Thomson problem on a spherical cap

TL;DR

This paper studies the arrangement of repelling charges on a spherical cap, combining numerical methods and analysis to understand defect formation and charge distribution for various configurations.

Contribution

It introduces an efficient numerical method for energy minimization of charges on a spherical cap, extending Kelvin's continuum solution to discrete cases.

Findings

Numerical results for different N and cap sizes.

Analysis of topological defect emergence.

Charge distribution patterns near boundaries.

Abstract

We investigate the low-energy configurations of N mutually repelling charges confined to a spherical cap and interacting via the Coulomb potential. In the continuum limit, this problem was solved by Lord Kelvin, who found a non-uniform charge distribution with an integrable singularity at the boundary. To explore the discrete analogue, we developed an efficient numerical method that enables energy minimization while maintaining the number of charges at the cap's edge fixed. Using this approach we have obtained numerical results for various values of N and cap angular widths. Based on these results, we analyze the emergence and behavior of topological defects as functions of both N and the cap's curvature.