Equilibria of the field generated by point charges

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.