Potential energy of a pair of identical coplanar uniformly charged circular rings

We study the features of the potential energy function for a system consisting of a pair of identical uniformly charged circular rings that are coplanar with each other. This model may emerge as a legitimate semi-classical approximation for a realistic two-dimensional electron system in a weak perpendicular magnetic field. An exact integral expression for the potential energy of the system is obtained as a function of the separation distance between the centers of the respective uniformly charged circular rings. It is found that the potential energy develops a cusp at a particular distance that corresponds to the two uniformly charged circular rings touching each other at one point. This observation has important implications for formation of clusters and/or aggregate systems.