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. 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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