Spherical integral representations for finite velocity conduction of an abruptly activated current through a crystal

Abstract

A source current is abruptly switched on within a crystalline medium with unaligned sets of permittivity and permeability principal axes. The current then propagates with finite velocity along an arbitrarily oriented conductor. The problem incorporates a partial zero condition and is tackled by Fourier analysis. The general solution for a scalar field is derived as a combination of four spherical integrals over the unit sphere. Two of these are time dependent and could represent the switch-on effect. Their time independent versions relative to the traveling current edge constitute both remaining spherical integrals. These must therefore participate in any eventual steady state and possibly contribute to ?erenkov radiation; their combination is expressible as a single spherical integral directly associated with a basic differential operator. A steady state is found to develop inside an expanding ellipsoid which retreats from the current edge.