Effective longitudinal dielectric constant of a rectangular lattice of parallel conducting prolate spheroids

Abstract

The effective longitudinal dielectric constant is calculated for an infinite, three-dimensional, rectangular lattice of identical, parallel, conducting prolate spheroids. The problem is formulated by means of a matrix equation. The matrix elements are explicitly worked out. The matrix equation embodies all the multipole interactions among the spheroids exactly, and is solved analytically by iteration. An exact solution is obtained formally as an infinite series. It is shown that the interaction effects can be quantified completely by a shift in value of the depolarization factor of the spheroids. In the spherical limit, the results reduce properly to those previously derived directly for a lattice of spheres. In the opposite, acicular limit, the interaction-shifted depolarization factor is obtained in closed form, and numerical results are presented. The calculation of the effective longitudinal magnetic permeability of a lattice of magnetic prolate spheroids is similar. The magnetic permeability is obtainable from the dielectric constant by simple variable substitutions.