Appendix Electromagnetic Units

The question of electromagnetic units has been a vexing one for students of electromagnetic theory for generations, and is likely to remain so for the fore- seeable future. It was thought by the reformers of the 1930s, Sommerfeld [32] and Stratton [33] in particular, that the rationalized system now encompassed in the standard Syst`eme International (SI) would supplant the older cgs sys- tems, principally the Gaussian (G) and Heaviside–Lorentz (HL) systems. This has not occurred. This is largely because the latter are far more natural from a relativistic point of view; theoretical physicists, at least of the high-energy variety, use nearly exclusively rationalized or unrationalized cgs units. The advantage of the two mentioned cgs systems (there are other systems, which have completely fallen out of use) is that then all the electric and magnetic fields, E, D, B, H, have the same units, which is only natural since electric and magnetic fields transform into each other under Lorentz transformations. Electric permittivities and magnetic permeabilities correspondingly are di- mensionless. The reason for the continued survival of two systems of cgs units lies in the question of “rationalization,” that is, the presence or absence of 4πs in Maxwell’s equations or in Coulomb’s law. The rationalized Heaviside– Lorentz system is rather natural from a field theoretic point of view; but if one’s interest is solely electromagnetism it is hard not to prefer Gaussian units. In our previous book [9] we took a completely consistent approach of using Gaussian units throughout. However, such consistency is not present in any practitioner’s work. Jackson’s latest version of his classic text [13] changes horses midstream. Here we have adopted what may appear to be an even more schizophrenic approach: Where emphasis is on waveguide and transmis- sion line descriptions, we use SI units, whereas more theoretical chapters are written in the HL system. This reflects the diverse audiences addressed by the materials upon which this book is based, engineers and physicists. Thus we must live with disparate systems of electromagnetic units. The problem, however, is not so very complicated as it may first appear. Let us start by writing Maxwell’s equations in an arbitrary system: 348 Appendix Electromagnetic Units

∇ · D = k1ρ, (A.1a) ∇ · B =0, (A.1b)

∇ × H = k2D˙ + k1k2J , (A.1c)

−∇ × E = k2B˙ , (A.1d) while the constitutive relations are

D = k3E + k1P , (A.2a)

H = k4B − k1M . (A.2b)

The Lorentz force law is

F = e(E + k2v × B) . (A.3) The values of the four constants in the various systems of units are displayed in Table A.1. Here the constants appearing in the SI system have defined

Table A.1. Constants appearing in Maxwell’s equations and the Lorentz force law in the different systems of units constant SI HL Gaussian k1 11 4π 1 1 k2 1 c c k3 ε0 11 1 k4 11 µ 0 values:

−7 −2 µ0 =4π × 10 NA , (A.4a) 1 √ = c ≡ 299 792 458 m/s , (A.4b) 0µ0 where the value of the speed of light is defined to be exactly the value given. (It is the presence of the arbitrary additional constant µ0 which seems objec- tionable on theoretical grounds.) Now we can ask how the various electromagnetic quantities are rescaled when we pass from one system of units to another. Suppose we take the SI system as the base. Then, in another system the fields and charges are given by

SI SI D = κDD , E = κEE , (A.5a) SI SI H = κH H , B = κBB , (A.5b) SI SI P = κP P , M = κM M , (A.5c) SI SI ρ = κρρ , J = κJ J . (A.5d) Appendix Electromagnetic Units 349

We insert these into the Maxwell equations, and determine the κsfromthe constants in Table A.1. For the Gaussian system, the results are ε κ = 0 , (A.6a) D 4π 1 κE = √ , (A.6b) 4πε0 1 κ = √ , (A.6c) H 4πµ 0 µ κ = 0 , (A.6d) B 4π √ κ = κ = κ = 4πε , (A.6e) P ρ J 0 4π κM = . (A.6f) µ0 The conversion factors for HL units are the same except the various 4πsare omitted. By multiplying by these factors any SI equation can be converted to an equation in another system. Here is a simple example of converting a formula. In SI, the skin depth of an imperfect conductor is given by (13.118), 2 δ = . (A.7) µωσ Converting into Gaussian units, the conductivity becomes J J σ = → 4πε =4πεσ . (A.8) E E Therefore, the skin depth becomes 2 c δ → = √ , (A.9) 4πεµσω 2πσω which is the familiar Gaussian expression. Let us illustrate how evaluation works in another simple example. The so- called classical radius of the electron is given in terms of the mass and charge on the electron, m and e, respectively, 2 2 2 e e e r0 = 2 = 2 = 2 . (A.10) 4πε0mc SI 4πmc HL mc G where the charges are related by κρ in (A.6e). Let us evaluate the formula in SI and G systems: 350 Appendix Electromagnetic Units

(1.602 × 10−19 C)2 × 10−7 NA−2 r = =2.818 × 10−15 m , (A.11a) 0 9.109 × 10−31 kg (4.803 × 10−10 esu)2 r = =2.818 × 10−13 cm . 0 9.109 × 10−28 g × (2.998 × 1010 cm/s)2 (A.11b)

It is even easier to evaluate this in terms of dimensionless quantities, such as the fine structure constant 2 2 2 e e e 1 α = = = = . (A.12) ¯hc G 4π¯hc HL 4πε0¯hc SI 137.036 The classical radius of the electron is then proportional to the Compton wave- length of the electron, ¯hc λ = =3.8616 × 10−13 m , (A.13) c mc2 where a convenient conversion factor ishc ¯ =1.97327 × 10−5 eV cm. Thus

−15 r0 = αλc =2.818 × 10 m , (A.14) which incidentally shows that the “classical radius” gives an unphysically small measure of the “size” of an electron. More discussion of electromagnetic units can be found in the Appendix of [9]. For a rather complete discussion see [34]. References

1. J. Mehra, K.A. Milton: Climbing the Mountain: The Scientific Biography of Julian Schwinger (Oxford University Press, Oxford, 2000) 2. K.A. Milton, editor: A Quantum Legacy: Seminal Papers of Julian Schwinger (World Scientific, Singapore, 2000) 3. M. Flato, C. Fronsdal, K.A. Milton, editors: Selected Papers (1937–1976) of Julian Schwinger (Reidel, Dordrecht, 1979) 4. M. Kac: In: Mark Kac: Probability, Number Theory, and Statistical Physics – Selected Papers, ed by K. Baclawski, M.D. Dowsker (MIT Press, Cambridge, 1979) 5. J. Schwinger, D.S. Saxon: Discontinuities in Waveguides: Notes on Lectures by Julian Schwinger (Gordon and Breach, New York, 1968) 6. N. Marcuvitz, editor: The Waveguide Handbook (McGraw-Hill, New York, 1951) 7. J. Schwinger, J.R. Oppenheimer: Phys. Rev. 56, 1066 (1939) 8. J. Schwinger: Quantum Mechanics: Symbolism of Atomic Measurement (Springer, Berlin, Heidelberg New York, 2001) 9. J. Schwinger, L.L. DeRaad Jr., K.A. Milton, Wu-yang Tsai: Classical Electro- dynamics (Perseus/Westview, New York, 1998) 10. J. Schwinger: Quantum Kinematics and Dynamics (W. A. Benjamin, New York, 1970) 11. J. Schwinger, editor: Selected Papers on Quantum Electrodynamics (Dover, New York, 1958) 12. J. Schwinger: Particles, Sources, and Fields, vols. I–III (Addison-Wesley [Perseus Books], Reading, MA, 1970, 1973, 1988) 13. J.D. Jackson: Classical Electrodynamics (McGraw-Hill, New York, 1998) 14. J.A. Wheeler, R.P. Feynman: Rev. Mod. Phys. 17, 157 (1945) 15. J. Schwinger: Found. Phys. 13, 373 (1983) 16. L.R. Elias, et al.: Phys. Rev. Lett. 36, 717 (1976); D.A.G. Deacon, et al.: Phys. Rev. Lett. 38, 892 (1977); H. Boehmer, et al.: Phys. Rev. Lett. 48, 141 (1982); M. Billardon, et al.: Phys. Rev. Lett. 51, 1652 (1983) 17. E.T. Whittaker, G.N. Watson: A Course in Modern Analysis (Cambridge Uni- versity Press, Cambridge, 1965) 18. S.Y. Lee: Accelerator Physics, 2nd edn. (World Scientific, Singapore, 2004) 352 References

19. Particle Data Group: Review of Particle Physics, Phys. Lett. B 592, 1 (2004), http://pdg.lbl.gov/2004/reviews/collidersrpp.pdf 20. A.W. Chao, M. Tigner, editors: Handbook of Accelerator Physics and Engineer- ing (World Scientific, Singapore, 2002) 21. A.W. Chao: Physics of Collective Beam Instabilities in High Energy Accelera- tors (Wiley, New York, 1993) 22. M. Reiser: Theory and Design of Charged Particle Beams (Wiley, New York, 1994) 23. P.A.M. Dirac: Proc. R. Soc. 167, 148 (1938). 24. Julian Schwinger Papers (Collection 371), Department of Special Collections, University Research Library, University of California, Los Angeles 25. J. Schwinger: Phys. Rev. 75, 1912 (1949) 26. D. Ivanenko, A.A. Sokolov: Dokl. Akad. Nauk SSSR [Sov. Phys. Dokl.] 59, 1551 (1948) 27. A.A. Sokolov, I.M. Ternov: Synchrotron Radiation (Akademie-Verlag, Berlin; Pergamon Press, Oxford, 1968) 28. H. Wiedemann: Synchrotron Radiation (Springer, Berlin, Heidelberg, New York, 2003) 29. A. Hofmann: The Physics of Synchrotron Radiation (Cambridge University Press, Cambridge, 2004) 30. A. Sommerfeld: Math. Ann. 47, 317 (1896); Zeits. f. Math. u. Physik 46,11 (1901) 31. P.M. Morse, P.J. Rubenstein: Phys. Rev. 54, 895 (1938) 32. A. Sommerfeld: Electrodynamics: Lectures in Theoretical Physics,vol.3(Aca- demic Press, New York, 1964) 33. J.A. Stratton: Electromagnetic Theory (Mc-Graw-Hill, New York, 1941) 34. F.B. Silsbee: Systems of Electrical Units, National Bureau of Standards Mono- graph 56 (U.S. Government Printing Office, Washington, 1962) Index

aberration 75 Bessel functions 43–47, 153–158, 165, accelerators 263–279 174–176, 217, 344 linear 263 addition theorem 45 action 9 asymptotic expansion 157, 287, 303, effective 346 324 relativistic particle 67 completeness relations 46 admittance differential equation 46, 154 arbitrary definition 120, 121 generating function 155 characteristic 137 integer order 155 field 121 integral representation 47, 155, 156, intrinsic 104 286, 289, 305 line 111 integrals 158, 160, 162, 300 matrix 239, 245, 252 modified 175, 176, 217, 218, 287 shunt 96, 245, 261 asymptotic expansion 176 airy functions 288 differential equation 218 Amp´ere’s law 97 integral representation 176 amplification 345 integrals 288, 294 analytic function 171, 311 pole expansion 219 angular momentum 4 small argument expansion 217, conservation 8, 14 219 magnetic charge 39 Wronskian 219 associated Legendre functions 45, 56, order 1/2 158 178 recurrence relations 154, 298 differential equation 46 series 154 small argument expansion 46 small argument expansion 45, 157 asymptotic correspondence Wronskian 154, 174 associated Legendre functions and zeros 160, 161 Bessel functions 47 Legendre polynomials and Bessel asymptotic 159 functions 45 Bessel inequality 196 attenuation constant 99, 258, 259, 261 Bessel’s equation 46, 153, 154, 174 betatron 277, 292 Babinet’s principle 89–91, 261, 324, bifurcated guide 326 equivalent circuit 253 354 Index boost 68 conductance 233, 248, 260 boundary conditions 81–88, 104, 107, coefficients 232 108, 128, 159, 164, 181, 185–187, conduction current 80, 86, 246, 247 189–194, 196, 201, 204, 205, 214, electric 119 250, 257 conductivity 1, 80, 98, 227–234, 256 current 228 conductors Dirichlet 108, 231 perfect 87, 96 Neumann 108, 231 conformal mapping 171 outgoing 93 conservation laws 12–14 retarded 93 angular momentum 8, 14 boundary displacement energy 8, 40 change of eigenvalue 211, 259 momentum 8, 40 boundary layer 84 conservation of charge 3, 67, 80, 87 Bragg scattering 326 magnetic 89 build-up time 265, 266 convergence of variation–iteration method 202 canonical momentum 11 conversion factor 350 capacitance 233 convolution theorem 319 lumped 127 coordinate displacement series 96, 110, 115, 247 scalar field 64 shunt 96, 110, 246, 247, 258 correlations 345 capacitive reactance 126 Coulomb potential 48 Cauchy’s theorem 311 cross section 37 Cauchy–Riemann conditions 171 absorption 314 Cauchy–Schwarz–Bunyakovskii aperture 326 inequality 190 dielectric sphere 37 cavity 128, 264–267 differential 297 circular cylinder 264 geometrical 300, 304, 307, 315 excitation by electrons 271–274 moving charge 76 Cerenkovˇ radiation 93, 94 radar 298 charge conservation 12, 65 slit 313 charge density 2 first approximation 314 charge relaxation time 86, 88, 92 principal correction 324 circular cylinder functions 153–158 strip classical limit correction to geometrical 307 free particle 338 total 297 classical radius of electron 37, 269, 349 current density 3, 227–231 closed set of eigenfunctions 195–200 edge behavior 309 coaxial line 99 relativistic particle 70 coherent states 329–334 current sheet 88, 90 commutation relations 331 cutoff wavelength 100, 110, 111, 134, comparison method 217 135, 137, 149, 151, 165, 246 complete set of eigenfunctions 195–200 cutoff wavenumber 100, 145 completeness asymptotic 167 coherent states 333 completeness relation 197 d’Alembertian 35 Bessel function 46 degeneracy 114, 149, 162 Legendre polynomials 46 delta function 2, 15–16, 145, 284 Compton wavelength of electron 350 destructive interference 292 Index 355 dielectric constant see permittivity cylindrical wedge 164 differential equation dominant 135, 148, 150, 159, associated Legendre function 46 164–167, 169, 189 Bessel function 46 equivalent circuits 246 diffraction 295–327 E plane obstacles 261 aperture 326 efficiency 268 approximate electric field 312 eigenfunctions differential cross section 313 Laplace operator 272 disk 326 transverse Laplacian 106, 107 Fourier transform method 307–324 eigenvalues 192, 240 Kirchhoff theory 315, 327 positivity 111 slit 307–324 transverse Laplacian 100, 106, 180 slits 326 eigenvectors 105, 129, 240, 244 straight edge 296, 327 Einstein summation convention 63 strip 298–307 electric field dipole moment edge behavior 308 electric 22, 48–51 electric field intensity see electric field magnetic 23 strength dipole radiation electric field strength 4 electric 28, 31, 36, 282 electric polarization 34 magnetic 28 electric wall 88 dipole–dipole interaction 50 electrodes 228 Dirac, P. A. M. 282 electromagnetic model of electron discontinuities 71–72 electrical 84, 130, 235, 260 electron interacting with oscillator geometrical 236–261 338–346 dispersion 127, 131, 329 classical solutions 342 displacement current 246, 247 extreme quantum limit 343 electric 118 ellipse 75, 172 magnetic 120 elliptic cylinder coordinates 169–174, dissipation 98–101, 227, 237, 248, 324 256–261, 264, 273, 282 elliptic cylinder function 173 E modes 257 energy 4 H modes 260 capacitive 249 dissipationless medium 83 electric 124, 125, 127, 248 dominant mode 137, 149, 159, 164, excitation 273 165, 200 inductive 249 Doppler effect 75 magnetic 124–126, 248 duality transformation 28, 38, 66, magnetostatic 31 89–91 nonpropagating modes 125 dyadic product 48 stored 264, 273 energy conservation 8, 13, 181, 182, E mode 106, 109, 115, 116, 118, 236, 237, 281, 314 133–135, 140–143, 148, 151, 179, energy density 5, 68 186–188, 193, 194, 204, 206, 211, dispersive 80, 92, 127 213–223, 235, 257, 272 linear 115–117, 121, 125 cavity 264 energy flux vector see Poynting circular guide 159–162 vector, 68 coax 165 energy spectrum 356 Index

free particle 338 cylinder 313 energy theorem 121, 127, 249 dielectric sphere 43 energy–momentum tensor 68 electrostatic 43 entire function 311 Helmholtz equation 18, 175 equivalent circuit Laplacian operator 212 bifurcated guide 253 retarded 19, 41, 295 equivalent voltage and current semi-infinite rectangular region 251 generators 247 toroidal coordinates 178 Euler’s constant 155, 175, 301 wave equation 18 expectation value 329 group velocity 6, 122, 123, 128, 131, exponential theorem 340 140, 258 guide wavelength 110 field strength tensor 65, 282 dual 66 H mode 106, 109, 116, 120, 135–143, figure of merit see Q 148, 151, 181, 186–188, 193, 194, fine structure constant 350 204, 206, 223–225, 235, 251 force circular guide 162–163 Lorentz 8, 11 coax 166 generalized 38 cylindrical wedge 164 on electric dipole 49 dominant 137, 149, 150, 159, 164, radiation reaction 281 165, 167, 169, 189 Fourier coefficients 195 equivalent circuits 246 Fourier integral theorem 16 H plane obstacle 261 free electron laser 77, 293 Hamiltonian free particle 336–338 electron interacting with oscillator spectrum 338 339 wavefunctions 338 free particle 336 Fresnel integrals 318 harmonic oscillator 334 asymptotic form 318, 321 Hankel functions 156 asymptotic expansion 157, 297, 306 Gamma function 288 differential equation 300 gauge Fourier representation 325 Lorenz 17, 271, 285 integral representation 295, 300, medium 93 305, 310 radiation 38, 39 integrals 300 gauge invariance 9 order 1/2 158 gauge transformation 9, 12, 69 recurrence relations 299 Gaussian distribution 336 small argument expansion 157 generating function harmonic function 96 Bessel function 155 harmonic oscillator 334–336 spherical harmonics 46, 52 classical limit 335 variation–iteration method 202, 224 excited by electron 338–346 generator 67, 331, 332 spectrum 335 constant current 247 wavefunctions 335 constant voltage 247 harmonic time dependence 79–94 Green’s function 231 Harold 324, 346 advanced 19, 41 Heaviside step function 75 causal or Feynman 19, 41 Hertz vectors 24, 112 Coulomb 41, 217 hyperbola 172 Index 357 impedance Levi-Civit`a symbol 65, 66 arbitrary definition 97, 118–121, LHC 284, 293 137, 235, 238, 240 light speed characteristic 97, 110, 111, 238 moving body 130 field 121 limit in the mean 195 intrinsic 97, 104, 111, 265 line element 177 line 111 local energy conservation 5 matrix 238, 239, 244 local momentum conservation 5 eigenvalues 241 Lorentz force 8, 11, 281 series 96, 245, 261 magnetic charge 38 waveguides 235–261 Lorentz invariance 70 incompatible variables 329 Lorentz transformation 41, 68, 73, 183, index of refraction 74, 180 347 inductance field strength 70 arbitrary definition 98 vector potential 69 lumped 127 lumped network 244–247 series 96, 110, 117, 246, 247, 259 dissipative 257 shunt 96, 110, 115, 246 inductive susceptance 126 Macdonald function 288 integral equation 296 macroscopic fields 32–34 charge on conductor 210, 211 magnetic charge 38, 67, 73, 89–91, 94 current 296, 308, 312, 319–321 angular momentum 39 electric field 319 magnetic field resonance 267 Jacobi, K. G. J. 47 magnetic field intensity see magnetic Joule heating 87, 227 induction junction between waveguides 236 magnetic flux 276 magnetic induction 4 Lagrange function 11 magnetic wall 90 electromagnetic field 67 magnetization 34, 130 interaction 65 Mathieu function 173, 324 Lagrange multipliers 192 Maxwell’s equations 1, 34 Lagrangian 9, 80–84 arbitrary units 348 relativistic 40 covariant form 66 Lagrangian operator 331 dyadic form 104, 129 Laplace operator harmonic form 80, 93, 103, 129, 185 eigenfunctions 272 microscopic 5 Laplace’s equation 55 Maxwell–Lorentz equations 5 toroidal coordinates 177 metric 63 Laplace’s first integral representation microtron 263–271 47 elementary theory 267–268 Laplace’s second integral representation phase focusing 270 47, 56 radiation losses 269 Larmor formula 282 vertical defocusing 268 relativistic 283 mode Legendre polynomials 43–47, 55, 56 ⊥ mode see Hmode addition theorem 45  mode see Emode completeness relations 46 momentum 4 integral representation 47 momentum conservation 8, 14 358 Index momentum density 5, 68 plunger 254 momentum flux dyadic see stress Poisson distribution 336 tensor Poisson sum formula 145 monopole potential function 95, 105 electric 48 periodicity 144 multipole expansion 48–61 potentials 9 energy 58, 60 four-vector 65, 75 force 58 Li´enard–Wiechert 21, 35, 76 torque 58 retarded 20, 285 multipole moments vector 272 electric 51 power potential 57 complex 117, 122, 124, 235 multipole radiation 21–31 dissipated 98, 257, 264 radiated by accelerated charge 36, Neumann function 154, 165 78, 283, 293 asymptotic expansion 157, 175 Poynting vector 5, 80, 90, 117, 239, integral representation 175 290, 314 order 1/2 158 complex 117, 127, 257 small argument expansion 155, 157, theorem 121 175 precession 283 non-Hermitian operator 331 probability distribution 334 eigenvalues 330 electron interacting with oscillator eigenvectors 332 340, 342, 345 nonpropagating mode 125 free particle 337 normal modes 128, 240, 272 Gaussian 336 normalization 96, 237, 272 harmonic oscillator 335 Poisson 336 optical theorem 297 propagation constant 97, 99, 100, 110, orthogonality relations 191, 192 118, 258, 261 orthonormality relations 240 nonpropagating mode 125 E modes 113, 159 propagator, Feynman or causal 19 H modes 114, 162 oscillators Q 258, 259, 261, 264, 266, 274 quantum 329–346 loaded 265 overcompleteness 146, 334 quadrupole moment electric 22, 50 parabola 173 quanta parabolic cylinder coordinates 169– absorbed or emitted 344 174 quantum action principle 339 parabolic cylinder function 173 quantum limit Parseval’s theorem 197 extreme 343 particle equations of motion 8 quantum uncertainty 94 permeability 1 vacuum 348 radiation fields 16–31 permittivity 1, 92 radiation losses 269 phase shift 240 radiation reaction 37, 281 phase space 334 rapidity 74 phase velocity 7, 122, 140, 258 Rayleigh’s principle 189, 193, 194, 196, plasma model 92, 129 198–200, 205, 208, 210 Index 359

bounds on arbitrary eigenvalue 208 space translation 13 second eigenvalue 205 speed of light reactance 248 moving body 130 reactive network 238 moving medium 74 reciprocity theorems 91, 92, 232, 237 vacuum 348 reflection 139, 150, 179–183 spherical harmonics 46, 51–61 specular 301 generating function 46 reflection coefficient 181, 182, 236, 237, square waveguide 152 240 stream function 105 refraction 179–183 stress dyadic see stress tensor relativistic particle stress tensor 5, 41, 68, 72 equations of motion 67, 275, 282 relativistic particle 71 relativity 63–78 Struve functions 299 resistance 98, 233, 248, 260, 266 differential equation 299 series 257, 259 sum rule shunt 266 inverse powers of eigenvalues 217, resonance 246, 267, 270 218, 220, 225 rotations 14 surface charge density 86, 87, 210, 211 surface current density 87, 88, 295 S-matrix see scattering matrix distribution on waveguide 138 scattering edge behavior 309 charged particles 74 surface displacement theorem 211, 259 E mode 295 susceptance 248 H mode 295 symmetry light by dielectric sphere 37 reflection 150 light by electron 37, 76 triangle 149 static limit 303 synchrotron 275–279, 283 strip 298–307 phase stability 277, 278 scattering amplitude 297 synchrotron radiation 269, 281–294 scattering matrix 236–243 angular distribution 288–291 symmetric and unitary 237 high harmonics 291 Schein, M. 292 angular power spectrum 290 Schr¨odinger equation characteristic harmonic number 287 free particle 337 polarization 290, 294 harmonic oscillator 334 power spectrum 287, 288 section radiation in mth harmonic 287 T section 245, 246 spectrum 284–294 Π section 245, 246 total power 284, 288 self field 282 separability 133, 153, 169, 177 T mode 95, 108, 109, 246 shadow 301 T section 245, 246 shift of reference point 242–247 TE mode see Hmode skin depth 98, 256, 264 TEM mode see Tmode Gaussian units 349 time displacement 12 slotted guide 138 time ordered product 345 Snell’s law 180 time reversal 90, 92 solid angle transformation 77 time-averaged quantities 80 solid harmonics 52 TM mode see Emode Sommerfeld, A. 296 toroidal coordinates 176–178 360 Index torque current 227–231 on electric dipole 50 eigenvalues 188–212 trace 7, 50 reactance 250 transformation function 332, 334, 338, scattering amplitude 297 340, 344 susceptance 249 transmission coefficient 181, 182, 236, vector 237 contravariant 63 transmission line equations 96, 104, covariant 63 106, 109 null 46 transmission lines 95–101, 103 virial theorem 7 equivalent 103–131, 235–261 voltage triangular waveguide imparted to electron 266 30◦,60◦,90◦ triangle 151 equilateral 143–151 isosceles 141–143, 152 wave equation 17, 35, 169 trilinear coordinates 147 circle 153 trinity nuclear test 263, 293 elliptic cylinder coordinates 173 parabolic cylinder equation 173 unbounded eigenvalues 197, 200 wave operator see d’Alembertian uncertainty relation 329 wavefunction unidirectional light pulse 7, 41, 128 free particle 338 uniqueness theorem 81, 82, 128 harmonic oscillator 335 units 4, 347–350 waveguide 103–131 Gaussian 347 bifurcated 251–256 Heaviside–Lorentz 35, 347 shorted 254 rationalized 347 circular 158–163, 213–225 SI 33, 34, 347 coaxial 165–167 variation–iteration method 200–212 coaxial wedge 167–169 nth approximation to lowest cylindrical wedge 163–165 eigenvalue 201, 214 dielectric 130 1 geometrical discontinuities 236–261 n + 2 th approximation to lowest eigenvalue 202 hexagonal 151 bound on second eigenvalue 206 lumped network description 244– circular guide 213–225 261 error estimate 203–210, 220–223, rectangular 133–140 225 triangular 141–151 lower bounds on eigenvalues 209, Weber–Hermite function 173 220 Wiggler 293 proof of convergence 202 work done by charges on field 29, 93, variational derivatives 10 284 variational principle 9–14, 80–92, work done on charges by field 227, 185–212 282, 285