Conduction in the small gap between two spheres

A solution to the conduction equation has been developed for two equal size, nonconducting spheres with the line between centers perpendicular to the applied field. The solution, valid when the gap between the spheres is small compared to their radius, is based on a matched asymptotic expansion. For the case when the conductivity is uniform everywhere (i.e., Laplace's equation), the solution agrees well with numerical results obtained from an infinite series solution in bispherical coordinates. An example with a nonuniform conductivity in the gap is presented to demonstrate how the method can be extended to more general conduction problems.