Binary circular inclusions in an effective medium approximation

We investigate the second order interactions of perfectly-conducting circular inclusions embedded in a medium of dielectric constant ε0 under the influence of an electric field E0. Using dipole theory we show that the potential difference for the composite system is written in terms of complicated continued fractions. The form of these continued fractions is rederived in terms of simple square matrices which simplify the computationally intense interaction terms between the two inclusions. Asymptotic expressions are derived from the continued fractions that are then used to calculate the conductance of the medium.