Electrostatic potential energy stored in a hemispherical surface with uniform surface charge distribution

Abstract

We report an exact result for the electrostatic potential energy stored in a hemispherical surface with uniform surface charge density. This system lacks the spherical symmetry of a uniformly charged spherical surface or its solid sphere counterpart. Consequently, a hemispherical surface with all its charge uniformly spread on its surface represents an interesting case study in electrostatics. In this work we combine the fact that the body has axial symmetry with clever mathematical tools to obtain an outcome that is remarkably simple. The final exact expression obtained for the electrostatic potential energy stored in a hemispherical surface with uniform surface charge density answers a long-standing question. Though the bulk of the treatment is analytical, the accuracy of the findings can also be ascertained by using symbolic computation software or numerical recipes. Hence, the present analytical results can be compared to ones obtained with various powerful numerical methods which nowadays are routinely used to check the correctness of analytical and semi-analytical calculations.