Electrostatic energy of interaction between uniformly charged hemispherical surfaces.

The purpose of this study is to gain insights on the nature of the electrostatic energy of interaction between two uniformly charged hemispherical surfaces with constant surface charge density. The present system of these two interacting bodies represents a very difficult problem to solve since hemispherical surfaces lack the spherical symmetry of a uniformly charged spherical surface or a solid sphere counterpart. For this reason, an exact analytical calculation of the electrostatic interaction energy is not possible for an arbitrary orientation. However, this work shows that analytical results are possible for scenarios where the system of the two interacting hemispherical surfaces manifests some form of inherent symmetry. We identify such situations where combination of suitable mathematical tools appropriate for the axial symmetry together with transformations that apply to a spherical system of coordinates can reduce the difficulty of the problem. We show that there are some special cases where this very difficult integral problem can be reduced to a simpler one of summation of an infinite series. The approach is illustrated by showing explicitly the calculation of the interaction energy between the two uniformly charged hemispherical surfaces when they are brought together so that they are touching each other across the "equator". [ABSTRACT FROM AUTHOR]