Appendix on Unit sand Dimensions

The question of units and dimensions in electricity and magnetismhas exercised a great number of physicists and engineers over the years .This situation is in marked contrast with the almost universal agreementon the basic units of length (centimeter or meter), mass (gram or kilogram),and time (mean solar second) . The reason perhaps is that the mechanicalunits were defined when the idea of "absolute" standards was a novelconcept (just before 1 800) , and they were urged on the professional andcommercial world by a group of scientific giants (Borda, Laplace, andothers) . By the time the problem of electromagnetic units arose therewere (and still are) many experts . The purpose of this appendix is to addas little heat and as much light as possible without belaboring the issue .

1 Units and Dimensions ; Basic Units and Derived Units

The arbitrariness in the number of fundamental units and in the dimensionsof any physical quantity in terms of those units has been emphasized byAbraham, Planck, Bridgman,* Birge,f and others . The reader interestedin units as such will do well to familiarize himself with the excellent series ofarticles by Birge .

The desirable features of a system of units in any field are convenienceand clarity. For example, theoretical physicists active in relativisticquantum field theory and the theory of elementary particles find it con-venient to choose the universal constants such as Planck's quantum o f

P. W. Bridgman, Dimensional Analysis, Yale University Press (1931) .~ R. T. Birge, Am. Phys. Teacher (now Am. J. Phys.), 2, 41 (1934) ; 3,102,171 (1935) .

611

612 Classical Electrodynamics

action and the velocity of light in vacuum to be dimensionless and of unitmagnitude. The resulting system of units (called "natural" units) has onlyone basic unit, customarily chosen to be length. All quantities, whetherlength or time or force or energy, etc ., are expressed in terms of this oneunit and have dimensions which are powers of its dimension . There isnothing contrived or less fundamental about such a system than oneinvolving the meter, the kilogram, and the second as basic units (mkssystem) . It is merely a matter of convenience . *

A word needs to be said about basic units or standards, considered asindependent quantities, and derived units or standards, which are definedin both magnitude and dimension through theory and experiment in termsof the basic units . Tradition requires that mass (m), length (1), and time{t} be treated as basic units . But for electrical quantities there is as yet nocompelling tradition . Consider, for example, the unit of current . The"international" ampere (for a long period the accepted practical unit ofcurrent) is defined in terms of the mass of silver deposited per unit time byelectrolysis in a standard silver voltameter . Such a unit of current isproperly considered a basic unit, independent of mass, length and timeunits, since the amount of current serving as the unit is found from asupposedly reproducible experiment in electrolysis .

On the other hand, the presently accepted standard of current, the"absolute" ampere, is defined as that current which when flowing in eachof two infinitely long, parallel wires of negligible cross-sectional area,separated by a distance of 1 meter in vacuum, causes a transverse forceper unit length of 2 x l0-' newton/meter to act between the wires . Thismeans that the "absolute" ampere is a derived unit, since its definition is

in terms of the mechanical force between two wires through equation(A .4) below.- The "absolute" ampere is, by this definition, exactly one-tenth of the em uzait of current, the abampere . Since 1948 the internationallyaccepted system of electromagnetic standards has been based on the meter,the kilogram, the second, and the above definition of the "absolute"ampere plus other derived units for resistance, voltage, etc . This seems to

* In quantum field theory, powers of the coupling constant play the role of other basicunits in doing dimensional analysis .

t The proportionality constant k . in (A.4) is thereby given the magnitude k2 = 10- 'in the mks system. The dimensions of the " absolute" ampere, as distinct from its magni-tude, depend on the dimensions assigned k , . In the conventional mks system ofelectromagnetic units, electric charge (q) is arbitrarily chosen as a fourth basic unit .Consequently the ampere has dimensions of qt-1, and k2 has dimensions of mlq-2 . Ifk2 is taken to be dimensionless, then current has the dimensions mlV'12 t-1 . The questionof whether a fourth basic unit like charge is introduced or whether electromagneticquantities have dimensions given by powers (sometimes fractional) of the three basic

mechanical units is a purely subjective matter and has no fundamental significance .

Appendix on Units and Dimensions 613

be a desirable state of affairs . It avoids such difficulties as arose when,in 1894, by Act of Congress (based on recommendations of an internationalcommission of engineers and scientists), independent basic units of current,

voltage, and resistance were defined in terms of three independent experi-ments (silver voltameter, Clark standard cell, specified column ofmercury) . * Soon afterwards, because of systematic errors in the experi-ments outside the claimed accuracy, Ohm's law was no longer valid, byAct of Congress !

2 E lectromagnet ic Units and Equations

In discussing the units and dimensions of electromagnetism we will takeas our starting point the traditional choice of length (1), mass (m), andtime (t) as independent, basic units . Furthermore, we will make thecommonly accepted definition of current as the time rate of change ofcharge (I = dqldt) . This means that the dimension of the ratio of chargeand current is that of time .t The continuity equation for charge andcurrent densities then takes the form

To simplify matters we will initially consider only electromagneticphenomena in free space, apart from the presence of charges and currents .

The basic physical law governing electrostatics is Coulomb's law on theforce between two point charges q and q', separated by a distance r. Insymbols this law is

F , = k, R4' (A .2)

r

The constant ki is a proportionality constant whose magnitude and dimen-sions either are determined by the equation if the magnitude and dimensionsof the unit of charge have been specified independently or are chosenarbitrarily in order to define the unit of charge . Within our present frame-work all that is determined at the moment is that the product (k1gq') hasthe dimensions (nzl3t-2) .

* See, for example, F . A. Laws, Electrical Measurements, McGraw-Hill, New York(1917), pp . 705-706 .

t From the point of view of special relativity it would be more natural to give currentthe dimensions of charge divided by length . Then current density J and charge density pwould have the same dimensions and would form a "natural" 4-vector . This is thechoice made in a modified Gaussian system (see the footnote (p . 621) for Table 4) .

614 Classical Electrodynamic s

The electric field E is a derived quantity, customarily defined to be theforce per unit charge. A more general definition would be that the electricfield be numerically proportional to the force per unit charge, with aproportionality constant which is a universal constant perhaps havingdimensions such that the electric field is dimensionally different from forceper unit charge, There is, however, nothing to be gained by this extrafreedom in the definition of E, since E is the first derived field quantity tobe defined . Only when we define other field quantities may it be convenientto insert dimensional proportionality constants in the definitions in orderto adjust the dimensions and magnitude of these fields relative to theelectric field. Consequently, with no significant loss of generality the electricfield of a point charge q may be defined from (A.2) as the force per unitcharge,

E = ki q (A.3)r

All systems of units known to the author use this definition of electric field .

For steady-state magnetic phenomena Ampe're's observations form abasis for specifying the interaction and defining the magnetic induction .According to Ampere, the force per unit length between two infinitely long,parallel wises separated by a distance d and carrying currents I and I' is ,

l2 = 2k2 d {A. 4}

The constant k2 is a proportionality constant akin to k i in (A .2). Thedimensionless number 2 is inserted in (A .4) for later convenience inspecifying k2 . Because of our choice of the dimensions of current andcharge embodied in (A.l) the dimensions of k2 relative to kl are determined,

From (A.2) and (A.4) it is easily found that the ratio kl/k2 has the dimension

of a velocity squared (12t-2) . Furthermore, by comparison of the magnitude

of the two mechanical forces (A .2) and (A.4) for known charges and

currents, the magnitude of the ratio k1 J k2 in free space can be found . Thenumerical value is closely given by the square of the velocity of light in

vacuum. Therefore in symbols we can writ e

k1 = c2 (A.5)k2

where c stands for the velocity of light in magnitude and dimension(c = 2.997930 ± 0 .000003 x 1010 cm/sec) .

The magnetic induction B is derived from the force laws of Ampere as

being numerically proportional to the force per unit current with a pro-portionality constant a which may have certain dimensions chosen for

Appendix on Units and Dimensions 615

convenience. Thus for a long straight wire carrying a current I, themagnetic induction B at a distance d has the magnitude (and dimensions )

B = 2k2a d (A.6 )

The dimensions of the ratio of electric field to magnetic induction can befound from (A. 1), (A .3), (A .5), and (A .6) . The result is that (EBB) has thedimensions (I/ta) .

The third and final relation in the specification of electromagnetic unitsand dimensions is Faraday's law of induction, which connects electric and

magnetic phenomena . The observed law that the electromotive forceinduced around a circuit is proportional to the rate of change of magneticflux through it takes on the differential form ,

where k3 is a constant of proportionality . Since the dimensions of Erelative to B are established, the dimensions of k3 can be expressed interms of previously defined quantities merely by demanding that bothterms in (A.7) have the same dimensions . Then it is found that k3 has thedimensions of a 1. Actua lly, k3 is equal to o1. This is established on thebasis of Galilean invariance in Section 6 .1 . But the easiest way to provethe equality is to write all Maxwell's equations in terms of the fields definedhere

V •E=4-, rk i p

V x B = 47rk2aJ +2ak at

v xE+k3aB=o

v •B - a

(A.8)

Then for source free reg ions the two curl equations can be combined intothe wave equat ion,

D 2 2B- a a a2B

0 (A.9)kl ate

The velocity of propagation of the waves described by (A .9) is related tothe combination of constants appearing there . Since this velocity isknown to be that of light, we may writ e

k,= c2 (A.10)

k3k2a

616 Classical Electrodynamics

Combining (A.5) with (A.10), we find

k3 =1

a

an equality holding for both magnitude and dimensions .

3 Various Systems of Electromagnetic Un its

(A . 11)

The various systems of electromagnetic units differ in their choices ofthe magnitudes and dimensions of the various constants above . Becauseof relations (A.5) and (A .11) there are only two constants (e .g., kj , k~,)

Table 1

Magnitudes and dimensions of the electromagnetic constants

for various systems of unit s

The dimensions are given after the numerical values . The symbol c stands forthe velocity of light in vacuum (c = 2 .998 x 1010 cm/sec = 2 .998 x 10' m/sec) .The first four systems of units use the centimeter, gram, and second as theirfundamental units (1, m, t) . The mks system uses the meter, kilogram, and second,plus charge (q) as a fourth unit .

System

Electrostatic

(esu)

Electromagnetic(emu)

Gaussian

Heaviside-Lorentz

Rationalizedmks

kl

Z

C 2 ( 12 t-2)

11

4Tr

= 10- Ic 2

rrE O4

lml3t^2q 2)

k2

Cf2(12j-2 )

1 1

2(,2r-2)

4rrc2 (t21-2

)

µo 10- 7

4 7r

(+nlq 2 )

k3

1

Wt-1} c1(tl-i)

1 1

that can (and must) be chosen arbitrarily . It is convenient, however, totabulate all four constants (k1, k2, (x , k3) for the commoner systems ofunits. These are given in Table 1 . We note that, apart from dimensions,the em units and mks units are very similar, differing only in variouspowers of 10 in their mechanical and electromagnetic units . The Gaussianand Heaviside-Lorentz systems differ only by factors of 4,7r . Only in the

Appendix on Units and Dimensions 617

Gaussian (and Heaviside-Lorentz) system does k3 have dimensions . It isevident from (A .7) that, with k3 having dimensions of a reciprocal velocity,E and B have the same dimensions . Furthermore, with k3 = c-1, (A.7)shows that for electromagnetic waves in free space E and B are equalin magnitude as well .

Only electromagnetic fields in free space have been discussed so far .Consequently only the two fundamental fields E and B have appeared.There remains the task of defining the macroscopic field variables D andH. If the averaged electromagnetic properties of a material medium aredescribed by a macroscopic polarization F and a magnetization M, thegeneral forms of the definitions of D and H ar e

D = EoE + ~P

frowhere co, µo, 2 , A' are proportionality constants . Nothing is gained bymaking D and P or H and M have different dimensions. ConsequentlyA and A ' are chosen as pure numbers (A = A ' = 1 in rationalized systems,2 = 2' = 47r in unrationalized systems) . But there is the choice as towhether D and P will differ in dimensions from E, and H and M differ fromB. This choice is made for convenience and simplicity, usually in order tomake the macroscopic Maxwell's equations have a relatively simple, neatform. Before tabulating the choices made for different systems, we notethat for linear, isotropic media the constitutive relations are alwayswritten D = EE

(A .13)B=µH

Thus in (A.12) the constants co and µo are the vacuum values of c and ,u .The relative permittivity of a substance (often called the dielectric constant)is defined as the dimensionless ratio (EEO), while the relative permeability(often called the permeability) is defined as {µlµo} .

Table 2 displays the values of E, and u,,, the defining equations for Dand H, the macroscopic forms of Maxwell's equations, and the Lorentzforce equation in the five common systems of units of Table 1 . For eachsystem of units the continuity equation for charge and current is given by(A.l), as can be verified from the first pair of Maxwell's equations in thetable in each case . * Similarly, in all systems the statement of Ohm's lawis J = QE, where a is the conductivity ,

* Some workers employ a modified Gaussian system of units in which current isdefined by I = (i/c)(dq/dt) . Then the current density J in the table must be replaced bycJ, and the continuity equation is V • J + (1/c)(ap/fit) = 0 . See also the footnote belowTable 4.

a kO-L00

Table 2Definitions of Ep, µo, D , H, macroscopic Maxwell 's eq uations, and Lorentz fo rce equ a tion in various systems of units

Where necessary the dimensions of quantities are given in parentheses . The symbol c stands for the velocity of light in vacuum with dimensions

LorentzSystem 450 µo D, H Macroscopic Maxwell's Equations Force pe r

Unit charg e

Elcctrostal ic(esu)

Elect ro-magnetic (t1 22)

(emu )

Gaussian

Heaviside-Lorentz

Rationalmks

C-z D= L+ 4 -aP a ll aB C}r21_2) H = c2 B - 4 7rM ~

• D =4rrp OX H =4aJ~--- V X E +-=0 4 • B =0 E -~v X B~ at of

D = 1 E + 4rrP 8D 8BI e- V • D =4 7rp VXH =4~rJ + OX E-~=0 0 • B =0 E+ v XB

at atH = S- 4arM

D = E + 47rP 4 7r 1 aD 1 aB v `camI V • D =47rp ~XH

S+ at AX E+ c at-0V • B =0 E}~ X B

H = B -- 47rM C c

D = E +P 1

( a

D~ 1 8S v1 H=S-M 0 • D =p Vk H=C .7 + at V}CE+c

at -00 - B =0 E~{-~ X B

107

4rrcZi 47r X 10-' D = EoE + P

Q'D=P(92tam-il- 8) (mlq 2) H B - M

P o

S7XH = J } ~D V X E -~ aB=O 0 • B =0 E-~ r XB

Appendix on Units and Dimensions 619

Table 3

Convers ion table for symbols and formulas

The symbols for mass, length, time, force, and other not specifically electro-magnetic quantities are unchanged. To convert any equation in Gaussianvariables to the corresponding equation in mks quantities, on both sides of theequation rep lace the relevant symbols listed below under "Gaussian" by thecorresponding "mks" symbols listed on the right . The reverse transformation isalso allowed. Since the length and time symbols are unchanged, quantities whichdiffer dimensionally from one another only by powers of length and/or time aregrouped together where possible .

Quantity Gaussian mks

Velocity of light

Electric field(potential, voltage)

Displacement

Charge densit y(charge, current density,current, polarization )

Magnetic induction

Magnetic field

Magnetization

Conductivity

Dielectric constant

Permeability

Resistance (impedance)

Inductance

Capacitance

E( (D , V) 1/47reo E( (D , V)

D DE p

p(q, J, I, P)

B

H

M

d

E

R(Z)

L

C

1p(q , J , 1, P)

✓47TEO

B~L o

A/¢zruo H

Po M

47r

a

47re 0

E

Ea

MO

47re OR(Z)

4rreaL

1C

4,rE O

620 Classical Electrodynamic s

Table 4

Conversion table for given amounts of a physical quantity

The table is arranged so that a given amount of some physical quantity,expressed as so many mks or Gaussian units of that quantity, can be expressed asan equivalent number of units in the other system . Thus the entries in each rowstand for the same amount, expressed in different units . Al] factors of 3 (apartfrom exponents) should, for accurate work, be replaced by (2 .997930 ± 0 .000003) ,arising from the numerical value of the velocity of light . For example, in the rowfor displacement (D), the entry {12Tr x l05} is actually (2 .99793 x 47T x 105) .Where a name for a unit has been agreed on or is in common usage, that name isgiven. Otherwise, one merely reads so many Gaussian units, or mks units .

Physical Quantity Symbol

Length

MassTimeForceWorkEnergyPowerChargeCharge density

CurrentCurrent densityElectric fieldPotentialPolarization

I

mtFW~UPq

PIJE

4), VP

Displacement

ConductivityResistanceCapacitanceMagnetic flux

Magnetic inductionMagnetic field

Magnetization

Rationalized mks

1 me ter (m)

1 kilogram (kg)I second (sec )I newto n

l joule

I wattI coulomb (soul)1 coul M-3

1 ampere (coul sec-1)1 amp r n-2I volt M-11 vol tl cool M-2

D I cool m-1

or I mho m-1R 1 ohmC I faradF I weber

B 1 weber M-2

H I ampere-turn m-'

M I ampere m-1

*Induc tance L 1 henry

10 2

1 03lIQ s

107

Gaussian

centimeters(cm)

grams (gm)second (sec)dynes

10'3 x 109

3 x 103

3 x ] 0'3 x 10 5

X 10-4

~-sao3 x lO s

]2~ x 105

9 x 10,y x 14- 119 x 101110 $

104

4~ x 1 0-3

1 0- 3

9x 10-11

ergs

ergs sec -1statcoulombsstatcoul cm -3statamperesstatamp ern zstatvolt cm -1statvolt

dipolemomentcm-3statvolt cm -1(statcoulcm-2)

sec- 1

sec cm 1cmgauss cm2 ormaxwells

gaussoersted

magneticmoment cm-3

Appendix on Units and Dimensions 621

4 Conversion of Equations and Amounts between Gaussian Units andmks Units

The two systems of electromagnetic units in most common use todayare the Gaussian and rationalized mks systems . The mks system has thevirtue of overall convenience in practical, large-scale phenomena, especiallyin engineering applications . The Gaussian system is more suitable formicroscopic problems involving the electrodynamics of individual chargedparticles, etc . Since microscopic, relativistic problems are emphasized inthis book, it has been found most convenient to use Gaussian units through-out. In Chapter 8 on wave guides and cavities an attempt has been madeto placate the engineer by writing each key formula in such a way thatomission of the factor in square brackets in the equation will yield theequivalent mks equation (provided all symbols are reinterpreted as inksvariables) .

Tables 3 and 4 are designed for general use in conversion from onesystem to the other. Table 3 is a conversion scheme for symbols andequations which allows the reader to convert any equation from theGaussian system to the mks system and vice versa . Simpler schemes areavailable for conversion only from the mks system to the Gaussian system,and other general schemes are possible . But by keeping all mechanicalquantities unchanged, the recipe in Table 3 allows the straightforwardconversion of quantities which arise from an interplay of electromagneticand mechanical forces (e.g., the fine structure constant e2/tic and theplasma frequency (0,2 = 4 7rne2/M) without additional considerations .Table 4 is a conversion table for units to allow the reader to express a givenamount of any physical entity as a certain number of mks units or cgs-Gaussian units .

* There is some confusion prevalent about the unit of inductance in Gaussian units .This stems from the use by some authors of a modified system of Gaussian units in whichcurrent is measured in electromagnetic units, so that the connection between charge andcurrent is Im = (Ile)(dg(dt) . Since inductance is. defined through the induced voltageV = L(dI/dt) or the energy U = ~L.I2, the choice of current defined in Section 2 meansthat our Gaussian unit of inductance is equal in magnitude and dimensions (t21- 1 ) to theelectrostatic unit of inductance. The electromagnetic current I„, is related to ourGaussian current I by the relation I. = (1 f c)I. From the energy definition of inductancewe see that the electromagnetic inductance L„, is related to our Gaussian inductance Lthrough L= c2L. Thus Lm has the dimensions of length . The modified Gaussiansystem generally uses the electromagnetic unit of inductance, as well as current . Then

the voltage relation reads V = (L„.lc)(dIy„ f dt) . The numerical connection between units ofinductance is

1 henry = s x 1 0-1 1 Gaussian (es) unit = 109 emu

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Tranter, C . J ., Integral Transforms in Mathematical Physics, 2nd ed., Methuen, London

(1956) .Van Vleck, J . H., Theory of Electric and Magnetic Susceptibilities, Oxford University

Press (1932) .Watson, G. N ., Theory of Bessel Functions, 2nd ed., Cambridge University Press (1952) .

Whittaker, E . T., History of the Theories of Aether and Electricity, 2 vols., Nelson,

London (1951, 1953 ).Williams, E . J., "Correlation of Certain Collision Problems with Radiation Theory,"

Kgl. Danske Videnskab . Selskab Mat-fys . Medd., XIII, No . 4 (1935) .

Index

Aberration of star light, 348, 361

Abraham-Lorentz equation of motion,582

Dirac's relativistic generalization of,609

Abraham-Lorentz model of electron,584

difficulties with, 589Absorption of radiation by oscillator,

602Absorption or total cross section of

radiation by oscillator, 605

integral over all frequencies, 606Acceleration, relativistic transformation

of, 388Acceleration fields of particle, 467

Addition of velocities, relativistic, 3 60Addition theorem for spherical har-

monics, 67

Adiabatic invariance, 419

of flux through orbit, 42 1of magnetic moment of particle, 421

Airy integrals, 48 4Alfven velocity, 331

Alfven waves, 33 0 , 331, 344

attenuation of, 333with displacement current, 334

Ampere's law, differential form of, 138

integral form of, 139Ampere's observations on forces be-

tween currents, 13 5Angular distribution of radiation by

accelerated charge, 473 ; see alsoBremsstrahlung, Multipole radia-tion, Radiation

Angular momentum, conservation of,

for particles and fields, 20 0in cylindrical wave guide, 576of circularly polarized plane wave,

201, 569of multipole radiation, 549of photon, 569selection rules for multipole radia-

tion, 549

Angular momentum operator, L, 542Anisotropic dielectrics, waves in, 233

Antenna, center-fed linear, 277

circular loop, 57 5half-wave and full-wave, 279, 566

multipole expansion for, 562radiation resistance of, 280short linear, 272

Associated Legendre polynomials, see

Legendre polynomial s

625

626 Index

Attenuation, general method of han-dling, 240

in cavity, 25 5

in wave guide, 24 9of waves in conducting medium, 224

of waves in plasma, 333, 341

Attenuation constant in wave guide, 25 1

Babinet's principle, for scalar diffrac-tion, 28 8

for vector fields, 291Bessel functions, 69

connection of, with Airy integrals,484

definitions of Iv, K,,, 75definitions of Jv, Ny, HP(1), HP (2),71definitions of j,, n,, h,(1), hz(2), 539expansions involving, 76, 86, 96, 97,

484, 501, 502, 535

Fourier-Bessel series, 73, 9 5Fourier transforms of KO and KI, 437integral representation of J., 96 , 294integral representations involving, 86,

92, 96, 29 5Kapteyn series, 74, 501limiting forms, 72, 75, 540Neumann series, 74orthogonality on finite interval, 73,

74, 95orthogonality on infinite interval,

77, 96recursion formulas, 71, 540Schlomilch series, 74series for J. , 71spherical, 53 9spherical, asymptotic forms, 540spherical, Wronskians, 541zeros of Jn(x), 72zeros of Jn'(x), 255

Beta decay, radiation emitted during,526

Bethe-Heitler bremsstrahlung formula,512

Biot and Savart law, 133Birefringence of the ionosphere, 229Bohr, N ., energy-loss formula of, 438Boundary conditions, and inconsistency

of Kirchhoff approximation, 282and types of partial differential equa-

tions, 17

Boundary conditions, at surface of goodconductor, 23 6

Cauchy, Dirichlet, Neumann, 15, 16for D and E, 11 0for dielectric wave guide, 26 0for TE and TM waves in guide, 243

magnetostatic, on H and B, 155mixed Dirichlet and Neumann, 90

normal and tangential E in free

space, 9, 1 0

Boundary-value problems, Green's-function solution of, 1 9

image method of solution of, 26 f .in cylindrical coordinates, 7 5 f.in dielectrics, 110 f ., 21 7in rectangular coordinates, 47 f.in spherical coordinates, 60 f .magnetostatics, 156 f .see also Diffraction, Resonant cavity,

Wave guideBremsstrahlung, angular and polariza-

tion distributions, 507 , 509, 5 1 6as energy-loss mechanism, 519as scattering of virtual quanta, 525classical, 510, 51 2effect of screening on, 51 6frequency spectrum, 511, 515, 517nonrelativistic, 509relativistic, 513 f.

Bremsstrahlung photon cross section,51 3

Brewster's angle, 22 0

Capacitance, 24

Cauchy boundary conditions, 15, 17,22 1

Causality, 185, 23 4in special relativity, 37 1lack of, with radiative reaction, 599

Cavities, see Resonant cavityCharge, effective magnetic, 158

electric, 2Lorentz invariance of, 377

Cherenkov angle B., 495, 498Cherenkov radiation, 49 4

angular and frequency distribution

of, 498

connection of, with energy loss, 448Fourier transforms of fields of, 445potentials of, 497

Index

Classical electron radius, 490, 5 89Clausius-Mossotti relation, 119Closure, see Completeness relationCollisions, between charged particle s

as energy-loss mechanism, 430relativistic kinematics of, 400 ; see

also Energy loss, ScatteringCollision time, 507

for fields of relativistic particle,

382Complementary screens, 288Completeness relation, for Bessel func-

tions on an infinite interval, 9 6for complex exponentials, 47, 84for spherical harmonics, 65general, 45

Compton scattering, 490Conduction in a moving fluid, 312Conductivity, effect of, on fields, 22 2

Said motion with infinite, 312model of, 22 5of plasma, 459tensor, in plasma, 345

Conductor, attenuation in, 224boundary conditions at, 236

definition of, 2 3fields at surface of, 236

fields inside, 222, 238

penetration depth in, 225surface resistance of, 240

Conservation, of angular momentum ofparticles and fields, 200

of electromagnetic energy, 18 9

of energy and momentum of particlesand fields, 190, 38 6

Conservation laws in covariant form,38 5

Constitutive relations, 17 9Continuity equation, for charge and

current, 133, 61 3for electromagnetic energy flow, 190,

38 6for fiend, 311, 330in covariant form, 37 8

Contour integration for retarded Green's

function, 184Contraction of length, see FitzGerald-

Lorentz contractio nConvective derivative, 172, 311

627

Conversion tables for electromagneticquantities between Gaussian andmks units, 619, 62 0

Correspondence principle of Bohr, 502Cosine integral, 279Coulomb gauge, 18 1

use of, in Darwin interaction, 409

Coulomb's law, 1Covariance, of conservation laws, 385

of electrodynamics, 377of equations of physics, 376of force equation, 384, 405of Lagrangian, 40 6of Maxwell's equations, 379

Cross section, classical, for bremsstrahl-ung, 51 2

classical, for scattering of radiationby a sphere, 572

classical, relation of, to impact pa-rameter, 452

Rutherford, 452for scattering by screened potential,

453Thomson, 48 9total, for scattering and absorption

of radiation by oscillator, 602

total, for scattering of particles byatoms, 45 5

see also Bremsstrahlung, ScatteringCurl, operating on vector spherical

harmonic, 57 0operator relation involving L, 546

Current, international unit of, 612

Current density, continuity equation for,

13 3

force on, in magnetic field, 137

magnetization caused by, 146Current flow in plasma column, 320

Current loop, vector potential and

fields of, 14 1Cutoff frequency, in dielectric wave

guide, 263in hollow wave guide, 245

Cutoff modes in wave guide, 245

Cylinder functions, see Bessel functions

Cylindrical coordinates, boundary-value problems in, 75, 89

delta function in, $4Green's function in, 84, 96

628

Cylindrical coordinates, Laplace'sequation in, 6 9

separation of variables in, 69waves in, 241

Index

Damping, of magnetohydrodynamicwaves, 33 3

of oscillations in cavity, 255of plasma oscillations, 340see also Radiative reactio n

Darwin-$reit interaction, 411Debye-Mickel screening radius or

Debye length, 342Debye wave number, 340Decay, of particle, relativistic kine-

matics of, 394of pi mesons, time dilatation in, 359

Delta function, charge densities interms of, 4, 36, 82, 83

current densities in terms of, 141,278, 563

definition of, 3equal to V2(1/r), 13in arbitrary coordinates, 79integral representations for, 47, 84,

9 6properties of, 4three-dimensional, in cylindrical co-

ordinates, 84

three-dimensional, in spherical co-ordinates, 79

Density effect in energy loss, 443

connection of, with Cherenkov radi-ation, 448

Dielectric constant, classical result for,234, 446

for Alfven waves, 334for plasma, 227 , 45 1

for plasma in magnetic field, 228Dielectrics, 108

anisotropic, waves in, 233boundary conditions, 110boundary-value problems with, l IQ f .electrostatic energy in, 123method of images for, 111

Dielectric wave guide, 259at optical frequencies, 264

cutoff frequency in, 26 3possible modes of propagation, 260

Differential operator relations, see

Gradient, Laplacian, etc .Diffraction, Babinet's principle in, 2$$

by circular aperture, 292, 3 07by half-plane, 30 6by rectangular opening, 307by small apertures, 297, 307, 308by sphere in short wavelength limit,

299comparison of scalar and vector ap-

proximations, 296Fresnel and Fraunhofer, 292Kirchhoff approximation in, 282scalar Huygens-Kirchhoff theory of,

280Smythe's vector theorem for, 287

use of Green's theorem in, 281, 283vector Kirchhoff approximation, 28 5

Diffusion of magnetic fields, 313Diffusion time of magnetic fields, 313Dilatation of time, see Time dilatationDimensions, discussion of, 6I 1

Dipole approximation, in energy loss,43 5

in radiation problems, 271, 274, 507Dipole fields, electrostatic, 100

magnetostatic, 143, 147of conducting sphere, 34of dielectric sphere, 1 1 5

of electrostatic dipole layer, 9of magnetized sphere, 157oscillating electric, 271oscillating magnetic, 274

Dipole moment, electrostatic, 100induced, 120interaction between two, 102magnetostatic, 13 2 , 146

magnetostatic, energy of, 150magnetostatic, force on, 149

magnetostatic, of current loop, 143,147

magnetostatic, relation of, to angularmomentum, 14 8

magnetostatic, torque on, 132, 150oscillating electric, 27 1oscillating magnetic, 27 4

Dipole sum rule for absorption of radi-ation, 606

Dirac delta function, see Delta function

Index

Disc's relativistic force equation, 609

Dirichelet boundary conditions, 16

Disc, potential of charged conducting,

89Discontinuity, of electric field, 9

of potential, 1 2Dispersion, and causality, 234

and propagation, 208 f.

anomalous, 21 1in dielectrics, 208in ionosphere, 229

in plasmas, 337Dispersion relation, for index of refrac-

tion, 234for plasma oscillations, 33 7 , 339

Displacement, definition of, 108Displacement current, 178Dissipative effects, in cavities, 255

in plasma oscillations, 332, 340

in wave guides, 248in wave propagation, 208

Divergence theorem, 6Doppler shift, relativistic, 36 3

transverse, 36 4

Drift of charged particles, in crossedE and B, 41 3

in inhomogeneous magnetic fields,41 7

Drift of lines of force in conductingfluid, 31 4

Drift velocity, caused by curvature of

lines of force, 41 8

caused by magnetic field gradient,

41 6Dual integral equations, 9 1

Dyadic, 19 3

Dynamics of relativistic charged par-

ticle in external fields, 404 f.

Ei3enfunctians, 87cxpamion of Green's function in

terms of, 88

for fields in cylindrical wave guides,244

for wave equation in box, 88

in cylindrical cavity, 25 4

in rectangular guide, 246

in spherical cavity, 576

629

Electric dipole, see Dipole fields, Di-pole moment

Electric field, definition of, 2derivable from potentials, 8, 179relativistic transformation of, 38 0

"Electric" waves, 243, 545Electromagnetic fields, multipole expan-

sion of, 54 3

of localized oscillating source, 270Electromagnetic stress - energy -momen-

tum tensor, 385Electromagnetic units, see Units

Electromotive force, definition of, 170Electrostatic force on conductor, 23

Electron, classical model of, 584difficulties with, 589radius of, 490, 589

Electron capture, radiation emitted in,528

Electrostatic units, see Units

Elliptic integrals, use of, 96, 142, 168

Energy, electrostatic potential, 8, 20in dielectrics, 12 3of charge distribution in external

field, 10 1magnetic, 173magnetic dipole, 150of magnetically permeable body, 176

of permanent magnets, 167self-, 22, 58 8

Energy conservation between particlesand fields, 190, 386

Energy density, electromagnetic, 189,

205

electrostatic, 2 14-4 element of stress-energy-momen-

tum tensor, 3$5Energy flow, 190

in wave guide, 248

velocity of, 208, 21 1Energy-level shift due to radiative reac-

tion, 6 00Energy loss, collisional, classical, 438

collisional, density effect in, 443collisional, in plasma, 450

coMisinnai, and method of virtualquanta, 536

collisional, quantum-mechanical, 440radiative, in accelerators, 471

630 Index

Energy loss, radiative, in nonrelativisticcollisions, 51 3

radiative, in relativistic collisions, 518Energy of particle, relativistic trans-

formation of, 39 1Energy transfer, in collision with bound

charge, 43 4in Coulomb collision, 430in discrete amounts, 439in elastic collision, 40 4

to oscillator, 436Equation of motion, integrodifferen-

tial, with radiative reaction, 597Equations of motion with radiative re-

action, 582, 583Ether, 347Ether drift, experiment on, 349Expansion, of circularly polarized plane

wave, 569of e~kVR, 54 1of retarded quantity, 586of scalar plane wave, 567

Expansion of Ix - x' j--1, in cylindricalcoordinates, 86, 96

in plane waves, 8 8in spherical coordinates, 62, 69

Expansions, see Orthonormal expan-

sions

Faraday's law, for moving circuit, 172in differential form, 1 73in integral form, 170

Fields, of moving particle, 46 7of relativistic particle, equivalence

of, to pulse of radiation, 382, 521Fields of uniformly moving charge, 381,

467

Fourier transforms of, 437in dielectric, Fourier transforms of,

445FitzGerald-Lorentz contraction, 352,

357graphical representation of, 373

Fizeau's experiment, 34 9Flow of viscous conducting fluid be-

tween parallel plates, 31 6Force, between charge and image

charge, 30, 32, 5 1

between current-carrying circuits, 136

Force, between point charge andsphere, 30, 31, 33

between two parallel wires, 136

Lorentz, 19 1on charged surface, 23on current distribution in magnetic

field, 137, 148on dielectric body, 126, 127, 13 1on magnetically permeable body, 177on magnetic dipole, 149on permanent magnets, 167, 168radiative reaction, 582

Force equation, Abraham-Lorentz, 582for fluid, 31 1in covariant form, 384, 405with radiative reaction, integrodiffer-

ential, 5974-vector character of radiation, 3904-vectors and tensors, 374Fourier integrals, 47Fourier series, 4 6Fourier transforms, of exponentially

damped wave, 25 6of fields of charge in uniform motion,

437

of wave packet, 209, 21 2Fraunhofer diffraction, see DiffractionFreezing in of lines of force, 313, 319

Frequency spectrum of radiation, 478,

480

qualitative aspects of, 47 6see also Bremsstrahlung, Radiation

Fresnel diffraction, see Diffraction

Fresnel formula for velocity of light i n

moving media, 349

Fresnel formulas for reflected and re-

fracted plane waves, 219, 220Fresnel integrals and diffraction, 3 06

Galilean invariance, 348, 369

and Faraday's law, 17 1

Galilean reference frames, 3 53Galilean relativity, 348, 369

Galilean transformation, 171, 388

Gauge, Coulomb or transverse, 140,182

Loren tz, 18 1

Gauge invariance, 181, 426

Index

Gauge transformations, for magneto-statics, 140

for time-varying fields, 181

Gaussian units, see Units

Gauss's law, applied to surface-chargedistribution, 9

differential form of, 6integral form of, 4

Gradient, in spherical vector form, 544of electric field and quadrupole in-

teraction, 101, 12 8of magnetic induction and force on

dipole, 149of magnetic induction, particle drift

in, 416

Green's first identity, 14Green's function for time-dependent

wave equation, 18 3retarded, 185, 269

Green's function for wave equation,spherical wave expansion of, 541

Green's function in electrostatics, 1 8examples of use of, 82, 83expansion of, in Bessel functions,

84, 96

expansion of, in Legendre polyno-mials, 62

expansion of, in spherical harmonics,69

for concentric spheres, 80

for cylindrical box, 97for rectangular box, 8 9for rectangular box, eigenfunction

expansion of, 8 8for sphere, 4 1

Green's reciprocation theorem, 25

Green's theorem, 15

use of, in diffraction, 28 0

use of, with wave equation, 188vector equivalent of, 285

Grounded conductor, definition of, 27Group velocity, 21 1

and phase velocity, 2] 1and phase velocity in guide, 249

in electronic plasma, 340in wave guide, 249

Guides, see Wave guide

Gyration frequency, 228, 411

631

Half-width of resonance, 601connection of, with Q in a cavity,

257Hamiltonian for relativistic charged

particle, 408Hankel function, see Bessel functionsHankel transform, 7 7Hartmann number, 317Heaviside-Lorentz units, see UnitsHelicity, definition of, 206

Hemispheres, at different potentials, 42potential from symmetry and unique-

ness, 62potential in terms of Legendre poly-

nomials, 6 1

Huygens's principle, 188, 280Hydrodynamics, see Magnetohydrody-

namicsHysteresis, magnetic, 15 3

Image charges, see ImagesImages, method of, 26 f .

method of, for conducting sphere inuniform field, 3 3

method of, for dielectrics, 111method of, for magnetic media, 167

method of, for point charge nearconducting sphere, 27

Impact parameter, and scattering

angle, 452minimum and maximum in brems-

strahlung, 510, 51 4minimum and maximum in energy

loss, 432, 440

minimum in method of virtualquanta, 520

Impedance of a transmission line, 265Index of refraction, analytical proper-

ties of, 234and phase and group velocities, 211dispersion relation for, 23 4sue also Dielectric constant

Inductance, 198, 199high-frequency, 225of transmission line, 265, 266

Induction, Faraday's law of, 170Initial-value problem, for scalar wave

equation, 18 6Poisson's solution of, 188

632 Index

Instabilities of a pinched plasma col-umn, 326

Internal electric field at molecule, 116Invariance, of length-time element in

special relativity, 369of separate length and time elements

in Galilean relativity, 369see also Adiabatic invariance, Rela-

tivistic invarianceInversion, method of, 3 5

method of, behavior of charge densi-ties under, 37

method of, examples of, 39, 40, 53Ionosphere, 226 f.Ives-Stilwell experiment, 364

Jacobian, in Lorentz transformation ofcoordinates, 376

in transforming delta functions, 7 9

Kinematics, relativistic, 394 f .Kirchhoff diffraction, see DiffractionKirchhoff's integral representation, 18 8

use of, in diffraction, 2$ 0vector equivalent of, 283

Klein-Nishina formula, 49 0

Lagrangian, for relativistic charged par-ticle, 407

for two interacting charged particles,41 1

Lamb shift, 602Landau damping, 340

Laplace's equation, 1 3

general solution of, in cylindricalcoordinates, 76, 77

general solution of, in spherical co-ordinates, 67

in cylindrical coordinates, 69in rectangular coordinates, 48in spherical coordinates, 54uniqueness of solution of, 1 5

Laplacian and the angular momentum

operator L, 543Larmor power formula, 469

relativistic generalization of, 470Legendre polynomials, 56

associated, 64expansion of inverse distance in, 62

Legendre polynomials, explicit formsof, 57

integrals involving, 60orthogonality, 58recursion relations, 59Rodrigues's formula for, 57see also Spherical harmonics

Lenz's law, 17 0L'Hospital's rule, 8 3Lienard's generalization of Larmor

power formula, 470Lienard-Wiechert potentials, 465

fields, 467Lifetime, of multipole transitions, 558

of pi mesons in motion, 359Light cone, 37 0Linear superposition, of electric fields, 3

of plane waves, 203, 20 8of potentials, 3 1

Line breadth due to radiative reaction,600

Localized source, see Multipole, Multi-

pole momen t

Longitudinal fields in conductor, 223

Lorentz condition, 18 1in covariant form, 378

Lorentz force, 19 1Lorentz force equation in covariant

form, 405Lorentz invariant, see Scalar, Relativ-

istic invarianceLorentz line shape, 436, 601, 604

for cavity, 256

Lorentz-Lorenz relation, 119

Lorentz transformation, 35 6

as orthogonal transformation in four

dimensions, 37 1

infinitesimal, 367noncommuting, 357, 387

successive, 357, 367see also Relativistic transformation

Loss, see Power loss

Macroscopic averages, 104, 194

Macroscopic equations, derivation of,

for electrostatics, 10 3derivation of, for magnetostatics,

150

Index

Macroscopic equations, derivation of,for time-varying fields, 19 4

Macroscopic fields, definition of B andH, 15 3

definition of E and D, 108Magnet, permanent, 161, 167Magnetic dipole, see Dipole fields, Di-

pole momentMagnetic field H, boundary conditions

on, 154definition of macroscopic, 153see also Magnetic induction

Magnetic flux density, see Magnetic in-

ductionMagnetic induction B, boundary con-

ditions on, 15 4definition of, 132, 134of circular loop, 141of current element, 134of long wire, 13 5of magnetized sphere, 156of nonrelativistic moving charge, 134relativistic transformation of, 380

unit of, defined in terms of force,136

Magnetic mirror, 149, 42 3Magnetic moment, adiabatic invariance

of, 42 1force on, in external field, 149of localized current distribution,

146radiation because of disappearance

of, 53 1radiation by time-varying, 274, 481

Magnetic-moment density, see Magnet-

ization

Magnet pressure, 315"Magnetic" waves, 243, 545

Magnetization, definition of macro-

scopic, 15 1

divergence of, as effective magnetic-

charge density, 158

effective current of, 1 52of current distribution, 146radiation by time-varying, 481, 553

Magnetohydrodynamic flow, 316

Magnetohydrodynamics, 309 f.

equations of, 311

633

Magnetohydrodynamic waves, 329, 344effect of finite conductivity and vis-

cosity, 333with displacement current, 334

Magnetosonic waves, 331Magnetostatics, basic equations of,

137 f .multipole expansion in, 145

Maximum and minimum impact param-

eters, see Impact paramete rMaximum and minimum scattering

angles, 453, 455Maxwell's equations, 177

in covariant form, 379in different systems of units, 618macroscopic, 194plane wave solutions of, 204spherical wave solutions of, 546

Maxwell stress tensor, 193Mean-square scattering angle, 456

Mean-value theorem, 2 5Mesons, time dilatation experiment

with, 359Method of images, see ImagesMethod of inversion, see Inversion

Michelson-Morley experiment, 349Microwaves, see Diffraction, Resonan t

cavity, Wave guideMinkowski diagram, 374Mixed boundary conditions, 16, 89

Mks units, see UnitsModes, in circularly cylindrical guide,

266in cylindrical cavity, 254, 267in dielectric guide, 260, 267in rectangular guide, 24 6

in triangular guide, 267TE and TM, in wave guide, 243

Momentum, canonical, for relativistic

charged particle, 408conservation of, between particles

and fields, 191

electromagnetic, definition of, 19 2of particle, relativistic transformation

of, 392Momentum density, electromagnetic, as

part of stress-energy-momentum

tensor, 385

$34 Index

Momentum impulse in Coulomb colli-sion, 43 1

Motion, see Particle motionMoving circuits and law of induction,

17 1Multiple scattering of particles, 456,

463Multipole, electrostatic, 98

electrostatic, expansion of interactionenergy in, 10 1

electrostatic, expansion of potentia lin, 9 8

electrostatic, rectangular, 100

magnetostatic, 14 5radiating, near, induction, and radia-

tion zones, 270time-varying, 271, 273, 54 5see also Dipole moment, Magnetic

moment, Multipole momentMultipole expansion, of E and B , 546

of electromagnetic fields, 543 f ,of energy of charge d istribution in

external field, 10 1of Green's function for wave equa-

tion, 54 1of radiation by linea r antenna, 562of scalar plane wave, 567of scalar potential, 98

of vector plane wave, 569Multipole fields, 543 f .

energy and angular momentum radi-ated, 548

properties of, 54 6Muftipole moment, electrostatic, 99

estimates of, for radiating atoms andnuclei, 558

magnetostatic, 14 5of linear antenna, 564of oscillating source, 271, 273, 556

Multipole radiation, angular distribu-tion of, 550 f.

by atoms and nuclei, 557by linear antenna, 562selection rules for, 54 9

Neumann boundary conditions, 16, 18Neumann function, see Bessel functionsNuclear forces, effect of, on scattering

cross sections, 454

Nuclear quadrupol e moment, 102interaction ene rgy of, 101 , 1 28 , 129

Ohm's law, 22 2generalization of, for plasma in mag-

netic field, 34 5in moving medium, 3 1 2validity of, for conducting fluid, 309

Operator relations, see Gradient, La-placian, etc .

Orthogonal functions, Bessel functions,

73general, 44

Legendre polynomials, 57spherical harmonics, 65

Orthogonality, of Bessel functions onfinite interval, 73, 9 5

of Bessel functions on infinite inter-val, 77

of Legendre polynomials, 58

of sines and cosines, 46of spherical harmonics, 65

Orthogonal transformations, 37 1Orthonormal expansions, 4 4

Fourier, on finite interval, 46

Fourier, on infinite interval, 47

Fourier-Bessel, 73, 95Legendre, 59spherical harmonic, 66

Oscillations, see Plasma oscillations,

WavesOscillator strength, 438, 60 1

Parity of multipole fields, 550

Parseval's theorem, 47 8

Particle motion, in crossed E and B , 412in dipole field of earth, 42 7in external fields, 411 f .in inhomogeneous B , 4 1 5, 421in uniform static B , 411

Penetration depth in plasma, 22 7see also Skin depth

Permeability, magnetic, 15 3Phase difference, and elliptic polariza-

tion , 205

between current and fie ld in plasma,

227between E and B in conductor, 224

Index

Phase of plane wave, relativistic in-variance of, 36 3

Phase shift for scattering by sphere,

57 1Phase velocity, and group velocity, 211

and group velocity in guide, 249in wave guide, 246of Alfven waves, 331, 334of plane waves, 203of plasma oscillations, 34 0

Photon, angular momentum in multi-pole, 549

angular momentum of, 201, 569Pinch effect, and instabilities, 326

dynamic models of, 32 2scaling law for, 325steady-state, 320

Plane wave, electromagnetic, 202 f .

electromagnetic, expansion in spheri-cal multipole waves, 56 9

electromagnetic, in conducting me-dium, 222 f.

electromagnetic, reflection and re-fraction of, 216 f .

scalar, general one-dimensional solu-tion, 200, 212, 23 3

scalar, propagation in dispersive me-dium, 210 f .

scalar, properties of, 203Plasma, conductivity of, 459

confinement of, by external fields ,329

confinement of, by magnetic mirrors,424

confinement of, by self-fields, 320definition of, 310dielectric constant of, 451energy loss in, 450instabilities in column of, 326longitudinal waves in, 337transverse waves in, 226, 33 9

Plasma frequency, 227, 33 7Plasma oscillations, and Boltzmann

equation, 34 5high-frequency, 335 f .in external magnetic field, 346Landau damping of, 340longitudinal, 33 7transverse, 339

635

Poincare stresses, 592

Poisson's equation, 12equivalent integral equation, 15general solution in spherical geom-

etry, 8 1uniqueness of solution of, 1 5

Polarizability, electronic, 120models of, 11 9

molecular, 118orientation, 122

Polarization, charge density, 107, 112current density from time-varying,

196definition of, 10 8surface-charge density, 112, 115, 117

Polarization effects in energy loss, 443Polarization of radiation, by reflection,

220circular, elliptical, linear, 205from accelerated charges, 468, 480from multipoles, 272, 274, 551from synchrotron, 484, 504scattered by sphere, 57 2X-rays, 509

Polarization vectors, 204, 207

Potential, electrostatic, 8in rectangular box, 49

of dipole layer, 1 1

of line charge, expansion in polarcoordinates, 8 6

of point charge, expansion in cylin-

drical coordinates, 86

of point charge, expansion in eigen-functions, 88

of point charge, expansion in spheri-

cal coordinates, 62, 6 9of point charge, in cylindrical box,

97

of point charge, in rectangular box,89

of surface-charge distribution, 10

Potential energy, see EnergyPower, radiated, angular distribution

of dipole, 272

radiated, angular distribution of half-and full-wave antenna, 279

radiated, distribution of (1, m) multi-

pole, 551

636 Index

Power, radiated, angular distribution

of quadrupole, 275, 55 2radiated, by charged particle, 470,

472radiated, by charged particle in ac-

celerators, 47 1radiated, by charge in arbitrary pe-

riodic motion, 5 U1radiated, by multipoles, 550f .radiated, Larmor's formula for, 469radiated, total, 272, 276, 55 3

Power flow, see Energy flo wPower loss, because of finite conduc-

tivity, 23 9in resonant cavities, 257per unit area at conducting surface,

240per unit length, in wave guides, 25 0

Power series solution, of Bessel's equa-tion, 70

of Legendre equation, 56Poynting's theorem, 189, 197

Poynting's vector, definition of, 19 0for plane wave, 205in wave guide, 248

Precession, Thomas, 364 f .Precession frequency, 228, 411Pressure, magnetic, 31 5

radiation, 20IPropagation, in anisotropic dielectric,

23 3in conducting medium, 223in dispersive medium, 212in hollow wave guide, 249in plasma, 226 f .

Proper time, 369

Q of resonant cavity, 25 6connection with half-width of reso-

nance, 257general expression for, 258

Q of spherical cavity, 576Quadrupole fields of oscillating source,

275see also Multipole fields

Quadrupole moment, electrostatic, 99,100

interaction of, with field gradient,101, 128

Quadrupole moment, nuclear, 102

of oscillating source, 275see also Multipole momen t

Radiated electromagnetic energy, trans-formation properties of, 39 0

Radiation, angular and frequency dis-tribution, for charge in periodic

motion, 50 1angular and frequency distribution,

for magnetic moments, 481angular and frequency distribution ,

for ultrarelativistic particle, 481 f .angular and frequency distribution,

general result for acceleratedcharge, 480

angular distribution of, for acceler-ated charge, 47 2

angular distribution of, for ultrarela-tivistic particle, 474

from creation of charge, 526from disappearance of charge, 528

from disappearance of magnetic mo-ment, 53 1

from electric dipole, 272from electric quadrupole, 275from full-wave antenna, 279from half-wave antenna, 279from linear antenna, 277 f ., 562 f .from localized source, 269from magnetic dipole, 274from orbital electron capture, 528 f .from short antenna, 27 3in beta decay, 526 f .in collisions, 506 f . ; see also Brems-

strahlungmultipole, see Multipole radiation

Radiation condition, 28 2Radiation cross section x(w), 510Radiation damping, see Radiative re-

actionRadiation length, 519Radiation pressure, 201Radiation resistance, 280Radiation zone, 269

in diffraction, 292Radiative energy loss, in accelerators,

47 1

in collisions, 513, 518

Index

Radiative reaction, 581 f .

and line breadth, 600and shift of energy level, 6 00

characteristic time T, 580effective force of, 582equations of motion including, 582,

597, 609Radius of the electron, 490, 589Rayleigh scattering law, 573, 6 03Reaction cross section, definition of, 60 6

Reaction threshold, 400Reactive effects of radiation, see Radia-

tive reactionReHectioa, from sphere in diffraction,

301, 302of charged particle from region of

large magnetic field, 42 3of plane waves, 216 f .of radio waves from ionosphere, 229total internal, 22 1

Refraction of plane waves, 216 f ., seealso Index of refraction

Relativistic effects in angular and fre-quency distributions of radiation,

474, 476, 484, 50 1

Relativistic invariance, of action inte-gral, 406

of 4-dimensional Laplacian, 37 5of 4-dimensional volume element, 3 76of 4-vector scalar products, 375of 4-vector scalar products, use in

kinematics, 395, 396, 39 8of phase of plane wave, 363, 383of products of fields, 38 9of radiated power, 469of radiation cross section, 515of wave equation, 3$ 8

Relativistic notation, 37 7Relativistic transformation, and Thomas

precession, 367from CM system to laboratory, 400 f.

of acceleration, 388

of charge and current densities, 378of coordinates, 35 7of electromagnetic fields, 380, 413,

41 4of electromagnetic fields, of moving

point charge, 38 1of 4-vectors and tensors, 374 f.

637

Relativistic transformation, of momen-tum and energy, 392

of potentials, 378of velocities, 360of wave vector and frequency, 363,

383Relativity, special theory of, 347 f .

special theory of, postulates of, 353Renormalization in quantum electrody-

namics, 594Resistance, see Conductivity, Ohm's law,

Radiation resistanc eResonance fluorescence, 604Resonant cavity, cylindrical, 254, 267

energy stored in, 25 7modes of oscillation in, 253power losses in walls of, 255 f .Q of, 256, 258, 57 6resonant frequencies of, 253spherical, 57 6

Resonant frequency, broadening of, dueto losses in cavity, 25 6

of atomic oscillator, 120, 234, 438of resonant cavity, 25 3shift of, due to radiation damping,

600Retarded Green's function, 185, 269,

283Retarded time, 1 85, 465Reynolds number, magnetic, 314Rotation of coordinates in successive

Lorentz transformations, 367Rutherford scattering, 45 2

Scalar, under Lorentz transformations,

374Scalar potential, electrostatic, definition

of, 8for time-varying fields, 179

magnetostatic, 156, 15 8Scattering cross section, classical, rela-

tion of, to impact parameter, 452for radiation, by conducting sphere,

304,572for radiation, by free charges, 489for radiation, by oscillator, 60 3for radiation, by quasi-free charges,

492for radiation, definition of, 489

838 Index

Scattering cross section, for radiation,

resonant, 604

Scattering of particles, by atoms, 451 f .effect of atomic screening on, 453

effect of nuclear size on, 454mean square angle of, 456multiple, 45 8single, 458total atomic cross section for, 455

Scattering of radiation, by conductingsphere, at long wavelengths, 569 f .

by conducting sphere, at short wave-lengths, 299 f .

by free charges, 488by oscillator, 602 f.by quasi-free charges, 491 f.coherent and incoherent, 493Thomson cross section for, 489

Screening by atomic electrons, effect of,on bremsstrahlung, 516 f .

effect of, on small-angle scattering,453

Selection rules for multipole transitions,549

Self-energy, classical, 588quantum mechanical, 593

Self-energy and momentum, 590 f .

covariant definition of, 594 f.

transformation properties of, 591

Self-fields of charged particle, 585Self-force, electromagnetic, 584 f .

Self-stress, and Poincar6 stresses, 592

Lorentz transformation of, 591of charged particle, 590

Separation of variables, in cylindrical

coordinates, 69in rectangular coordinates, 47

in spherical coordinates, 54Shielding, electrostatic, with hollow di-

electric, 12 9magnetic, with permeable shell, 162,

166Skin depth, 225, 238

and Q of a cavity, 25 8

and surface resistance, 240

in plasma, 227

Shell's law, 216

Solenoid, 16 5Sources of multipole radiation, 553,

556Space-like and time-like separations, 370Special theory of relativity, see Rela-

tivitySpecular reflection from a sphere, 302Sphere, conducting, and point charge,

27, 31, 3 3general solution with Green's func-

tion, 40 f ., 8 1 f.in uniform electric field, 33

of inversion, 3 5scattering of radiation by, 299 f .,

569 f .uniformly magnetized, 156uniformly magnetized, in external

field, 160Spherical Bessel functions, see Bessel

functionsSpherical coordinates, 54

delta function in, 79Laplace's equation in, 5 4

Spherical harmonics, Y1M, 64 f .

addition theorem for, 6 7and angular momentum, 542completeness relation for, 65explicit forms of, 66in magnetostatics, 144, 150operations of L on, 542orthogonality of, 6 5sum rule for, 69vector, see Vector spherical har-

monicsSpherical scalar waves, 538 f .Spherical vector waves, 543 f .Spherical wave expansion, of E and B,

546of plane scalar wave, 567

of plane vector wave, 569

Spin-orbit interaction, 368

Stabilization of plasma column, 327 f .

Standards, units and, 612

Standing waves in a resonant cavity,252

Step function, representation by Le-gendre polynomials, 5 9

B(r) , definition of, 234

Index

Stokes's theorem, 9Stored energy in resonant cavity, 25 6Stress, 19 3Stress-energy-momentum tensor, 193,

385and conservation laws, 386vanishing trace of, 385see also Self-stress

Superposition principle, see Linear su-perposition

Surface-charge density, and disconti-nuity in normal E and D, 9, 110

and electrostatic force, 23effective magnetic, 159on conducting sphere, 29, 34on perfect conductor, 23 6

on sphere with line charge inside, 84on thfn charged disc, 9 3polarization, 112, 115, 117potential of, 1 0

transformation of, in method of in-version, 37

Surface current, effective, 240effective magnetic, 15 9for perfect conductor, 236in diffraction, 289

Surface distributions, of charge, 9of dipole moment, 10

Susceptibility, electric, 109, 11 9simple models for, 119 f .

Synchrotron radiation, 481 f.angular and frequency distribution of,

484, 486, 487from Crab nebula, 488, 504polarization of, 484, 504

Tension along lines of magnetic field,315, 328

Tensor, electromagnetic field-strength,379

Lorentz transformation of, 375Maxwell's stress, 1 93relation to dyadic, 193stress-energy-momentum, 385

Thermonuclear plasmas, 320 , 326, 329,343

containment of, by magnetic mirrors,423

639

Thermonuclear plasmas, instabilities in,326 f .

Thomas factor, 368Thomas precession, 364 f .Thomson cross section, 489Thomson scattering, 48 8

quantum modifications to, 490Thomson's theorem, 25Time dilatation, 357 f.

experiment on, with pi mesons, 359graphical representation of, 374

Time-like and space-like separations,37 0

Torque, on current distribution, 137

on magnetic dipole, 132, 150Transformation, see Relativistic trans-

formation, Galilean transformationTransition probability, 55 8

in hydrogen-like atoms, 502, 608Transmission coefficient of circular

aperture, 294, 308Transmission line, dominant mode in,

243examples of, 265, 266relation between L and C for, 199

Transverse electric (TE) waves, atten-uation of, in wave guides, 25 1

connection of, with multipole mo-ments, 553 f .

cylindrical, 24 3in dielectric wave guide, 261in rectangular wave guide, 246spherical, 545

Transverse electromagnetic (TEM)waves, 243

absence of, in hollow wave guides,244

on transmission lines, 264, 265Transverse magnetic (TM) waves, at-

tenuation of, in wave guides, 25 1connection of, with multipole mo-

ments, 553 f.cylindrical, 24 3in cylindrical cavity, 25 4in dielectric wave guide, 263spherical, 545

Transverse waves, in conducting me-dium, 223

640 Index

Transverse waves, in magnetohydrody-namics, 33 1

in plasma, 226, 339

plane, 204Traveling wave solutions, 203, 212

in wave guide, 244

Uncertainty principle, 209, 21 5use of, to obtain quantum-mechanical

modifications, 440, 442, 453, 455,

511, 527, 532, 59 9Uniqueness theorem, 1 5

use of, with Legendre polynomial ex-

pansion, 61, 63Units, and relative dimensions of elec-

tromagnetic quantities, 613 f .appendix on, 611 f.conversion between Gaussian and

mks, 62 1

different system of electromagnetic,61 6

different systems of electromagnetic,important equations in, 61 8

table for conversion of, 619, 620

Van Allen belts, problems illustratingprinciples, 427

Vector field, decomposition of, intolongitudinal and transve rse parts,182, 19 9

Vector Green's theorem, 283 f .Vector potential, for t ime-varying fields,

179in magnetostatics, 139 f .in non-Cartesian coordinates, 14 1of loca l ized oscillating source, 269 f,of magnetic dipole, 14 6

of oscillating electric dipole, 27 1of oscillating electric quadrupole, 275of oscillating magnetic dipole, 274

Vector spherical harmonics, 545 f .

absolute square of, 55 1explicit forms for l = 1, 2, 3, 551,

565, 57 3integral theorems involving, 556, 568sum rule for, 553

Vector theorem, divergence, 6Green's, 283, 28 5Stokes's, 9

Vector theorem, with surface and vol-ume integrals, 284

with vector spherical harmonics, 556,568

Velocity addition law, relativistic, 360Velocity fields of point charge, 467Velocity of light, constancy of, 35 3

numerical value of, 61 4Velocity of particle, relativistic, 393

Velocity selector, 414, 42 6Virtual quanta, method of, 5 20 f .

method of, and bremsstrahlung, 525

method of, examples of use of, 520,525, 536

spectrum of, 524Viscosity, coefficient of, 31 1

effect of, on magnetohydrodynamicflow, 316 f ., 332

magnetic, 31 4

Wave equat ion, 180, 183covariant solution of, 388for scalar and vector potentials, 180from Maxwell's equations, 203, 543Green's function for, 185initial-value problem for, 186, 200one-dimensional solution of, 200transverse, in wave guide, 24 1

Wave guide, 244 f .attenuatian in, 249 f .boundary conditions in, 243

cutoff frequency in, 245dielectric, 259 f .modes in circularly cylindrical, 266modes in rectangular, 246modes in triangular, 267possible modes of propagation in, 243

Wavelength in wave guide, 24 5Wave number, and frequency, as 4-ve---

tar, 383connection of, with frequency, 203,

208, 227, 338, 33 9

Debye, 340imaginary part of, due to losses, 249,in wave guide, 24 5spread of, in wave packets, 209

Wave packets, in one dimension, 20Ff .

propagation of, in a dispersive me-dium, 210, 212

Index

Wave packets, spreading in time, 215

Waves, Alfven, 331, 34 4

AIfven, attenuation of, 33 3Alfven, with displacement current,

334magnetohydrodynamic, 329 f .magnetosonic, 33 1see also plane wave, Spherical waves,

Transverse waves

641

Weizsacker-Williams method, 520 f .Williams-Weizsacker method, 520 f .Work, relation of, to potential energy,

8, 20Work function of metal, and images,

3 2Wronskian, 8 5

for Bessel functions, 8 6for spherical Bessel functions, 541