Magnets in an electric field: hidden forces and momentum conservation

Abstract

In 1967 Shockley and James addressed the situation of a magnet in an electric field. The magnet is at rest and contains electromagnetic momentum, but there was no obvious mechanical momentum to balance this for momentum conservation. They concluded that some sort of mechanical momentum, which they called "hidden momentum", was contained in the magnet and ascribed this momentum to relativistic effects, a contention that was apparently confirmed by Coleman and Van Vleck. Since then, a magnetic dipole in an electric field has been considered to have this new form of momentum, but this view ignores the electromagnetic forces that arise when an electric field is applied to a magnet or a magnet is formed in an electric field. The electromagnetic forces result in the magnet gaining electromagnetic momentum and an equal and opposite amount of mechanical momentum so that it is moving in its original rest frame. This moving reference frame is erroneously taken to be the rest frame in studies that purport to show hidden momentum. Here I examine the analysis of Shockley and James and of Coleman and Van Vleck and consider a model of a magnetic dipole formed in a uniform electric field. These calculations show no hidden momentum.

An exact formula for electromagnetic momentum in terms of the charge density and the Coulomb gauge vector potential

“…These concepts have also been much discussed in connection with hidden momentum ideas (see e.g. Aguirregabiria et al [14,15], Hnizdo [16], Johnson et al [17], Babson et al [18], Boyer [19], Redfern [20]).…”