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 τ/tc ≪ 1, the ratio of the light-travel time τ across the sphere and the characteristic time scale tc 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.
References
The electromagnetic angular momentum of a spinning shell with constant angular velocity was previously calculated by de Castro (Ref. 23).
The stress tensor is sometimes defined with plus signs, with the alternative convention that is the momentum crossing into the element of surface per unit time.
The reduction of order works even when ε is taken to be of order unity. In this case, the procedure relies entirely on the assumption that τ/tc ≪ 1, and it produces a very similar form for the reduced equation of motion, with coefficients that acquire ε-dependent corrections.
