The effective electrical conductivity of a
two-phase material consisting of a lattice of identical
spheroidal inclusions in a continuous matrix is determined
analytically. The inclusions are located at the node points of a
simple-cubic lattice and the axis of rotation of each spheroid coincides
with one of the lattice vectors, such that the spheroids are aligned with
each other and with the lattice. With an electric field applied in the
direction of the rotation axes of the spheroids, the electric potential is
found by solving Laplace's equation. The solution is found by analytically
continuing the interstitial field into the particle domain and replacing the
particles with singular multipole source distributions. This yields an
expression for the potential in the interstitial domain as a multipole
expansion. Using Green's theorem, it can be shown that only the first
coefficient in this expansion is required to determine the effective
conductivity of the composite. The coefficients are determined by applying
continuity conditions at the particle-matrix boundary and, in a novel
approach, this is achieved by transforming the multipole expansion into an
expansion in spheroidal harmonics. Results for spheres and prolate and
oblate spheroids are compared with experimental data and previous
theoretical work, and excellent agreement is observed.