A numerical verification of the Clausius-Mossotti relation

The results of the approximation formula deduced by Clausius and Mossotti to calculate the effective permittivity of an arrangement of a large number of spheres with a permittivity ξ_s filling a medium with a permittivity ξ_s are compared with a highly accurate three-dimensional finite-element method solution. This formula gives the effective permittivity as a function of ξ_s and σ_m as well as the ratio of the two volumes. The influence of different structures with the same volume ratios is investigated. It is shown that there is only an insignificant difference between the Clausius-Mossotti approximation formula and the highly accurate finite-element solution for different lattice structures. The differences between the two solutions become more pronounced for higher values of the volume ratio and of the conductivity ratio of the two materials