A Generalization of the Dielectric Ellipsoid Problem

Abstract

The classical problem of the dielectric ellipsoid involves the determination of the field within a homogeneous, isotropic, dielectric ellipsoid when it is placed in a uniform electric field. In the present generalization, both the ellipsoid and the medium in which it is placed, although still homogeneous, are anisotropic and also possess conductivities which are anisotropic. The principal axes of the ellipsoid, of the two dielectric tensors, and of the two conductivity tensors, may all be differently oriented. The external field, although uniform in space, varies sinusoidally with time. The condition specified in the last sentence is consistent with the electromagnetic field equations only in a region whose maximum dimension is small compared with $\frac{\lambda }{2\pi }$ where $\lambda$ is the wave-length which corresponds to the frequency in question. Thus the solution given here is restricted by the condition that the maximum dimension of the ellipsoid must be small compared with $\frac{\lambda }{2\pi }$.