# charges on sphere

When charges are released on sphere, what is the shape made by charges?

Two charges are on opposite points of one diameter of the sphere.

Three charges make a shape of an equilateral triangle.

Four charges gives tetrahedron.

What shall five and more give?

• Related MO.SE question: mathoverflow.net/q/187063/13917 Oct 8 '12 at 15:20
• @Qmechanic I'll just add that the main important result from that question was $N^2/2$ is the $1/r$ summation for uniformly distributed, infinite, points. I'm having fun with some of the links posted here, they do indeed limit to the N^2/2, but there are some interesting lower order terms, not to mention some deviations that can't be approximated with calculus approaches, which is fascinating. Oct 8 '12 at 15:58

This problem with $N$ point charges on a sphere is a famous problem in electrostatics known as the Thomson problem. For large $N$, it is in general an open problem still under active research.
This has been a problem since Thomson proposed the arrangement of electrons and positive charges (nucleus was not known at that time) in rigid electron shells of atom which is what called Plum-pudding model of atom. He suggested that electrons are arranged in a symmetrical pattern with respect to the center of sphere which is applicable only to smaller elements in periodic table (Old-timer wandered a lot after discovering the $e/m$ ratio).