness of the Solution with Dirichlet or Boundary Conditions 37 Formal Solution of Electrostatic Boundary-Value Problem with Green Function 38 Electrostatic Potential Energy and Energy Density; Capacitance Variational Approach to the Solution of the Laplace and Poisson Equations 43 Relaxation Method for Two-Dimensional Electrostatic Problems References and Suggested Reading 50 Problems 50

Chapter 2 I B0undary- Value Problems in Electrostatics: I 2.1 2.2 2.3 2.4 2.5 2.6 2.7

40

47

57

Method of Images 57 Point Charge in the Presence of a Grounded Conducting Sphere 58 Point Charge in the Presence of a Charged, Insulated, Conducting Sphere 60 Point Charge Near a Conducting Sphere at Fixed Potential 61 Conducting Sphere in a Uniform Electric Field by Method of Images 62 Green Function for the Sphere; General Solution for the Potential 64 Conducting Sphere with Hemispheres at Different Potentials 65 XV

XVI

Contents

2.8 2.9

Orthogonal Functions and Expansions 67 Separation of Variables; Laplace Equation in Rectangular Coordinates 70 A Two-Dimensional Potential Problem; Summation of Fourier Series 72 Fields and Charge Densities in Two-Dimensional Corners and Along Edges 75 Introduction to Finite Element Analysis for Electrostatics References and Suggested Reading 84 Problems 85

2.10 2.11 2.12

Chapter 3 ! Boundary- Valu cs 3?c:§:~ 5?:§Ea

F‘.

ZS

79

E11 @--Iescs=7!=5E3S2 Fx E3: =1

Laplace Equation in Spherical Coordinates 95 Legendre Equation and Legendre Polynomials 96 Boundary-Value Problems with Azimuthal Symmetry 101 Behavior of Fields in a Conical Hole or Near a Sharp Point Associated Legendre Functions and the Spherical Harmonics

3.1 3.2 3.3 3.4 3.5

Y,,,,(l9, qfi) 3.6 3.7 3.8 3.9 /'\4f\

.5.1U

3.11 3.12 3.13

Addition Theorem for Spherical Harmonics 110 Laplace Equation in Cylindrical Coordinates; Bessel Functions Boundary-Value Problems in Cylindrical Coordinates 117 Expansion of Green Functions Spherical Coordinates 119 Solution of Potential Problems with the Spherical Green Function Expansion 112 Expansion of Green Functions in Cylindrical Coordinates 125 Eigenfunction Expansions for Green Functions 127 Mixed Boundary Conditions, Conducting Plane with a Circular Hole 129 References and Suggested Reading 135 Problems 135

Multipole Expansion

I45

45

Multipole Expansion o P-‘I1 1:?r,_,re Energy of Q" (3::rQ-"" ge Dlstliuuiluii P-

in an External Field 150 Elementary Treatment of Electrostatics with Ponderable Media Boundary-Value Problems with Dielectrics 154 Molecular Polarizability and Electric Susceptibility 159 Models for Electric Polarizability 162 Electrostatic Energy in Dielectric Media 165 References and Suggested Reading 169 Problems 169

4.3 4.4 4.5 4.6 4.7

Chapter 5 !

104

107

Chapter 4! Multipoles, Electrostatics of Macroscopic Media, uieiectrzcs 4.1 4.2

\:>Ln

PIg

L

~= 0% SR E3£3 $2 11iiii. ~55 [*4 L ‘/‘I

S

I’

-*-—\>P° E9 —\N 41> Lowest Order Relativistic Corrections to the Lagrangian for Interacting Chm Charged Particles: The Darwin Lagrangian 596 12.7 Lagrangian for the Electromagnetic Field 598 12.8 Proca Lagrangian; Photon Mass Effects 600 12.9 Effective “Photon” Mass in Superconductivity; London Penetration Depth 603 12.10 Canonical and Symmetric Stress Tensors; Conservation Laws 605 12.11 Solution of the Wave Equation in Covariant Form; Invariant Green Functions 612 References and Suggested Reading Problems 61 7

Chapter A1 Q0 “ =. Cherenkov anQ5 I-/w

y.._x $.13 y.._x

_L\/A-r‘

2»2-» 2»s».»QU.)\)-I3LI“!

615

stuns, Energy LOSS, ana Scattering o_‘I-s C. =3 Q rged Farticles, ' ~ ' U 41/1 S S§ in M Cb n Radiation UL“!

_°_~_°____.1

5:.s

.l.

II!

!§.

I'7_- _____-

T

_

I n

‘I

"rm

I

Q‘.

1'\(‘1‘FQ_1*" -:1»\ fF1\111r\-1»-v\1-\ rfi/'\]1:/-1:1-\4»\ 1§,.4.---,.,._.. TTA;._____ T__ __f_1_ .-¢ T1. H4’ 1 11.11611"fi-tr gy T1" 11a11D1.D1 111 k./ULl1U111U Culilbiull DCSLWCCII neavy lI1L1Ll611[ I’a1"[1Cl€

-I:

and Free Electron; Energy Loss in Hard Collisions 625 Energy Loss from Soft Collisions; Total Energy Lo O0 U0 cm[*0 \1 1J6I1S1[y Ettect in Collisional Energy Loss 631 Cherenkov Radiation 637 Elastic Scattering of Fast Charged Particles by Atoms 640

l-l--* —l—\l --\-—*

Mean Square Angle of Scattering; Angular Distribution of Multiple

13.7

Transition Radiation

Scattering

643

646

References and Suggested Reading Problems 655

654

Chapter 14 I Radiation by Moving Charges 14.1 14.2 14.3 14.4

661

Liénard—Wiechert Potentials and Fields for a Point Charge 661 Total Power Radiated by an Accelerated Charge: Larmor’s Formula and Its Relativistic Generalization 665 Angular Distribution of Radiation Emitted by an Accelerated Charge 668 Radiation Emitted by a Charge in Arbitrary, Extremely Relativistic Motion 671 lJ1SI1‘1[)L1I10l1 in rrequency and Angle of Energy Radiate by Accelerated Charges; Basic Results 673 “O

l. _.x HZ; U”!

‘I

A

01

A0

0

1%

1

4

1

qt?‘

w»

4:

1

Contents

14.6 14.7 14.8

Frequency Spectrum of Radiation Emitted by a Relativistic Charged Particle in Instantaneously Circular Motion 676 Undulators and Wigglers for Synchrotron Light Sources 683 Thomson Scattering of Radiation 694 7),.L‘,.-.,.-..,.,.,, ,..-- ,1 C‘ . . A A ,.,,4,.J

T),.,..,J.'..,.,.

I\C_/CICILLCA LHLLL i)l/LSSCALCLL I\CLl(/Llllg

Problems

cs\::\1

698

Chapter 15 I Bremsstrahlung, Method of Virtual Quanta, Radiative Beta Processes 15.1 15.2 15.3 15.4 15.5 15.6 15.7

708

Radiation Emitted During Collisions 709 Bremsstrahlung in Coulomb Collisions 714 Screening Effects; Relativistic Radiative Energy Loss 721 Weizsacker—Williams Method of Virtual Quanta 724 Bremsstrahlung as the Scattering of Virtual Quanta 729 Radiation Emitted During Beta Decay 730 Radiation Emitted During Orbital Electron Capture: Disappearance of Charge and Magnetic Moment 732 References and Suggested Reading Problems 737

737

Chapter 16 I Radiation Damping, Classical Models of Charged Particles

745

16.1 16.2 16.3 16.4 16.5

Introductory Considerations 745 Radiative Reaction Force from Conservation of Energy Abraham-Lorentz Evaluation of the Self-Force 750 Relativistic Covariance; Stability and Poincare Stresses Covariant Definitions of Electromagnetic Energy

16.6

and lvlomentum /5/ Covariant Stable Charged Particle

16.7 16.8

Level Breadth and Level Shift of a Radiating Oscillator Scattering and Absorption of Radiation by an Oscillator

I\I

4

747 755

F-II-P-I

References and Suggested Reading Problems 769

759

763 766

768

Appendix on Units and Dimensions

775

Units and Dimensions, Basic Units and Derived Units Electromagnetic Units and Equations 777 ‘7I')1"‘;f\I1O Qtrofnnqo r\'F Dlonfrnnqqnnofifl T Tfiifc V CI.1.1\JLlD

l\J Lnl-—-l*>-

L)-yDL\./.l1lD

\JL

1_JL\/\-/L1.\).l11(»l51l\./L1.\./

kJll1l.-O

\1

\0 l— Units Conversion of Equations and Amounts Betwee D\1CLM and Gaussian Units 782

Bibliography In .1 .1-’ IIIUCX

XXi

-JVD p-A

785

775

1

_

*;:< -: CD (D ion of the agr CD CD E3 CD :3 t of the standard model with experiment in atomic and particle physics where relativistic quantum mechanics rules is truly astounding. Classical mechanics and classical electrodynamics served as progenitors of our current understanding, and still play j-A . O0

important roles in practical life and at the research frontier.

This book is self-contained in that, though some mathematical background (vector calculus, differential equations) is assumed, the subject of electrodynam-

ics 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, and 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 its domain of validity, and to elucidate som GD1 CD fthe 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 electro E3 so gnetism for the first time will not follow all the arguments or see their significance. For others, however, this introduction will serve as a springboard into the later parts of the book, beyond Chapter 5, and will remind them of how the subject stands as an experimental science.

I1

Maxwell Equations in Vacuum, Fields, and Sources The equations governing electromagnetic phenomena are the Maxwell

equations,

v.[):p V >< H — @ = J 31‘

(I.1a)

V>25: cu 5;. "CD-' 7 1'-PET‘. and current density,

C-4

p-i -

("F

p-A .

ri Pd

:'$

§1A jj

ea

'-iiin1:D 3 (ID ‘Z: Ed i-CD: IICTIIT

(D c. QD 1-‘F CD ::s O’)

Q3 .

p-A .

|—.|A O’)

CID

hi 0

Ill

::rco ca CD 1:: :22".1:: ;:

0"‘?

j-i .

,42?FD : E 55 "-5 n: IIII

hi I

("F

‘ l: I-1 0:-3 (T< B)

(I.3)

which gives the force acting on a point charge q in the presence of electromagnetic fields. These equations have been written in SI units, the system of electromagnetic units used in the first 10 chapters of 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. Essential to electrodynamics is the speed of light in vacuum, given in SI units by c = (/.t0e0)"1’2. As discussed in the

Appendix, the meter is now defined in terms of the second (based on a hyperfine transition in cesium-133) and the speed of light (c = 299 792 458 m/s, exactly). These definitions assume that the speed of light is a universal constant, consistent with evidence (see Section 11.2.C) indicating that to a high accuracy the speed

of light in vacuum is independent of frequency from very low frequencies to at least 1/ = 1084 Hz (4 GeV photons). For most practical purposes we can approximat_ _ = 3 >< 108 m/s or to be considerably more accurate, c = 2.998 >< 108 m/s. l-3 re E? D cu cu $20 :3 CD._ 'E3 $20 93(D tic fields E and B in (1.1) ‘1:cu l-: cu CD ,.3.0‘ ~ E3$20 :3 c: P"!-.~ ca.I (D- 00 (D.l-3 to r-F' caI E: E: 0'-F(D :1. I ca l"h l-3 (D $20 ca P-1C < 10_19 C

Sect. 1.2 Inverse Square Law or the Mass of the Photon

5

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 102°). The experiments are discussed in Section 11.9, where the question of the CD>-: CDC3 N ' C3 -: ' soC3 CD CDCD>-:1 CDCT‘ no>-: (IQ CD' so lso tr CDso (D CD. The discreteness of electric charge does not need to be considered in most macroscopic applications. A 1-microfarad capacitor at a potential of 150 volts, for example, has a total of 1015 elementary charges on each electrode. A few t—l

F-F

pd

pd

pd

DO

P-

Q‘-F

thousand electrons more or less would not be noticed. A current of 1 microam-

pere corresponds to 6.2 X 1011 elementary charges per 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 the Maxwell equations (I.1a). 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 6.11, 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 (I.1a) emerge. In this sense it is somewhat a matter of con-

vention 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 de-

scribed in Chapter 6.

I2

Inverse Square Law or the Mass of the Photon TLA

/J-:n-I-_r\1-\r-sr\

111U Ll1Dl.(111L/U UUPUIIUUIILU

A CP-‘h G-‘FCr‘ (T

r\1r\n4—1nr\n4—nI-zn

Inxvv

UIULLIUDLGLIL 1Cl.\/V

r\~p

'pr\1»~r\r\

UL LULLU

wvrnn

nl~\r\w111»1

r~|11r\1»-\+-:4-r\

WCID D11UW11 L1Ll(111L1Ld.'

tively 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 (I.1b). The original experiments had an accuracy of only a few percent and, furthermore, were at a laboratory length scale. Experi1

11'

1

"

‘

1‘.

_‘l

‘___-

_1f£L'______4_.-__f-____

_.C

_'

_1___.._

‘I_____

.____

II1€I1IS HI I11g[l€I' pI'€ClS1OH HHQ 1l'lVO1V1Ilg (.l1ll€I€Ill I€glII1CS U1 SIZ6 H21V6 [)6CI1 pfif

formed over the years. It is now customary to quote the tests of the inverse square law in one of two ways: /_\

A

. A1__A_ .1-.1:_..__.

- ..:__. __ 4 /_-2,+g ___ .1

'

_.___.t_

1 __n1__-

___ 1:-_-:2 1:__-

_

(21)

1‘iSSLlII1€ Ulkll II1€ 1OI'L€ VEll'1€S 21$ I/I"

21I1L1 qLlULC 21 V211Ll6 UI 11II11L IUI 6.

(b)

Assume that the electrostatic potential has the “Yukawa” form (see Section 12.8), r_1e_'” and quote a value or limit for /.L or ,uf1. Since fl. = myc/h, ~: here my . CZ f-F:1- (‘D CDC0 C2Ed :3’(‘D ca. 3 J--L CD CD CD of the photon, the test of the inverse square ‘EN

law is sometimes phrased in terms of an upper limit on my. Laboratory experiments usually give e and perhaps /.t or my; geomagnetic experiments

give it or my. *Quarks have charges % and —% in these units, but are never (so far) seen individually.

-I-I1"IQl'\IIIIl‘\“":l‘\I'I

Q'\I4III

Q I I K i 1 A "1

lllllU\lllLllUll auu lJlllVC_y

/Z‘

A

fl\

-

%/ bi 5// ../ \./12 //A ’%\@:(~\ /"If;

"W 7%//Z1‘$%¢-7)fi'__'%¢/I/' K‘3%\ S’§"_/< 1075. Two other noteworthy laboratory experiments based on Gauss’s law are those of Plimpton and Lawton,1 which gave |e| < 2 >< 10-9, and the recent one of Williams, Faller, and Hill.§ A schematic drawing of the apparatus of the latter experiment is shown in Fig. L2. Though not a static experiment (v = 4 >< 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 10-12 V could have been detected. Their null result, when interpreted by means of the Proca equations (Section 12.8), gives a limit of e = (2.7 i 3.1) >< 10-16. Measurements of the earth’s magnetic field, both on the surface and out from the surface by satellite observation, permit the best direct limits to be set on e or equivalently the photon mass my. 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 12.15), namely,

my < 4 >< 10"“ kg or --—1 \

IL

Z

p-x CD

O0

W.

v->-|--

For comparison, the electron mass is me = 9.1 >< 10-11 kg. The laboratory experiment of Williams, Faller, and Hill can be interpreted as setting a limit my < 1.6 >< 10-50 kg, only a factor of 4 poorer than the geomagnetic limit. 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 = mycz, suggests that the photon mass must satisfy an inequality, my < hvo/c2, where 1/0 is any electromagnetic resonant frequency. The lowest Schumann resonance has 1/O = 8 Hz. From this we calculate my < 6 >< 10-511 kg, a very small value only one order of magnitude above the best limit. While this argument has crude validity, more careful consideration (see Section 12.8 and the references given there) shows that the limit is roughly (R/H)1/2 ~ 10 times larger, R = 6400 km being the radius of the earth, and H = 60 km being the height of the iono*H. Cavendish, Electrical Researches, ed. J . C. Maxwell, Cambridge University Press, Cambridge (1879), pp. 104—113. lIbid., see note 19. 1S. J. Plimpton and W. E. Lawton, Phys. Rev. 50, 1066 (1936). §E. R. Williams, J. E. Faller, and H. A. Hill, Phys. Rev. Lett. 26, 721 (1971).

8

Introduction and Survey

|_

Transmitter

I

|

I Cr Y stal and oven

.._ m. in __ Comparator and buffer amp

7

I F’"='snirtsiiiiearly =:5.?.SIri"err@*9£J.iL{I-

\r%-I-Lmkl

\r\.I

%§IIr§@LI%

16

Introduction and Survey

the domain E5’ CT so nge a C3 ca. E? cu =2 cu En C5 Cl. till; .l leading to the saturation of the 'Z at' E =3agnetization. Removal o" ' I ' ‘ 'eaves E: E? a considerable fraction of the moments still al' .. (T)1-r53-ca. E3 ~=-..an CQ C3 D :3 ‘R *-I5;--L 1-+ Q’to cm E3. CIQ '13 P-I“.hrE:' UT? r-1 Iat65"‘-1 CDE2 CD CD. ' ’::n CDd to, for example, 0

p-A -

P|-A - ,-4"

pd

§j

3 F-3

=2;>=

£2“at D 5 2"

e charge density p is singular at the interface so as

I-4 P-'11 :1-,.5"

to produce an idealized surface charge density o-, then the integral on the right

in (1.13) is I‘

Jlpd3X=(TL\(1

v

Thus the normal components of D and B on either side of the boundary surface are related according to (D2 — D1) - n = 0-

(1.17)

(B2 — B1) - n = 0

(1.18)

In words, we say that the normal component of B is continuous and the discontinuity of the normal component of D at any point is equal to 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 short arms of the contour C in Fig. 1.4 are of negligible length and each long arm is parallel to the surface and has length Al, then the left-hand integral of (1.16) is < (H2 — H1) = K

(1.20)

In (1.20) it is understood that the surface current K has only components parallel to the surface at every point. The tangential component of E across an interface is continuous, while the tangential component of H is discontinuous by an amount

whose magnitude is equal to the magnitude of the surface current density and whose direction is parallel to K >< n. The discontinuity equations (1.17)—(I.20) are useful in solving the Maxwell

Sect. 1.6

Some Remarks on Idealizations in Electromagnetism

19

equations in different regions and then connecting the solutions to obtain the fields throughout all space.

I 6 Some Remarks on Idealizations in Electromagnetism In the preceding 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 electromagnetism. In electrostatics, for example, we speak of holding objects at a fixed potential with respect to some zero of potential usually called “ground.” The relations 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 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 l

T

P vrl P11

j_J\/J \.l\/LL

i ‘SI re

JLLLLJ

-

_

Q hie Q '1nee 1.11-L) LJO]\l.L\/\/L)

an

-

q

¢

u

4-

q

1

A

q

Q

q

1

F r-hnroe thin xxrireq no Pnnrlnntnrq nnrl qnqnenrlino \/LLI.-|,.l.b\/, DLLLLL VV.L-I-\/"L; you \/'\./L-l\-I-\-4\/"I-w\1-1'»-1, \-4-.lA\-1 'u\-4urI\I'-1-l\-IA-nab

OL

1

I

1

the objects in the room, Cavendish measured the amounts of cnarge on Cyll11Cl6I'S, discs, etc., held at fixed potential and compared them to the charge on a sphere (the same sphere shown in Fig. I.1) at the same potential. His values of capaci-

tance, 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 th ,.$ C3 1-1 -1 E3 CD no‘C3 C) E3 an intain potentials, comparison methods using beams of charged particles intermittently, for example. p-1 - CO

DO

O0

1'-F

("F

("F

("F

U’)

(“F

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

nected 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 the surface that is macroscopically small, but , 53'cw “§CD E-0 8"SE'rsally large. An exampl (D CT‘ GD "C3 CID {:5 etration 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 called the skin depth, and that for high enough frequencies and good enougn conductivities 5 can be macroscopically very small. It is then appropriate to inr the direction perp FD C3 CD... -1 to the surface to (TO >-e ("D E3" ("D -F P-1 CD =2 S-F,3"(D definitions above it is evider e-+ Q-+Cr‘ $1 P-

Q .1.

JF

I-

“-5 /'\ >< \-./ C50 /'“\ >
.>
\.-/

L11 \./

x — x’ ,13vr r\ _________ }

L11

‘F

fli

JD

-n

3 E? land, viewed as a function of x, is the negative The vector factor in the .. gradient of the scalar 1/ Ix — 9".(To '1i.. hi 0

X—X

,

|v_v'|3 |A 1\|

/

< Vi)? Z 0, for all 1/'1), (1.14) follows immediately from (1.15). Note that V >< E = O 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 worthwhile concentrating on the scalar function and giving it a name. Consequently we define the scalar potential (x) by the equation:

E = —V

(1.16)

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

by (X) = —1— I I-5-(3-‘%| d3x’ 477-50

(1.17)

"”

where the integration is over all charges in the universe, and (I) is arbitrary only fn flan cnvfcnnf flnof 0 nnnofonf no-n Ian orlriarl fn +]»\Q 1~;n-.|*\f_I"\r\1*\r1 cw-irlci n-F /1 L\J

L].].\/

\/AL\/1.lL

L11ClL

Cl

\./\_)1.ls)LCl1.lL

ball

lJ\/

Cl\.l\-l\/\-1

LKJ

L11\/

.|.].511L

11Cl11\.l

D1\.1\/

\J.l.

'l'7\

\.L-_|_ I)-

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 I1‘ _- "IT 1‘ —- qu

so that the work done in moving the charge from A to B is B

B

W:-—) F-dl=—q) E-dl A

(1.18)

A

The minus sign appears because we are calculatin CT‘ e work done on the charge Clgdlllbl. L116 Cl‘./LIUII U1 L116 IICIU. VVILII UC11I11L1OIl kl ;_i9°o\ [I16 WOIK (¢ClI1 DC WHLLCI1 ,_._,_:__._,4.

4.1-,-.

,-...=4.:,._..

ll

4.1-...

£1.-.1,-1

“TfA_'I_

.J._.C§__I1_I.___

/'1

A_'I__

______1_

_"_ __

1- _

____-:_A..A-___

\_/ (-‘F

FB_..

__

(B __

.-

_ .

W = q JA | vq> - at = q JA | a = q(,, — A) B

§1.1/ < E = 0.

‘I

1

0

I

Surface uistrwunons o_] Lnarges ana uzpoies ana Discontinuities in the Electric Field and Potential (7

P.

.

I\'_.A._§I_

.

J9 _

._ _.

_p IT’-

-. ___

___

____ -1

I\§.__

_'__

_____ -1

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 coulombs per square meter) and electric fields E1 and E2 on either side of the surface, as shown in Fig. 1.4, then Gauss’s tells us

diately that (E2, — E1) - n = 0'/60

(1.22)

This does not determine E1 and E2 unless there are no other sources of field and the geometry and form of 0 are especially simple. All that (1.22) says is that there Side 2

1:25???i?§5§?i?§?€?§5;:.-.

i--1*

7

S‘

"

Fivure 1.4 Discoininiiiiy 111 the normal layer of charge.

(R IQ

(3CT‘ so "C3

("Y-C D

1-1

1

Introduction to Electrostatics- OD r-1

is a discontinuity of o/so in the normal component of electric field in crossing a (3

0Q2 F at

61> 15ith a surface-cha >-1 cm co density o-, the crossing being made in the direction

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 nece C0 CD :1:1-1: ‘-: "C3 so C3‘ 1:ith negli (IQ CT‘ (D (D E5C1. (JO and one side on either side of the boundary. An expression for the potential (hence the field, by differentiation) at any point in space (not just at the surface) can be obtained from (1.17) by replacing f"+

|-11

p-A .

[-11

p d3x by cr

(I)(X) = L I

4'2re0 s|x—x|

air

(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 on a surface S. A 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 str>- (D C3 gth D(x) is formed by letting S’ approach infinitesimally close to S while the surface-charge density o(x) becomes infinite in such a manner that the product of o(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)+()

rdCr‘ e direction c> |"'h t-'1'“Cr‘ o> c>_ “ole moment C" I‘-‘H Q-‘P-~,:r e layer is norm“sx "C '

§ 0

._

ji

C

P‘?-

cc Cl uriace is anu

_.L

(“P-mi P-I

,_-___.B.__._

U

____1

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 '13 (‘D >-: I-hCD :1 E3.C: g mathematically the limiting pr CD (D (D U’) U’) CL(D Ch (‘D >-1 bed in Words above on surface-density expression (1.23). The first way is perhaps simpler, but the second gives useful practice in vector calculus. Consequently we proceed with

5%"e‘

)-ii .

H.

:R\

\\, \,§€L)

/T\\

d(x) _

\\ \\ \ ‘S S’

Figure 1.5 Limiting process involved in creating a dipole layer.

Sect. 1.6

Surface Distributions of Charges and Dipoles

33

I I1

X r X / -, X aw d(X') —4"___¢* t

'7 ,4 _ i '_,—’ $0 ,

0

dan

1)

S

'

S’ Figure 1.6

E= C3 $3‘ P. E5‘ cn Fl‘ " -(D” d

e unit normal t ti5ElEal(IQ9': I--\A ag”,_,.1:;€,=="E;3 tential due to t 2iiiE.

Si 5:it-»éntiOP-h,_‘CDQ..

)-i

Ocd c

_ (x) —-

1

o(x')

g

L

, da I 41760 sIx—xI

__

1

_i

Dipole-layer geometry.

my

5° Q; rface U31 ca.

(D

o(x’)

_

‘B1:‘ CD =-1 |"'b1-1 9 :3

2.," (D I

Wfifi

,

I __

I

4’7TE0 s'Ix—-x +ndI

da I!

For small d we can expand Ix — x’ + ndI‘1. Consider the general expression Ix + aI_1, where IaI O the potential becomes (x) = 2% IS D(x’)n - V’(|—;%—’T|) da’

In passing note with dipole momen

(1.24)

e integrand in (1.24) is the potential of a point dipole 'a=s = n D da’. The potential at x caused by a dipole p at x’ is 4-\

Q-‘F

CT‘

1-?-

..

iN

1 11-(X-X’) V

47760

Ix—-XIr3

/‘\

)_.\ [\)Q11 \_./

Equation (1.24) has a simple geometrical interpretation. We note that 1

I

ii-v’ -Z da’: |X*X'I

°°S6d,“,- do |X—X|

where dfl is the element of solid angle subtended at the observation point by the area element da’, as indicated in Fig. 1.7. Note that dfl has a positive sign if 6 is

sx

\

,./\V> \

__

4n /Q/ /

, //,i/ /\\ da’

E gQ/ \\\ Z - 'T/ \\\ /4’ 5' X,‘ Pf‘ F? ...

Figure 1.7

The potential at P due to the

dipole layer D on the area element da’ is just the negative product of D and the solid angle element dfl subtended by da’ at P.

34

Chapter 1

Introduction to Electrostatics—SI

an acute angle (i.e., when the observation point views the “inner” side of the dipole layer). The potential can be written: 1

*E>*/*\ >4 \_,/ I — 4_7T?O Jfs D(X’) dfi

For a constant surface-dipole-moment density D, the potential is just the product of the moment divided by 41re0 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 D/so in crossing from the inner to the outer side, being —D/260 on the inner side and +D/260 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: (D2

_‘

(D1

I

D/6Q

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; it 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 : p/€()

and

VxE=O

(mo

the latter equation being equivalent to the statement that E is the gradient of a scalar function, the scalar potential Q): E=—V@

/"\ P-A

l-A

CJ\ \../

Equations (1.13) and (1.16) can be combined into one partial oiiieientiai i

equation for the single function (x): V2 = —p/60

(1.28)

This equation is called the Poisson equation. In regions of space that lack a charge density, the scalar potential satisfies the Laplace equation:

var =()

(129)

Sect. 1.8

Green’s Theorem

35

We already have a solution for the scalar potential in expression (1.17): 1 /"\

>-i \-/

_ _

I‘

/X_r\

I

4¢re(~,

(__t

)

6'/13x!

I < ..—x|

/'\

l-X

l-X

~.1\-/

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 liiniting procedure. Define the “a-potential” J.

/

\

1

(Pa(X) Dy

1

[

p(X')

,

q)a(X) — 4'/T60 -' ‘\/(X ~ X’)'2+ a2 d3x The actual potential (1.17) is then the limit of the “a-potential” as a e O. Taking the Laplacian of the “a-potential” gives 1 1 V2(I)a(X) = -i I p(X')V2(—"'—'"i) d3x’ 47760 \/ r2 + a2

1

:__ _

I

41760

3a2 __W_,,W__

I

d3

p(X )l(r2 + a2)5/2]

(130) I

X

where r == Ix — x’ The square-bracketed expression is the negative Laplacian of 1/'\/ r2 + a2. It is well-behaved everywhere for nonvanishing a, but as a tends to zero it becomes infinite at r = 0 and vanishes for r at O. It has a volume integral equal to 41¢ for arbitrary a. For the purposes of integration, divide space into two regions by a sphere of fixed radius R centered on x. Choose F such that p(x') changes little over the interior of the sphere, and imagine a much smaller than R and tending toward zero. If p(x') is such that (1.17) exists, the contribution to the integral (1.30) from the exterior of the sphere will vanish like a2 as a —> O. 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 4

/-R

v 2 (1)a(X) =_iJ 60 O

A

U,

0

|_

+5”“a,)5,,

o

Lp(X) +

“I

C 6 v2p

+

J r 2 dl’ + 0(a)2

T\l..,\,,4- 10.4-/.~..,.4-1,... --I,.1.-1,, .|.J11 UL/L 1111.551 CILIUII YICIUD

1

V2a(x) = —— p(x) (1 + O(a2/R2)) + O(a2, azlog a) Vzp + E0 ' ' ' 1-i D ' S /1 ")Q\ " '-r e limit a —> 0, we obtain tiie i oisson equation (i LO)

I-4 I- I-

P"?-

The singular nature of the Laplacian of 1/r can be exhibited formally in terms of a Dirac delta function. Since V2(1/r) = O for r # O and its volume integral is

-411", we can write the formal equation, V2(1/r) = —41r8(x) or, more generally, /

.

\

v (IX _ X,l) — 1

in OD

I‘V____ ___- I_

\

___

41To(x A

c\/

x) ;\

(1.31) /4

/\4\

FIT]. _ ________

l1l'€€ll'S 1 IlCUl'€lTl

If electrostatic problems always involved localized discrete or continuous distributions of charge with no boundary surfaces, the general solution ( 1.17) would

36

Chapter 1

Introduction to Electrostatics--SI

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 simnle cases /e " "2 ethod of *'na"es‘ To handle the boundary conditions it is necessary to develop some new mathematical tools, namely, the identities or the CD>-i (D E3 on CD. CI("D to George Green (1824). These follow as simple applications of the divergence theorem. The divergence theorem: 1

11.ltJ1\/ \./

DU

\

.5.’

I- P- DL11

L1

.l. 11

5

)-

[V-Ad3x=§A-nda v 5

applies to any well-behaved vector field A defined in the volume V bounded by the closed surface S. Let A = Q5 Vi/1, where qb and mp are arbitrary scalar fields. Now

v - -i 51> .> 23 C!) normal derivative at th G!) c:>-i |""bSD 0 c1> CD. l"‘__l

/A\

8n

A l

R 8n

1

r

QQ

|;

rt;E. point x lies within the volume v, we obtain: T7

P-4 |'-15

(D

1 ’ 1
O5

1"-1

i

AI;

unique. Similarly, for Neumann boundary conditions, the solution is unique, .\.1.\...¢ 1?

°

4-

.\_.L'

_..\_.-

,.,J,]1ti- /\

,.,..,.,-.4-“flit

apait 110111 2111 il1“i11TipO1‘t€11'1t 'clI'U1i.I"c1I'y 'clULlI v'c uuiistai

From the right-hand side of (1.38) it is evident that there is also a unique O0 tion to a problem 15Fl‘ C3“ =2 ,:..>4 CD CL CT‘-1 “-1 I- =3‘CUCD/"\ "CZ? SD >-1 CD1-1 all of which may be at infinity, of course). On1\T J

1 10

]"\\T Ll)’ L/11 1\./1JJ.\/ L

/II/‘ ‘NTQ1‘I1’Y\')1’\1’\ ll I 1 "1 \/ 11111011111

]'\f\1‘I1"l(q¢)1‘YT Ll\J Ll1.1\.il-(.11 J

\/\J11\.l1 lJ.\J1.1D

f\1"l \J1.l

O Cl

P] \-/1

("F

Formal Solution of Electrostatic Boundary- Value _.-_I_I__.__

_..-_24.I_

I"V_.-___._

I1“..- _4__'__._

l'l'()Dl€lTl Wllll LII'€€l'l Plll'lClll)l'l

The solution o 1-1» CE. (T "U C‘isson or Laplace equation in a finite volume V with either Dirichlet or Neumann boundary conditions on the bounding surface S can be 1-‘

UULCl111\/Ll

1-\r1 1*v\Qn-r\n

Uy

r\'F f1YQA1fi’fi

111k/C1116 U1 \J1L/U11 D

f-1»\r\r\1~r'\1»v\

L11UU1U111 \1.JJ)

n1»\r-1

nr\_r\n11r\r1

Qwmnm

‘F111/\r\+~;r\1/\r1

61111.1 DU L¢Cl11ULl \J1UU11 1Ll11L/L1\J11D-

In obtaining result (1.36)—not a solution—we chose the function lb to be

El‘ 3?

— ll v'lI’ it Imflinn tlwp nntnntial nf Q unit nn I C3 (-+ Lb LI\/L.l..l& L.l1.\/ t/\JL\/.l.lL.I.LlL \JL (J \»l1..l.L\- tJ\J pd

1 \

V'2(

\|X - xv1 —

GO

CD :2.‘>-1 (3 5'0

CO SD

ticfvi LLLILJL C3

(ID

tr (‘D (‘D

1'-P

#2

atinn$2.. C-l\v.l\J1.l|

41r5(x — X’)

(1.31)

The function 1/Ix —- x’! is only one of a class of functions depending on the ___1-.:_11-1,.,__ __ _..__ ,1 _-/ _A__.1 ___,_11- .1 r*v_-_ ___ .C____ __1_-_____ ___1-:__1- ____1:__1~__ VclI1clU1C3b X cl11U X , dl1Ll 1/d11C3U LII"€€l’l _]l/ll’l(,llUl’li5, Wlllbll bdllbly

1>1

’) = —

42>

=1O0 /"\ >1

/_\

)_.\ U.) '1 1-\ \_/

T__ _ -__ -____1 1Il gCIlCIcl1,

~ X")

/—\ ji-A L») \CD \_,/

where

G(X ii’) = 7

li

+ F(X X) 7

(1.40)

with the function F satisfying the Laplace equation inside the volume V: V"2F(x, x’) = U

(1.41)

Sect 1 10

Formal Solution of Electrostatic Boundary-Value Problem with Green Function

39

In facing the problem of satisfying the prescribed boundary conditions on (l)

or 6(l)/6n, 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 (I) and i‘)’~i')i’i5‘n appear in the surface integral. It is at best an integral relation for (1). 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 it: = G(x, x’) and choose F(x, x’) to eliminate one or the other of the two surface integrals, obtaining a result that 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 nr lri iiave h ljttlie 5\/.l1\/ mfinpr a litv_ "on... J 1.AS i \'x"ll T111. he \J\/ qpgn LJ\/ ii immeriintelxr ...................,, thiq ..-... iQ ... nnt -I--l\-IV 1‘F*.(111i1‘F‘.(1 .-q....-..,

and G(x, x’) satisfies rather simple boundary conditions on S.

With Green’s theorem (1.35), qb = (I), (I: = G(x, x’), and the specified properties of G (1.39), it is simple to obtain the generalization of (1.36): (l)(x) = -A-gr; IV p(x’)G(x, x’) d3x' 1 6(1) 8G(x x’) + ——§ |:G(x x’) w —— (I)(x') 417s

(1.42)

da’

’

The freedom available in the definition of G (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’) = 0

for x’ on S

(1.43)

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

1

F

-

1 f

aG,,

417 Js

dn

~61 X \_./ == -+ | _o(x')GD(x, x’) d3x' — .— (P (I)(x') /"\

41760 Jv

’

_

'

(1.44)

For Neumann boundary conditions we must be more careful. The obvious choice of boundary condition on G(x, x’) seems to be 8GN Z(x x’)=0 8n’ ’

forx’onS

since that makes the second term in the surface integral in (1.42) vanish, as de-

sired. But an application of Gauss’s theorem to (1.39) shows that

3’ 3

iii

;-ee. 4.

4

4

Q

4

44

4

4

Q1

44-

Q

4

J5.

4

=1 4-

n

I-5-1

n

Consequently the simplest allowable boundary condition on UN 1S aG,M ,

..

477

A , (x, x’) = —-7 on S

-

.

.,

for x’ on S

/_4

.-.

(1.43)

where S is the total area of the boundary surface. Then the solution is 1 1 8(1) (I)(x) = ((I))S + if p(X')GN(X x’) d3x’ + —§ —, GN da’ 41760 v ’ 417 séln

(1.46)

where ((l))S is the average value of the potential over the whole surface. The customary problem is the so-called exterior problem in the vol-

- ..

Chapter 1

Introduction to Electrostatics—Sl

ume V 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 ((1))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 determine G(x, x’) because of its dependence on the shape of the surface S. We will encounter such problems in Chapters 2 and 3. The mathematical symmetry property G(x, x’) = G(x", x) can be proved for

the Green functions satisfying the Dirichlet boundary condition (1.43) by means of Green’s theorem with ()5 = G(x, y) and i/1 = 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 source, 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’)/41760.

It is a solution of the Laplace equation inside V 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 chosen 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’)/4n60 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 G &1 1 YGG

Z &1 %

—r,

DULU cc; puiiii A .

1 I1

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 (l) (which vanishes at infinity), the work done on the charge (and hence its potential energy) is given by

Wt = (Ii-‘P(Xi-)

(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 xv,-. Then 6*

/'\

\4/

Ffi

1 F:-.1 Q; = .i .————. 41769 J-=1 IX, — xi]

*See K.-J. Kim and J. D. Jackson, Am. J. Phys. 61, (12) 1144-1146 (1993).

(1.48)

Sect. 1.11 oo CD

CT‘ an ("F

F‘?-

CT‘ C D

("F

Electrostatic Potential Energy and Energy Density; Capacitance

D 12$CT.an "C3 C) ("FC

fij

C D 12$ C D >-1 0'0 ‘
-1 CT‘CD P-1»C. :1 ctional F“?-

p-.i -

j-|—i

I[ilr] =

JV Vi// - Vip d3x — JV gt/1 d3x

(1.63)

where the function t/i(X) is wel‘- CF‘(TCF‘ “ Q1"ed inside the volume V and on its surface < P-

S (which may consist of several separate surfaces), and g(X) is a specified

“source” function without singularities within V. We now examine the first-order change in the functional when we change ip —> ip + 51p, where the modification 6i/;(X) is infinitesimal within V. The difference 61 = I[ip + 61p] — I [i//] is 51 = L Vii - V(5i/1) d3x — L g5ilr d3x + ---

(1.64)

The neglected term is semipositive definite and is second order in Sip. Use of Green’s first identity with qb = 51/1 and ii = QU yields 8

51 = I [—V2¢ — g] (Sip d3x + § 6¢—¢ cla v

s

(1.65)

8n

Provided 8i/1 = 0 on the boundary surfac that the surface integral vanishes), the first-order chan U12 (D E3 [[1,9] vanish |""h (D Q3 CT. C1’) (D €I~ C/3 a?‘ /\ U’)

352
93