CHAPTER 4 TWO STANDARD SHORTCUTS USED TO TRANSFORM ELECTROMAGNETIC EQUATIONS 4.1 THE FREE-PARAMETER METHOD

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1 CHAPTER 4 TWO STANDARD SHORTCUTS USED TO TRANSFORM ELECTROMAGNETIC EQUATIONS The last several chapters have explained how the standard rules for changing units apply to electroagnetic physical quantities. Having becoe failiar with these rules, we are now sure that electroagnetic equations and forulas transfor in a way that akes sense when going fro one syste of units to another. We also know, however, that following these rules can be algebraically cubersoe, forcing us always to watch for the appearance or disappearance of constants ε 0 and µ 0 as we recognize or refuse to recognize charge as a new diension. Engineers and physicists are no ore eager than anyone else to do unnecessary work; consequently, they have coe up with both the free-paraeter ethod and substitution tables, two shortcuts that can greatly reduce the tie required to convert electroagnetic equations and forulas fro one syste of units to another. Unfortunately, neither shortcut is perfect: substitution tables can give abiguous answers in unusual situations, and to apply the free-paraeter ethod we ust first relate our equation or forula to one or ore of a predefined list of equations and forulas. Nevertheless, these shortcuts often provide a quick and easy way of transforing electroagnetic expressions; and whenever there is any doubt about the result, the transforation can be checked using the procedures explained in the previous chapters. 4.1 THE FREE-PARAMETER METHOD Table 4.1 lists Maxwell s equations and the Lorentz force law for the six ajor electroagnetic systes discussed in this book. As pointed out at the beginning of Chapter 3, any classical electroagnetic forula can be derived fro Maxwell s equations and the Lorentz force law. This eans we can consult Table 4.1, select the appropriate equations in the desired set of units, and fro the derive the forulas we need to know. Although this process gets the job done, it usually requires a lot of work. To avoid the unpleasant prospect of deriving all of our forulas and equations fro Maxwell s equations and the Lorentz force law, we construct instead a long list of basic electroagnetic equations that contains everything (including Maxwell s equations and the Lorentz force law) likely to be useful. Instead of providing six long lists one for every electroagnetic syste we use the four free paraeters ε, µ, k 0,and shown in Table 4.2 to reduce the six lists to one

2 226 CHAPTER 4 As an exaple of how this works, consider what happens when we disregard the h and f prefixes and write Maxwell s equations as D = ρ Q, (4.1a) B (4.1b) ( H = k 0 J + ) D, (4.1c) B E + k 0 (4.1d) where and D = ε E + P H = 1 µ B µ M H = 1 µ B M I. (4.1e) (4.1f) As always, E and D are the electric field and electric displaceent, respectively; H and B are the agnetic field and agnetic induction, respectively; ρ Q is the volue charge density; J is the volue current density; P is the electric dipole density; M H is the peranent-agnet dipole density; and M I is the current-loop agnetic dipole density. Clearly, Eqs. (4.1a f) reduce to the correct set of equations in Table 4.1 when ε, µ, k 0,and are given the appropriate values fro Table 4.2. The sae thing can be done to the Lorentz force law; if it is written as F = Q E + k 0 Q ( v B ), (4.2) then it too reduces to the correct equation in Table 4.1 when k 0 is given the appropriate value fro Table 4.2. As we have just seen, the free-paraeter ethod works ost easily and naturally when we neglect the distinction between rationalized and unrationalized electroagnetic quantities that is, neglect the h and f prefixes which so far we have been careful to preserve. It should be ephasized that the distinction between a change of units and a rescaling of an electroagnetic physical quantity is just as iportant as before; the free-paraeter ethod just akes it inconvenient to keep track of this distinction using a single table. If we want to preserve the distinction between rationalized and unrationalized physical quantities, we can consult Tables 4.3(a) or 4.3(b) after putting an equation or forula into the rationalized ks or Heaviside-Lorentz systes, respectively. As pointed out in Section 3.6 of Chapter 3, ost textbooks written today using the rationalized ks syste say that they are using SI units.

3 TWO STANDARD SHORTCUTS USED TO TRANSFORM ELECTROMAGNETIC EQUATIONS 227 Table 4.1 Maxwell s equations and the Lorentz force law in the rationalized ks syste (which is also called SI units), the unrationalized ks syste, Gaussian cgs units, the Heaviside-Lorentz cgs syste, esu units, and eu units. rationalized ks syste, also called SI units unrationalized ks syste f D = ρ Q, B E + B f H = J + f D, f D = f ε 0 E + P, f H B = M I, F = Q E + Q( v B) f µ 0 D = 4πρ Q, B E + B, D = ε 0 E + 4π P, B H = 4π M I, F = Q E + Q( v B) µ 0 Gaussian cgs units D = 4πρ Q, B E + 1 B c H = 4π J c + 1 D c, D = E + 4π P, H = B 4π M I, F = Q E + Q c ( v B) Heaviside-Lorentz cgs syste h D = h ρ Q, h B h E + 1 h B c h H = 1 c h J + 1 h D, hd c = h E + h P, hh = h B h M I, F = h Q h E + h Q c ( v hb) esu units D = 4πρ Q, B E + B, D = E + 4π P, H = c 2 B 4π M I, F = Q E + Q( v B) eu units D = 4πρ Q, B E + B, D = 1 c 2 E + 4π P, H = B 4π M I, F = Q E + Q( v B) To show how the free paraeter ethod, with or without prefixes, works, we apply the first row of Table 4.2 to Eq. (4.1e), reducing it to the rationalized ks syste: ( D = 4πccgs 2 ) E + P. (4.3a)

4 228 CHAPTER 4 Table 4.2 Free-paraeter values for the rationalized ks syste which is also called SI units, the unrationalized ks syste, Gaussian cgs units, the Heaviside-Lorentz cgs syste, esu units, and eu units. rationalized ks syste, also called SI units unrationalized ks syste Rationalization free paraeter = 1 ε = Perittivity free paraeter 4πc 2 cgs = 4π ε = c 2 cgs Pereability free paraeter µ = 4π 10 µ = 10 7 henry 7 henry Light-Speed free paraeter Is this a rationalized syste? k0 = 1 Yes, use ε = f ε0, µ = f µ0,and consult Table 4.3(a) to see where the other prefixes go. k 0 = 1 No, use ε = ε 0, µ = µ0, and there are no prefixes. Gaussian cgs units = 4π ε = 1 µ = 1 k0 = 1 c No Heaviside-Lorentz cgs syste = 1 ε = 1 µ = 1 k0 = 1 c Yes, consult Table 4.3b to see where the prefixes go. esu units = 4π ε = 1 µ = 1 c 2 k 0 = 1 No eu units = 4π ε = 1 c 2 µ = 1 k 0 = 1 No

5 TWO STANDARD SHORTCUTS USED TO TRANSFORM ELECTROMAGNETIC EQUATIONS 229 Table 4.3(a) Rationalized and unrationalized physical quantities in the rationalized ks syste, which is also referred to as SI units. agnetic vector potential volue current density pereance Unrationalized, A Unrationalized, J Rationalized, f P agnetic induction surface current density charge Unrationalized, B Unrationalized, J S Unrationalized, Q capacitance inductance resistance Unrationalized, C Unrationalized, L Unrationalized, R electric displaceent peranent-agnet dipole oent reluctance reluctance Rationalized, f D Rationalized, f H Rationalized, f R electric field current-loop agnetic dipole oent volue charge density Unrationalized, E Unrationalized, I Unrationalized, ρ Q dielectric constant peranent-agnet dipole density resistivity Rationalized, f ε Rationalized, f M H Unrationalized, ρ R relative dielectric constant current-loop agnetic dipole density elastance Unrationalized, ε r Unrationalized, M I Unrationalized, S perittivity of free space agnetic pereability surface charge density Rationalized, f ε 0 Rationalized, f µ Unrationalized, S Q agnetootive force relative agnetic pereability conductivity Rationalized, f F Unrationalized, µ r Unrationalized, σ agnetic flux agnetic pereability of free space electric potential Unrationalized, B Rationalized, f µ 0 Unrationalized, V conductance agnetic pole strength agnetic scalar potential Unrationalized, G Rationalized, f p H Rationalized, f H agnetic field Rationalized, f H current Unrationalized, I electric dipole oent Unrationalized, p electric dipole density Unrationalized, P Fro the fifth entry of row one, we note that ε πccgs 2 = f ε 0 ; and fro Table 4.3(a) we see that D f D,