The Temperature Field or Electric Potential Around Two Almost Touching Spheres

Abstract

The temperature field or electric potential around two equal, perfectly conducting spheres which are almost touching is studied using the method of matched asymptotic expansions. The dominant “outer” approximation of the calculational scheme applies in the narrow gap between the spheres while the “inner” approximation applies in the remaining volume outside the gap. The purpose of the calculation is to investigate the properties of certain singularities whose existence has been indicated by earlier solutions of the Laplace equation around two spheres. For example, the electrostatic forces acting on the spheres or the heat flux between them can become infinite when the spheres touch. The explicit forms of the singularities are found and used to assess the accuracy of earlier solutions. In the appendix the corresponding two-dimensional problems for almost touching cylinders are analysed.