Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 80 / 2 / iv / Solution

Let $k=(a_1-a_0)/(a_1+2a_0)$. Far from all the small spheres, their leading dipole contributions have the same direction and their leading $|x{|}^{-2}$ potential terms add:Translations of their centres change only higher far-field multipoles. Since the outer body is a sphere of radius $R_0$,Replacing the whole body by an isotropic effective sphere of conductivity $a^{\ast }$ gives its leading dipole coefficient $R_0^3(a^{\ast }-a_0)/(a^{\ast }+2a_0)$. Matching coefficients yieldsSolving this equation gives the Maxwell approximation for conductivity:For positive phase conductivities and $0\le p\le 1$, the denominator is positive. This rational approximation is obtained by dipole matching; its extrapolation beyond the dilute regime is a closure assumption, not an exact formula for arbitrary inclusion arrangements.

For the consistency check, $\beta =3a_{0}k$, so its dilute expansion isThe isotropic interaction relation supplied in part (iii) givesThus the Maxwell approximation and the interaction correction agree through second order in volume fraction.