Analytical expression for the electric field enhancement between two closely-spaced conducting spheres

An external electric field applied to two conducting spheres in close approach is enhanced (by charge separation on the spheres) in the region between the spheres. For spheres of equal size, this enhancement is a universal function of the ratio of the separation of the spheres to their radius, and increases without limit as this ratio decreases. We calculate the enhancement factor analytically for perfectly conducting spheres, providing a simple formula valid when the spheres are close together, that is when the enhancement is large and the known series solution is difficult to evaluate. The same methods allow us to find the close-approach forms of the longitudinal and transverse polarizabilities of the two-sphere system.