Potential and field singularity at a surface point charge

TL;DR

This paper analyzes the magnetic potential near a point charge on a curved boundary, revealing additional singular terms related to surface curvature that differ from volume charge behavior, impacting vortex interactions in superconductors.

Contribution

It introduces a detailed analysis of surface charge singularities, identifying curvature-dependent terms that modify the classical inverse distance singularity.

Findings

Presence of logarithmic and asymmetric singular terms related to surface curvature.

Additional singularities are proportional to principal curvatures of the boundary.

Results are relevant for understanding vortex interactions in superconductors.

Abstract

The behavior of the magnetic potential near a point charge (fluxon) located at a curved regular boundary surface is shown to be essentially different from that of a volume point charge. In addition to the usual inverse distance singularity, two singular terms are generally present. The first of them, a logarithmic one, is axially symmetric with respect to the boundary normal at the charge location, and proportional to the sum of the two principal curvatures of the boundary surface at this point, that is, to the local mean curvature. The second term is asymmetric and proportional to the difference of the two principal curvatures in question; it is also bounded at the charge location. Both terms vanish, apparently, if the charge is at a planar point of the boundary, and only in this case. The field in the charge vicinity behaves accordingly, featuring generally two singular terms proportional to the inverse distance, in addition to the main inverse distance squared singularity. This result is significant, in particular, for studying the interaction of magnetic vortices in type II superconductors.