A new look at two old problems in electrostatics, or much ado with hemispheres

In this paper, we take a look at two electrostatics problems concerning hemispheres. The first problem concerns the direction of the electric field on the flat cap of a uniformly charged hemisphere. We show that the symmetry and principle of superposition coupled with Gauss's law gives a delightfully simple solution and then go on to examine how the more 'powerful' methods of partial differential equations can be used to arrive at the same solution. A second problem that deals with the force of repulsion between the two hemispheres of a uniformly charged sphere is also dealt with from the same viewpoint. We feel that this approach could be quite instructive to students