Equilibria of the field generated by point charges

TL;DR

This paper analyzes the equilibrium points of Riesz potentials generated by positive charges at regular polygon vertices, revealing their locations and behavior as functions of the number of charges and the Riesz parameter.

Contribution

It provides a detailed characterization of equilibrium points for Riesz potentials in a symmetric charge configuration, including their asymptotic behavior and uniqueness near a specific parameter value.

Findings

Equilibrium points lie on perpendicular bisectors of the polygon sides.

Asymptotic behavior of equilibrium points depends on the number of charges and Riesz parameter.

Near β=1, there is exactly one non-origin equilibrium point on each bisector.

Abstract

We consider a special case of Maxwell's problem on the number of equilibrium points of the Riesz potential $1/r^{2\beta}$ (where $r$ is the Euclidean distance and $\beta$ is the Riesz parameter) for positive unit point charges placed at the vertices of a regular polygon. We show that the equilibrium points are located on the perpendicular bisectors to the sides of the regular polygon, and study the asymptotic behavior of the equilibrium points with regard to the number of charges $n$ and the Riesz parameter $\beta$. Finally, we prove that for values of $\beta$ in a small neighborhood of $\beta=1$ the Riesz potential has only one equilibrium point different from the origin on each perpendicular bisector, and one equilibrium point at the origin.