Electromagnetic radiation and the self torque of an oscillating magnetic dipole

A uniformly-charged spherical shell of radius R, mass m, and total electrical charge q, having an oscillatory angular velocity Ω(t) around a fixed axis, is a model for a magnetic dipole that radiates an electromagnetic field into its surrounding free space at a fixed oscillation frequency ω. An exact solution of the Maxwell-Lorentz equations of classical electrodynamics yields the self-torque of radiation resistance acting on the spherical shell as a function of R, q, and ω. Invoking the Newtonian equation of motion for the shell, we relate its angular velocity Ω(t) to an externally applied torque, and proceed to examine the response of the magnetic dipole to an impulsive torque applied at a given instant of time, say, t = 0. The impulse response of the dipole is found to be causal down to extremely small values of R (i.e., as R → 0) so long as the exact expression of the self-torque is used in the dynamical equation of motion of the spherical shell.