Radiating Systems, Multipole Fields and Radiation — John David Jackson
Physics — E&M · covariant-ohms-law
Problem
In the rest frame of a conducting medium the current density satisfies Ohm's law, J′=σE′, where σ is the conductivity and primes denote quantities in the rest frame.
(a) Taking into account the possibility of convection current as well as conduction current, show that the covariant generalization of Ohm's law is
Jα−c21(UβJβ)Uα=cσFαβUβ
where Uα is the 4-velocity of the medium.
(b) Show that if the medium has a velocity v=cβ with respect to some inertial frame that the 3-vector current in that frame is
J=γσ[E+β×B−β(β⋅E)]+ρv
where ρ is the charge density observed in that frame.
(c) If the medium is uncharged in its rest frame (ρ′=0), what is the charge density and the expression for J in the frame of part b? This is the relativistic generalization of the equation J=σ(E+v×B) (see p. 320).