Multipole expansions in two dimensions

Abstract

In this paper we consider two-dimensional electrostatics. We derive the multipolar expansions for the potential due to an arbitrary distribution of charge, and for the energy of a charge distribution in a spatially non-uniform external electric field. We also derive the multipole expansion for the energy of two rigid, non-overlapping charge distributions. The results are expressed in both cartesian tensor and circular tensor forms. Some interesting differences from the three-dimensional case emerge : for example, in any order the multipole moment has at most two independent components; and there is no unique preferred relative orientation for two interacting multipoles, which implies that in some respects, at least, the physics of a 2D fluid may differ appreciably from that of its 3D analogue.