Past exam of the mathematics course of the University of Cambridge 2006 iii Paper 80 2 ii Solution Created 2026-10-03 Updated 2026-10-14
Let and retain only the independent response of each well-separated small sphere. In the leading dilute spherical-inclusion conductivity approximation,Averaging gives , where . Since ,The spherical response holds for every , soThis retains the order- term and neglects interactions. A precise dilute regime can be obtained by shrinking the spheres while holding separated centres fixed. Volume fraction alone is not enough if the inclusions instead remain in close clusters whose collective polarization persists at leading order.