Electromagnetic radiation and the self field of a spherical dipole oscillator

TL;DR

This paper derives an exact solution for the electromagnetic self-field and radiated field of a spherical dipole oscillator, clarifying causality issues in self-force calculations and providing insights into radiation resistance.

Contribution

It presents an exact analytical solution to Maxwell's equations for a spherical dipole, addressing causality concerns in self-force and radiation resistance.

Findings

Self-field does not cause acausal behavior in the dipole dynamics.

Departure from causality occurs only when the charge is considered in isolation.

Exact self-force expression suggests the impulse-response remains causal.

Abstract

For an oscillating electric dipole in the shape of a small, solid, uniformly-polarized, spherical particle, we compute the self-field as well as the radiated electromagnetic field in the surrounding free space. The assumed geometry enables us to obtain the exact solution of Maxwell's equations as a function of the dipole moment, the sphere radius, and the oscillation frequency. The self field, which is responsible for the radiation resistance, does not introduce acausal or otherwise anomalous behavior into the dynamics of the bound electrical charges that comprise the dipole. Departure from causality, a well-known feature of the dynamical response of a charged particle to an externally applied force, is shown to arise when the charge is examined in isolation, namely in the absence of the restraining force of an equal but opposite charge that is inevitably present in a dipole radiator. Even in this case, the acausal behavior of the (free) charged particle appears to be rooted in the approximations used to arrive at an estimate of the self-force. When the exact expression of the self-force is used, our numerical analysis indicates that the impulse-response of the particle should remain causal.