 Spherical conductivity inclusion

A sphere of conductivity $a_1$ in a host $a_0$ under a uniform incident gradient $E$ has interior gradient $3a_{0}E/(a_1+2a_0)$. Outside a sphere of radius $R$, its potential correction is $-R^3(a_1-a_0)(E\cdot x)/[(a_1+2a_0)|x{|}^3]$. Its polarization is $\beta E$ inside and zero outside. Temperature and normal-flux continuity determine these coefficients.