Electromagnetic radiation and the self torque of an oscillating magnetic dipole

TL;DR

This paper derives an exact expression for the self-torque due to electromagnetic radiation on an oscillating magnetic dipole modeled as a charged spherical shell, analyzing its response to impulsive torques.

Contribution

It provides an exact solution for the self-torque in classical electrodynamics for a spherical shell magnetic dipole, including its causal response as the radius approaches zero.

Findings

Self-torque expression is valid for very small radii.

The dipole's response to impulsive torque is causal.

Exact solutions align with classical electrodynamics principles.

Abstract

A uniformly-charged spherical shell of radius $R$, mass $m$, and total electrical charge $q$, having an oscillatory angular velocity $\Omega(t)$ 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 $\omega$. 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 $\omega$. Invoking the Newtonian equation of motion for the shell, we relate its angular velocity $\Omega(t)$ 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 $R \to 0$) so long as the exact expression of the self-torque is used in the dynamical equation of motion of the spherical shell.