Radiating Systems, Multipole Fields and Radiation — John David Jackson
Physics — E&M · steady-state-currents-radiation
Problem
As an idealization of steady-state currents flowing in a circuit, consider a system of N identical charges q moving with constant speed v (but subject to accelerations) in an arbitrary closed path. Successive charges are separated by a constant small interval Δ.
Starting with the Liénard–Wiechert potentials for each particle, and making no assumptions concerning the speed v relative to the velocity of light show that, in the limit N→∞, q→0, and Δ→0, but Nq=constant and q/Δ=constant, no radiation is emitted by the system and the electric and magnetic fields of the system are the usual static values.
(Note that for a real circuit the stationary positive ions in the conductors neutralize the bulk charge density of the moving charges.)