www.TheSolutionManual.com www.TheSolutionManual.com MCtfxcvn www.TheSolutionManual.com Classical Electrodynamics by JOHN DAVID JACKSON Professor of Physics, University of Illinois CLASSICAL www.TheSolutionManual.com ELECTRODYNAMICS John Wiley & Sons, Inc., New York • London • Sydney febi-axov- www.TheSolutionManual.com ii 20 Copyright © 1962 by John Wiley & 19 18 17 16 15 14 Sons, Inc Ail rights reserved This book or any part thereof must not be reproduced in any written permission form without the of the publisher Printed in the United States of America Library of Congress Catalog Card Number: 62-8774 ISBN 471 43131 To the memory of my father, www.TheSolutionManual.com Walter David Jackson www.TheSolutionManual.com tum quanClassical electromagnetic theory, together with classical and for training theoretical present-day of core the forms mechanics, undergraduate and graduate physicists A thorough grounding in these training subjects is a requirement for more advanced or specialized magnetism Typically the undergraduate program in electricity and two or perhaps three semesters beyond elementary physics, with and elaborathe emphasis on the fundamental laws, laboratory verification phenomena, and tion of their consequences, circuit analysis, simple wave involves radiation The mathematical tools utilized include vector calculus, series, ordinary differential equations with constant coefficients, Fourier equations, differential and perhaps Fourier or Laplace transforms, partial Legendre polynomials, and Bessel functions As a general rule a two-semester course in electromagnetic theory is such a course that my book given to beginning graduate students in electromagnetism is course graduate is designed My aim in teaching a subject matter as a basic the present to least threefold The first aim is It is for at magnetic coherent whole, with emphasis on the unity of electric and mathematical of mode in the and basis physical their phenomena, both in number description The second, concurrent aim is to develop and utilize a electromagnetic of topics in mathematical physics which are useful in both and Green's theorems Green's include These mechanics wave and theory and functions, orthonormal expansions, spherical harmonics, cylindrical purimportant A third and perhaps most spherical Bessel functions of pose is the presentation of new material, especially on the interaction www.TheSolutionManual.com Preface VIII Preface relativistic charged particles with electromagnetic fields In this last area personal preferences and prejudices enter strongly My choice of topics is governed by what I feel is important and useful for students interested in theoretical physics, experimental nuclear and high-energy physics, and that as yet ill-defined field of plasma physics The book begins in the traditional manner with electrostatics The first six chapters are devoted to the development of Maxwell's theory of Much of the necessary mathematical apparatus is constructed along the way, especially in Chapters and 3, where boundaryvalue problems are discussed thoroughly The treatment is initially in terms of the electric field E and the magnetic induction B, with the derived macroscopic quantities, D and H, introduced by suitable averaging over ensembles of atoms or molecules In the discussion of dielectrics, simple classical models for atomic polarizability are described, but for magnetic materials no such attempt is made Partly this omission was a question of space, but truly classical models of magnetic susceptibility are not possible Furthermore, elucidation of the interesting phenomenon of ferromagnetism needs almost a book in itself The next three chapters (7-9) illustrate various electromagnetic pheno- mena, mostly of a macroscopic Plane waves in different media, including plasmas, as well as dispersion and the propagation of pulses, are treated in Chapter The discussion of wave guides and cavities in Chapter is developed for systems of arbitrary cross section, and the problems of attenuation in guides and the Q of a cavity are handled in a very general sort way which emphasizes the physical processes involved The elementary theory of multipole radiation from a localized source and diffraction occupy Chapter in many Since the simple scalar theory of diffraction is covered optics textbooks, as well as undergraduate books on electricity and magnetism, I have presented an improved, although theory of diffraction based on vector rather than scalar still approximate, Green's theorems' The subject of magnetohydrodynamics and plasmas receives increasingly more attention from physicists and astrophysicists Chapter 10 represents a survey of this complex field with an introduction to the main physical ideas involved The first electricity nine or ten chapters constitute the basic material of classical graduate student in physics may be expected and magnetism to have been exposed to lower A much of this material, perhaps at a somewhat an undergraduate But he obtains a more mature view of it, understands it more deeply, and gains a considerable technical ability in analytic methods of solution when he studies the subject at the level of this book He is then prepared to go on to more advanced topics The advanced topics presented here are predominantly those involving level, as the www.TheSolutionManual.com electromagnetism 62S Index Cylindrical coordinates, Laplace's in, Gradient, Laplacian, separation of variables waves in, Differential operator relations, see 69 in, 69 241 Damping, of magnetohydrodynamic waves, 333 of oscillations in cavity, 255 of plasma oscillations, 340 see also Radiative reaction Darwin-Breit interaction, 411 Debye-Hiickel screening or radius Debye length, 342 Debye wave number, 340 Decay, of particle, relativistic kine- matics of, 394 of pi mesons, time dilatation in, 359 Delta function, charge densities in terms of, 4, 36, 82, 83 current densities in terms of, 141, 278, 563 V (l/r), 79 84, 96 properties of, three-dimensional, in cylindrical coordinates, 84 three-dimensional, in Time 313 fields, 313 dilatation loss, in radiation problems, 271, 274, 507 100 magnetostatic, 143, 147 of conducting sphere, 34 Dipole spherical co- ordinates, 79 Density effect in energy loss, 443 connection of, with Cherenkov radi- 448 Dielectric constant, classical result for, 234, 446 fields, electrostatic, of dielectric sphere, 115 of electrostatic dipole layer, of magnetized sphere, 157 oscillating electric, 271 274 oscillating magnetic, Dipole moment, electrostatic, 100 induced, 120 interaction between two, 102 for Alfven waves, 334 for plasma, 227, 451 for plasma in magnetic field, magnetostatic, 132, 146 magnetostatic, energy of, 150 magnetostatic, force on, 149 228 Dielectrics, 108 anisotropic, fields, 435 integral representations for, 47, waves magnetostatic, of current loop, 233 boundary conditions, 110 boundary-value problems with, electrostatic energy in, 123 method of images for, 111 Dielectric wave guide, 259 at optical frequencies, 264 cutoff frequency in, 263 possible Diffusion of magnetic Diffusion time of magnetic Dimensions, discussion of, 611 Dipole approximation, in energy 13 in arbitrary coordinates, ation, 288 by circular aperture, 292, 307 by half-plane, 306 by rectangular opening, 307 by small apertures, 297, 307, 308 by sphere in short wavelength limit, 299 comparison of scalar and vector approximations, 296 Fresnel and Fraunhofer, 292 Kirchhoff approximation in, 282 scalar Huygens-Kirchhoff theory of, 280 Smythe's vector theorem for, 287 use of Green's theorem in, 281, 283 vector Kirchhoff approximation, 285 Dilatation of time, see definition of, equal to etc Diffraction, Babinet's principle in, 143, 147 in, magnetostatic, relation of, to angular 10 f modes of propagation, 260 momentum, 148 magnetostatic, torque on, 132, 150 oscillating electric, 271 oscillating magnetic, Dipole sum ation, 274 rule for absorption of radi- 606 Dirac delta function, see Delta function www.TheSolutionManual.com equation 629 Index Dirac's relativistic force equation, 609 boundary conditions, 16 Disc, potential of charged conducting, Discontinuity, of electric field, and propagation, 208 234 plasma oscillations, 337, 339 Displacement, definition of, 108 Displacement current, 178 in plasma in wave guides, 248 wave propagation, 208 oscillations, 332, 340 magnetic fields, Drift of lines of force in conducting 314 by curvature of 418 caused by magnetic field gradient, 416 Drift velocity, caused lines of force, charged par- external fields, 404 f 88 wave 244 in, 528 Elliptic integrals, use of, 96, 142, 168 Energy, electrostatic potential, 8, 20 in dielectrics, 123 of charge distribution in external field, 101 magnetic, 173 magnetic dipole, 150 of magnetically permeable body, 176 of permanent magnets, 167 self-, 22, 588 Energy conservation between particles and fields, 190, 386 Energy density, electromagnetic, 205 189, 21 tum tensor, 385 Energy flow, 190 in wave guide, 248 tion, guides, 600 loss, collisional, classical, collisional, density effect in, 438 443 plasma, 450 and method of quanta, 536 collisional, in in box, 254 rectangular guide, 246 spherical cavity, 576 in cylindrical cavity, in Electron capture, radiation emitted Energy for fields in cylindrical in 584 velocity of, 208, 211 Energy-level shift due to radiative reac- expansion of Green's function in wave equation of, 4-4 element of stress-energy-momen- Eigenfunctions, 87 for model 589 490, 589 electrostatic, integral equations, 91 Dyadic, 193 relativistic Electron, classical Electrostatic units, see Units E and B, 413 inhomogeneous 417 Electromagnetic units, see Units Electromotive force, definition of, 170 Electrostatic force on conductor, 23 radius of, Divergence theorem, Doppler shift, relativistic, 363 transverse, 364 Drift of charged particles, in crossed of, multipole expan- difficulties with, Dissipative effects, in cavities, 255 terms fields, of localized oscillating source, 270 tion, ticle in potentials, 8, 179 transformation of, 380 E lectromagnetic stress - energy - momentum tensor, 385 Dispersion relation, for index of refrac- Dynamics of from sion of, 543 anomalous, 211 208 in ionosphere, 229 in plasmas, 337 Dual Electric field, definition of, Electromagnetic f in dielectrics, fluid, Di- fields, "Electric" waves, 243, 545 of potential, 12 in Dipole moment relativistic Dispersion, and causality, 234 in pole derivable 89 for dipole, see Electric 88 collisional, collisional, radiative, virtual quantum-mechanical, 440 accelerators, 471 in www.TheSolutionManual.com Dirichelet 630 Index loss, radiative, in nonrelativistic collisions, sphere, 30, 31, 33 radiative, in relativistic collisions, Energy of Energy 518 relativistic trans- transfer, in collision with bound particle, formation Force, between point charge and 513 391 of, between two parallel wires, 136 Lorentz, 191 on charged surface, 23 on current distribution charge, 434 in Coulomb in discrete field, in elastic collision, tial, of 404 436 to oscillator, Equation 430 collision, amounts, 439 motion, integrodifferen- with radiative reaction, 597 Equations of motion with radiative reaction, 582, 583 Ether, 347 Ether drift, experiment on, 349 Expansion, of circularly polarized plane wave, 569 of e***/18, 541 of retarded quantity, 586 of scalar plane wave, 567 Expansion of |x — x'| _1 in cylindrical coordinates, 86, 96 for fluid, 311 in co variant form, 384, 405 with radiative reaction, integrodiffer- ential, see 4-vectors and tensors, 374 Fourier integrals, 47 Fourier series, 46 Fourier transforms, of exponentially damped wave, 256 of fields of charge in uniform motion, 437 Orthonormal of expan- sions wave packet, 209, 212 Fraunhofer diffraction, see Diffraction Freezing in of lines of force, 313, 319 Faraday's law, for moving circuit, 172 in differential form, 173 in integral form, 170 Fields, of moving particle, 467 of 597 4-vector character of radiation, 390 88 in spherical coordinates, 62, 69 Expansions, magnetic on dielectric body, 126, 127, 131 on magnetically permeable body, 177 on magnetic dipole, 149 on permanent magnets, 167, 168 radiative reaction, 582 Force equation, Abraham-Lorentz, 582 , in plane waves, in 137, 148 relativistic of, to pulse particle, equivalence of radiation, 382, 521 Fields of uniformly moving charge, 381, Frequency spectrum of radiation, 478, 480 qualitative aspects of, 476 see also Bremsstrahlung, Radiation Fresnel diffraction, see Diffraction Fresnel formula for velocity of light in moving media, 349 Fresnel formulas for reflected and re- 467 Fourier transforms of, 437 in dielectric, Fourier transforms of, fracted plane waves, 219, 220 Fresnel integrals and diffraction, 306 445 FitzGerald-Lorentz 352, contraction, Galilean invariance, 348, 369 and Faraday's law, 171 357 Galilean reference frames, 353 graphical representation of, 373 Fizeau's experiment, 349 Galilean relativity, 348, 369 Flow of Galilean transformation, 171, 388 Gauge, Coulomb or transverse, viscous conducting fluid be- tween parallel plates, 316 Force, between charge and image Lorentz, 181 charge, 30, 32, 51 between current-carrying 182 circuits, 136 Gauge invariance, 181, 426 140, www.TheSolutionManual.com Energy 631 Index statics, magneto- for 140 257 Hamiltonian for time-varying fields, 181 Gaussian units, see Units Gauss's law, applied to surface-charge distribution, form of, form of, differential integral Half-width of resonance, 601 connection of, with Q in a cavity, Gradient, in spherical vector form, 544 of electric field and quadrupole in- particle, for Hankel function, see Bessel functions Hankel transform, 77 Hartmann number, 317 Heaviside-Lorentz units, see Units Helicity, definition of, Hemispheres, teraction, 101, 128 charged relativistic 408 206 at different potentials, 42 from symmetry and unique- of magnetic induction and force on potential dipole, 149 of magnetic induction, particle drift potential in terms of Legendre poly- in, Green's Green's 416 first identity, 14 function time-dependent for wave equation, 183 function for wave of, Bessel in functions, 84,96 expansion of, in Legendre polynomials, 62 expansion of, in spherical harmonics, 69 expansion of, eigenfunction 88 of, for of, 452 in brems- in energy 510, 514 minimum and maximum 432, 440 minimum method of in virtual quanta, 520 Green's reciprocation theorem, 25 Impedance of a transmission Green's theorem, 15 use of, with wave equation, 188 vector equivalent of, 285 Grounded conductor, Group velocity, 211 definition of, and phase velocity, 211 and phase velocity in guide, 249 in electronic plasma, 340 in wave guide, 249 guide Gyration frequency, 228, 411 line, 265 Index of refraction, analytical properties of, 234 and phase and group velocities, 211 dispersion relation for, 234 use of, in diffraction, 280 Wave 111 magnetic media, 167 for point charge near of, for dielectrics, conducting sphere, 27 Impact parameter, and scattering loss, for sphere, 41 Guides, see method method method in 33 field, strahlung, for rectangular box, 89 box, uniform minimum and maximum for cylindrical box, 97 rectangular Image charges, see Images Images, method of, 26 f method of, for conducting sphere angle, for concentric spheres, 80 for nomials, 61 Huygens's principle, 188, 280 Hydrodynamics, see Magnetohydrodynamics equation, spherical wave expansion of, 541 Green's function in electrostatics, 18 examples of use of, 82, 83 expansion 62 Hysteresis, magnetic, 153 retarded, 185, 269 Green's ness, 27 see also Dielectric constant Inductance, 198, 199 high-frequency, 225 of transmission line, 265, 266 Induction, Faraday's law of, 170 Initial-value problem, for scalar equation, 186 Poisson's solution of, 188 wave www.TheSolutionManual.com Gauge transformations, 632 Index of a pinched plasma col- umn, 326 Internal electric field at molecule, 116 Invariance, of length-time element in 369 of separate length and time elements in Galilean relativity, 369 special relativity, see also Adiabatic invariance, Relativistic Inversion, method ties invariance method of, 35 examples Ionosphere, 226 f orthogonality, 58 recursion relations, 59 Rodrigues's formula for, 57 see also Spherical harmonics Lenz's law, 170 Lienard's of, of, forms L'Hospital's rule, 83 behavior of charge densiunder, 37 method Legendre polynomials, explicit of, 57 integrals involving, 60 53 of, 39, 40, generalization of Larmor power formula, 470 Lienard-Wiechert potentials, 465 fields, 467 Lifetime, of multipole transitions, 558 of pi mesons in motion, 359 364 Ives-Stilwell experiment, Light cone, 370 Jacobian, in Lorentz transformation of coordinates, 376 in transforming delta functions, Kinematics, relativistic, 394 79 Linear superposition, of electric fields, of plane waves, 203, 208 of potentials, Line breadth due to radiative reaction, 600 f Kirchhoff diffraction, see Diffraction Localized source, see Multipole, Multi- Kirchhoff's integral representation, 188 pole moment use of, in diffraction, 280 Longitudinal vector equivalent of, 283 Lorentz condition, 181 Klein-Nishina formula, 490 fields in in covariant form, conductor, 223 378 Lorentz force, 191 Lagrangian, for ticle, relativistic charged par- 407 two interacting charged particles, 411 Lamb shift, 602 Landau damping, 340 Laplace's equation, 13 solution force equation in covariant form, 405 for general Lorentz of, Lorentz invariant, see Scalar, Relativistic invariance Lorentz line shape, 436, 601, 604 for cavity, 256 Lorentz-Lorenz relation, 119 in cylindrical coordinates, 76, 77 general solution of, in spherical co- as orthogonal transformation in four dimensions, 371 ordinates, 67 69 48 in spherical coordinates, 54 uniqueness of solution of, 15 Laplacian and the angular momentum operator L, 543 Larmor power formula, 469 in cylindrical coordinates, in rectangular coordinates, relativistic generalization of, Lorentz transformation, 356 470 Legendre polynomials, 56 associated, 64 expansion of inverse distance 367 noncommuting, 357, 387 successive, 357, 367 infinitesimal, see also Relativistic transformation Loss, see Power loss Macroscopic averages, 104, 194 Macroscopic equations, derivation for electrostatics, 103 derivation of, for magnetostatics, in, 62 150 of, www.TheSolutionManual.com Instabilities 633 Index Macroscopic fields, definition of B of, and H, 153 definition of E and D, 108 Magnet, permanent, 161, 167 Magnetic dipole, see Dipole fields, Di- moment pole Magnetic field H, boundary conditions on, 154 definition of macroscopic, 153 see also Magnetic induction Magnetic flux density, see Magnetic in- duction Magnetic induction B, boundary conditions on, 154 Magnetohydrodynamic waves, 329, 344 effect of finite conductivity and viscosity, 333 with displacement current, 334 Magnetosonic waves, 331 Magnetostatics, basic equations of, 137 f multipole expansion in, 145 Maximum and minimum impact parameters, see Impact parameter Maximum and minimum scattering angles, 453, 455 Maxwell's equations, 177 in covariant form, 379 in different systems of units, 618 macroscopic, 194 definition of, 132, 134 plane wave solutions of circular loop, 141 spherical wave of, 204 solutions of, 546 of current element, 134 Maxwell of long wire, 135 Mean-square scattering angle, 456 Mean-value theorem, 25 Mesons, time dilatation experiment with, 359 Method of images, see Images of magnetized sphere, 156 of nonrelativistic moving charge, 134 relativistic unit of, transformation of, 380 defined in terms of force, Method 136 Magnetic mirror, 149, 423 Magnetic moment, adiabatic invariance of, 421 force on, in external field, 149 of localized current distribution, radiation of, because of disappearance 531 radiation by time-varying, 274, 481 Magnetic-moment density, see Magnet- ^ ization Magnet pressure, 315 "Magnetic" waves, 243, 545 Magnetization, definition of macro- scopic, 151 divergence of, as effective magnetic- charge density, 158 effective current of, 152 of current distribution, 146 by time-varying, 481, 553 Magnetohydrodynamic flow, 316 Magnetohydrodynamics, 309 f radiation equations of, 311 193 of inversion, see Inversion Michelson-Morley experiment, 349 Microwaves, see Diffraction, Resonant Wave cavity, guide Minkowski diagram, 374 Mixed boundary conditions, Mks 146 stress tensor, units, see Modes, in 266 16, 89 Units circularly cylindrical guide, 267 267 in rectangular guide, 246 in triangular guide, 267 TE and TM, in wave guide, 243 in cylindrical cavity, 254, in dielectric guide, 260, Momentum, canonical, for relativistic charged particle, 408 conservation of, between particles and fields, 191 electromagnetic, definition of, 192 of particle, of, relativistic transformation 392 Momentum part density, electromagnetic, as stress-energy-momentum 385 of tensor, www.TheSolutionManual.com Macroscopic equations, derivation for time-varying fields, 194 634 Index Momentum impulse in Coulomb 431 colli- Motion, see Particle motion circuits and law of induction, Moving 171 Multiple of particles, 456, 463 netic field, in in, electrostatic, expansion of potential energy 98 100 electrostatic, rectangular, magnetostatic, 145 radiating, near, induction, tion zones, and radia- 270 moment, Multipole moment Multipole expansion, of E and B, 546 of electromagnetic fields, 543 f of energy of charge distribution in external field, 101 of Green's function for wave equa- 541 of radiation by linear antenna, 562 of scalar plane wave, 567 of scalar potential, 98 of vector plane wave, 569 Multipole fields, 543 f 558 of linear antenna, 564 of oscillating source, 271, 273, 556 Multipole radiation, angular distribu- finite interval, 73, 95 of Bessel functions on infinite interval, 77 of Legendre polynomials, 58 of sines and cosines, 46 of spherical harmonics, 65 Orthogonal transformations, 371 Orthonormal expansions, 44 Fourier, on finite interval, 46 Fourier, on infinite interval, 47 Fourier-Bessel, 73, 95 Legendre, 59 see Plasma Oscillator strength, 438, 601 Parity of multipole fields, 550 Parseval's theorem, 478 in dipole field of earth, E and B, 412 427 411 f inhomogeneous B, 415, 421 uniform static B, 411 in external fields, in Penetration depth in plasma, 227 see also Skin depth Permeability, magnetic, Neumann boundary conditions, 16, 18 Neumann function, see Bessel functions on oscillations, Waves in tion of, 550 f by atoms and nuclei, 557 by linear antenna, 562 selection rules for, 549 454 spherical harmonics, 65 Particle motion, in crossed magnetostatic, 145 effect of, placian, etc Orthogonal functions, Bessel functions, 73 general, 44 Legendre polynomials, 57 spherical harmonic, 66 estimates of, for radiating atoms and cross sections, 309 Operator relations, see Gradient, La- Oscillations, energy and angular momentum radiated, 548 properties of, 546 Multipole moment, electrostatic, 99 Nuclear forces, moving medium, 312 Orthogonality, of Bessel functions on time-varying, 271, 273, 545 see also Dipole moment, Magnetic nuclei, mag- 345 validity of, for conducting fluid, expansion of interaction 101 electrostatic, tion, Ohm's law, 222 generalization of, for plasma in scattering Multipole, electrostatic, 98 in, Nuclear quadrupole moment, 102 interaction energy of, 101, 128, 129 scattering 153 Phase difference, and elliptic polarization, 205 between current and field in plasma, 227 between E and B in conductor, 224 www.TheSolutionManual.com sion, 635 Index 571 Phase velocity, and group velocity, 211 and group velocity in guide, 249 in wave guide, 246 of Alfven waves, 331, 334 of plane waves, 203 of plasma oscillations, 340 Photon, angular momentum in multipole, 549 angular momentum of, 201, 569 Pinch effect, and instabilities, 326 dynamic models of, 322 scaling law for, 325 steady-state, 320 Plane wave, electromagnetic, 202 f electromagnetic, expansion in spherical multipole waves, 569 electromagnetic, in conducting medium, 222 f electromagnetic, reflection and refraction of, 216 f scalar, general one-dimensional solution, 200, 212, 233 scalar, propagation in dispersive medium, 210 f scalar, properties of, 203 Plasma, conductivity of, 459 confinement of, by external fields, 329 confinement of, by magnetic mirrors, 424 confinement of, by self-fields, 320 definition of, 310 dielectric constant of, 451 energy loss in, 450 instabilities in column of, 326 longitudinal waves in, 337 transverse waves in, 226, 339 Plasma frequency, 227, 337 Plasma oscillations, and Boltzmann equation, 345 high-frequency, 335 f in external magnetic field, 346 Landau damping of, 340 longitudinal, 337 transverse, 339 Poincare stresses, 592 Poisson's equation, 12 equivalent integral equation, 15 general solution in spherical geometry, 81 uniqueness of solution of, 15 Polarizability, electronic, 120 models of, 119 molecular, 118 orientation, 122 Polarization, charge density, current 107, 112 from time-varying, density 196 definition of, 108 surface-charge density, 112, 115, 117 Polarization effects in energy loss, 443 Polarization of radiation, by reflection, 220 circular, elliptical, linear, 205 from accelerated charges, 468, 480 from multipoles, 272, 274, 551 from synchrotron, 484, 504 scattered by sphere, 572 X-rays, 509 Polarization vectors, 204, 207 Potential, electrostatic, in rectangular box, 49 of dipole layer, 11 of line charge, expansion in polar coordinates, 86 of point charge, expansion in cylindrical coordinates, 86 of point charge, expansion in eigenfunctions, 88 of point charge, expansion in spherical coordinates, 62, 69 of point charge, in cylindrical box, 97 of point charge, in rectangular box, 89 of surface-charge distribution, 10 Potential energy, see Energy Power, radiated, angular distribution of dipole, 272 radiated, angular distribution of half- and full-wave antenna, 279 radiated, distribution of (l,m) multipole, 551 www.TheSolutionManual.com Phase of plane wave, relativistic invariance of, 363 Phase shift for scattering by sphere, 636 Index Power, angular radiated, distribution of quadrupole, 275, 552 by charged radiated, 470, particle, Quadrupole moment, nuclear, 102 of oscillating source, 275 see also Multipole moment 472 by charged particle in ac471 by charge in arbitrary pe- celerators, radiated, riodic motion, 501 by multipoles, 550 radiated, f radiated, Larmor's formula for, 469 radiated, total, 272, 276, 553 Power Power flow, see tivity, Energy flow because of loss, conduc- finite 240 series solution, of Bessel's equa- general for result accelerated charge, 480 Poynting's theorem, 189, 197 Poynting's vector, definition of, 190 for plane wave, wave 205 ment, 531 guide, 248 Precession, Thomas, 364 f Precession frequency, 228, 411 Pressure, magnetic, 315 radiation, 201 Propagation, in ated charge, 472 angular distribution of, for ultrarelativistic particle, 474 from creation of charge, 526 from disappearance of charge, 528 from disappearance of magnetic mo- 70 of Legendre equation, 56 in f angular distribution of, for acceler- per unit length, in wave guides, 250 tion, for ultrarelativistic particle, 481 angular and frequency distribution, 239 in resonant cavities, 257 per unit area at conducting surface, Power Radiated electromagnetic energy, transformation properties of, 390 Radiation, angular and frequency distribution, for charge in periodic motion, 501 angular and frequency distribution, for magnetic moments, 481 angular and frequency distribution, anisotropic dielectric, 233 conducting medium, 223 in dispersive medium, 212 in hollow wave guide, 249 in plasma, 226 f Proper time, 369 in from from from from from from from from from electric dipole, electric 272 quadrupole, 275 full-wave antenna, 279 half-wave antenna, 279 linear antenna, 277 f., 562 f localized source, 269 magnetic dipole, 274 orbital electron capture, 528 f short antenna, 273 in beta decay, in collisions, 526 f 506 f.; see also Brems- strahlung multipole, see Multipole radiation Q of resonant cavity, 256 connection with half-width of reso- nance, 257 general expression for, 258 Q action of spherical cavity, 576 Quadrupole 275 fields of oscillating source, see also Multipole fields Quadrupole moment, electrostatic, 99, 101, 128 Radiation Radiation Radiation Radiation length, 519 pressure, 201 resistance, 280 zone, 269 in diffraction, 292 Radiative energy loss, in accelerators, 100 interaction Radiation condition, 282 Radiation cross section x( w )> 510 Radiation damping, see Radiative re- of, with field gradient, 471 in collisions, 513, 518 www.TheSolutionManual.com radiated, 637 Index and and line breadth, shift Relativistic transformation, of f tum and 600 of energy level, 600 580 effective force of, 582 equations of motion including, 582, 597, 609 Radius of the electron, 490, 589 Rayleigh scattering law, 573, 603 Reaction cross section, definition of, 606 Reaction threshold, 400 characteristic time t, Reactive effects of radiation, see Radiative reaction from sphere Reflection, in diffraction, 301, 302 of charged particle from region of large magnetic field, 423 of plane waves, 216 f of radio waves from ionosphere, 229 total internal, 221 Refraction of plane waves, 216 f.; see also Index of refraction and frequency distributions of radiation, 474, 476, 484, 501 Relativistic effects in angular momen- energy, 392 of potentials, 378 of velocities, 360 of wave vector and frequency, 363, 383 Relativity, special theory of, 347 f special theory of, postulates of, 353 Renormalization in quantum electrodynamics, 594 Resistance, see Conductivity, Ohm's law, Radiation resistance Resonance fluorescence, 604 Resonant cavity, cylindrical, 254, 267 energy stored in, 257 modes of oscillation in, 253 power losses in walls of, 255 f Q of, 256, 258, 576 resonant frequencies of, 253 spherical, 576 Resonant frequency, broadening of, due to losses in cavity, 256 of atomic oscillator, 120, 234, 438 of resonant cavity, 253 shift of, due to radiation damping, kinematics, 395, 396, 398 of phase of plane wave, 363, 383 600 Retarded Green's function, 185, 269, 283 Retarded time, 185, 465 Reynolds number, magnetic, 314 Rotation of coordinates in successive Lorentz transformations, 367 Rutherford scattering, 452 of products of fields, 389 of radiated power, 469 Scalar, under Lorentz transformations, Relativistic gral, invariance, of action inte- 406 of 4-dimensional Laplacian, 375 of 4-dimensional volume element, 376 of 4-vector scalar products, 375 of 4-vector scalar products, use in of radiation cross section, 515 of wave equation, 388 Relativistic notation, of, 377 Relativistic transformation, and Thomas for time-varying fields, 179 magnetostatic, 156, 158 precession, 367 from 374 Scalar potential, electrostatic, definition CM system to laboratory, 400 f Scattering cross section, classical, relation of, to impact parameter, of acceleration, 388 452 of charge and current densities, 378 for radiation, by conducting sphere, of coordinates, 357 of electromagnetic 304, 572 for radiation, by free charges, 489 fields, 380, 413, 414 for radiation, by oscillator, 603 by quasi-free charges, of electromagnetic fields, of moving point charge, 381 for radiation, of 4-vectors and tensors, 374 for radiation, definition of, 489 f 492 www.TheSolutionManual.com Radiative reaction, 581 63S Index Scattering cross section, for radiation, Scattering of particles, by atoms, 451 atomic screening on, 453 454 456 square angle of, tivity 458 atomic cross section for, 455 Scattering of radiation, by conducting sphere, at long wavelengths, 569 f by conducting sphere, at short wave- 299 27, 31, 33 Selection rules for multipole transitions, scattering of radiation momentum, 590 covariant definition of, 594 f f transformation properties of, 591 585 Self-force, electromagnetic, 584 f Self-stress, and Poincare stresses, 592 Lorentz transformation of, 591 Self -fields of charged particle, 590 Separation of variables, in cylindrical coordinates, 69 in rectangular coordinates, 129 magnetic, with permeable shell, 162, and Q of a cavity, 258 and surface resistance, 240 in plasma, 227 Snell's law, 216 f , external functions Spherical coordinates, 54 delta function in, 79 Laplace's equation in, 54 Y lm , 64 f completeness relation for, 65 explicit forms of, 66 in magnetostatics, 144, 160 operations of L on, 542 orthogonality of, 65 sum rule for, 69 vector, see Vector spherical har- monics Spherical scalar waves, 538 f Spherical vector waves, 543 f 47 Shielding, electrostatic, with hollow di- Skin depth, 225, 238 299 Spherical Bessel functions, see Bessel Spherical in spherical coordinates, 54 166 by, 569 f uniformly magnetized, 156 uniformly magnetized, in field, 160 and angular momentum, 542 quantum mechanical, 593 particle, 33 field, addition theorem for, 67 Self -energy, classical, 588 electric, 40 f., 81 f uniform electric tion, in Spherical harmonics, 549 of charged general solution with Green's func- of inversion, 35 f by free charges, 488 by oscillator, 602 f by quasi-free charges, 491 f coherent and incoherent, 493 Thomson cross section for, 489 Screening by atomic electrons, effect of, on bremsstrahlung, 516 f effect of, on small-angle scattering, 453 Self-energy and 553, Space-like and time-like separations, 370 total lengths, radiation, Specular reflection from a sphere, 302 Sphere, conducting, and point charge, multiple, 458 single, multipole Special theory of relativity, see Rela- effect of nuclear size on, mean of 556 f wave expansion, of E and B, 546 of 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 gendre polynomials, 59 0(0, definition of, 234 by Le- www.TheSolutionManual.com effect of Solenoid, 165 Sources resonant, 604 Stokes's theorem, Stored energy in resonant cavity, 256 Stress, 193 193, Stress-energy-momentum tensor, 385 and conservation laws, 386 vanishing trace of, 385 see also Self-stress Superposition principle, see Linear superposition Surface-charge density, nuity in normal and electrostatic force, effective magnetic, and disconti- E and D, 9, 110 23 159 on conducting sphere, 29, 34 on perfect conductor, 236 on sphere with line charge inside, 84 on thin charged disc, 93 polarization, 112, 115, 117 potential of, 10 transformation of, in method of in- Transformation, see Relativistic transformation, Galilean transformation Transition probability, 558 in hydrogen-like atoms, 502, 608 Transmission coefficient of circular aperture, 294, 308 Transmission version, 37 in diffraction, 289 Surface distributions, of charge, moment, 10 ments, 553 Susceptibility, electric, 109, 119 simple models for, 119 in dielectric of, polarization of, 484, 504 relation to dyadic, 193 stress-energy-momentum, 385 Thermonuclear plasmas, 320, 326, 329, spherical, by magnetic mirrors, f wave guide, 261 wave guide, 246 545 (TEM) electromagnetic Transverse waves, 243 of, in hollow wave guides, 244 on transmission lines, Transverse magnetic 264, 265 (TM) waves, at- tenuation of, in wave guides, 251 connection of, with multipole mo- ments, 553 cylindrical, f 243 in cylindrical cavity, in dielectric spherical, of, in, in rectangular absence Tension along lines of magnetic field, 315, 328 Tensor, electromagnetic field-strength, 379 Lorentz transformation of, 375 Maxwell's stress, 193 343 containment 423 dominant mode 243 cylindrical, f Synchrotron radiation, 481 f angular and frequency distribution 484, 486, 487 from Crab nebula, 488, 504 line, 243 examples of, 265, 266 relation between L and C for, 199 Transverse electric (TE) waves, attenuation of, in wave guides, 251 connection of, with multipole mo- Surface current, effective, 240 effective magnetic, 159 for perfect conductor, 236 of dipole Thermonuclear plasmas, instabilities in, 326 f Thomas factor, 368 Thomas precession, 364 f Thomson cross section, 489 Thomson scattering, 488 quantum modifications to, 490 Thomson's theorem, 25 Time dilatation, 357 f experiment on, with pi mesons, 359 graphical representation of, 374 Time-like and space-like separations, 370 Torque, on current distribution, 137 on magnetic dipole, 132, 150 wave 254 guide, 263 545 Transverse waves, in conducting medium, 223 www.TheSolutionManual.com 639 Index 640 Index Transverse waves, in magnetohydrodynamics, 331 in plasma, 226, 339 Vector theorem, with surface and vol- ume plane, 204 wave 284 568 Traveling wave solutions, 203, 212 in integrals, with vector spherical harmonics, 556, guide, Velocity addition law, relativistic, 360 Velocity fields of point charge, 467 244 light, constancy of, 353 numerical value of, 614 Velocity of particle, relativistic, 393 Velocity of Uncertainty principle, 209, 215 use of, to obtain quantum-mechanical modifications, 440, 442, 453, 455, Velocity selector, 414, 426 511, 527,532, 599 Virtual quanta, Uniqueness theorem, 15 use of, with Legendre polynomial ex- method method pansion, 61, 63 of, method of, 520 f and bremsstrahlung, 525 examples of use of, 520, and relative dimensions of electromagnetic quantities, 613 f appendix on, 611 f between and Gaussian mks, 621 system of electromagnetic, 616 spectrum of, 524 Viscosity, coefficient of, 311 effect of, on magnetohydrodynamic 316 f., 332 magnetic, 314 flow, different different systems of electromagnetic, important equations in, Wave equation, 180, 183 covariant solution of, 388 and vector potentials, 180 from Maxwell's equations, 203, 543 618 for scalar table for conversion of, 619, 620 Green's function for, 185 Van problems illustrating principles, 427 Vector field, decomposition of, into longitudinal and transverse parts, Allen belts, 182, 199 fields, in non-cartesian coordinates, 141 of localized oscillating source, 269 of magnetic dipole, 146 f of oscillating electric dipole, 271 of oscillating electric quadrupole, 275 of oscillating magnetic dipole, 274 Vector spherical harmonics, 545 f absolute square of, 551 forms for /= 1,2,3, 551, 565, 573 integral theorems involving, 556, sum rule for, 553 Vector theorem, divergence, Green's, 283, 285 Stokes's, 568 problem for, 186, 200 one-dimensional solution of, 200 transverse, in wave guide, 241 Wave guide, 244 f 249 f boundary conditions in, 243 cutoff frequency in, 245 dielectric, 259 f attenuation Vector Green's theorem, 283 f Vector potential, for time-varying 179 in magnetostatics, 139 f explicit initial- value modes modes modes in, in circularly cylindrical, 266 246 in triangular, 267 in rectangular, possible modes of propagation in, 243 Wavelength in wave guide, 245 Wave number, and frequency, as 4-vector, 383 connection of, with frequency, 203, 208, 227, 338, 339 Debye, 340 imaginary part in wave spread Wave guide, of, in of, due to losses, 249 245 wave packets 209 packets, in one dimension, 208 f propagation of, in a dispersive medium, 210, 212 www.TheSolutionManual.com 525, 536 Units, conversion of, 641 Index packets, spreading in time, 215 Alfven, attenuation of, 333 Alfven, with displacement We izsacker-Williams method, 520 f Williams-Weizsacker method, 520 f Work, relation of, to potential energy, Waves, Alfven, 331, 344 current, 334 magnetohydrodynamic, 329 f magnetosonic, 331 see also Plane wave, Spherical waves, Transverse waves 8, Work 20 function of metal, and images, 32 Wronskian, 85 for Bessel functions, 86 for spherical Bessel functions, 541 www.TheSolutionManual.com Wave www.TheSolutionManual.com ... www.TheSolutionManual.com Classical Electrodynamics by JOHN DAVID JACKSON Professor of Physics, University of Illinois CLASSICAL www.TheSolutionManual.com ELECTRODYNAMICS John Wiley & Sons, Inc.,... of my father, www.TheSolutionManual.com Walter David Jackson www.TheSolutionManual.com tum quanClassical electromagnetic theory, together with classical and for training theoretical present-day... when moving The special theory of relativity had its origins in classical electrodynamics even after almost 60 years, classical electrodynamics and delights still impresses example of the co variance