Introduction

It has be supposed by ``Ace Schallger'' on the newsgroup sci.astro that, based on Maxwell's equations, a magnetic flux loop in a charge- and current-free region of space could remain a stable object. He reasons that such a loop would oscillate alternately between magnetic-dipole-like and electric-dipole like behaviors, but otherwise could exist stable for an infinite amount of time.

While my understanding of electricity and magnetism suggests that such a configuration of fields would rapidly dissipate as electromagnetic waves at the speed of light, I found the problem intriguing at a technical level. Namely, could the problem be solved using Maxwell's equations, and the question resolved?

The following is a discussion of the solution of Maxwell's equations for the particular geometry in question. The first part is an exercise in solving differential equations and separation of variables. The second part is an application of the solution to the problem, using a reasonably realistic set of initial conditions. I provide some still images and a movie which show the time evolution of the solution.

The final result is that a toroidal flux loop does not appear to be a stable configuration of fields. This is not a huge surprise to me, but the movie shows how the wave front propagates outward with the speed of light, leaving the original area magnetic field-free.