Self-torque and angular momentum balance for a spinning charged sphere

TL;DR

This paper analyzes angular momentum balance in a spinning charged sphere, calculating electromagnetic fields, self-torque, and radiation effects to demonstrate how angular momentum is conserved through radiation and self-torque interactions.

Contribution

It provides a detailed calculation of electromagnetic fields, self-torque, and angular momentum transfer for a spinning charged sphere with variable angular velocity.

Findings

Electromagnetic field angular momentum is partly radiated away.

Self-torque accounts for part of the angular momentum loss.

Angular momentum balance is explicitly demonstrated.

Abstract

Angular momentum balance is examined in the context of the electrodynamics of a spinning charged sphere, which is allowed to possess any variable angular velocity. We calculate the electric and magnetic fields of the (hollow) sphere, and express them as expansions in powers of $\tau/t_c \ll 1$, the ratio of the light-travel time $\tau$ across the sphere and the characteristic time scale $t_c$ of variation of the angular velocity. From the fields we compute the self-torque exerted by the fields on the sphere, and argue that only a piece of this self-torque can be associated with radiation reaction. Then we obtain the rate at which angular momentum is radiated away by the shell, and the total angular momentum contained in the electromagnetic field. With these results we demonstrate explicitly that the field angular momentum is lost in part to radiation and in part to the self-torque; angular momentum balance is thereby established. Finally, we examine the angular motion of the sphere under the combined action of the self-torque and an additional torque supplied by an external agent.