Maxwell approximation for conductivity ($a^{\ast }=a_0(1+2pk)/(1-pk),k=(a_1-a_0)/(a_1+2a_0)$)

Matching the far dipole of a spherical composite with the sum of isolated spherical-inclusion dipoles gives this approximation. Its dilute expansion is $a_0+p\beta +p^2{\beta }^2/(3a_0)+O(p^3)$. The rational expression is a useful closure, not the exact conductivity of every spherical-inclusion microstructure.