  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.
