When electric charges of equal magnitude and sign are released on a regular sphe

Question

When electric charges of equal magnitude and sign are released on a regular sphere (and assume that they stick to the surface of the sphere, but they are free to move along its surface), what is the shape of the figure made by the charges as vertexes when they come to a state of equilibrium?

Case 1 - Only one charge is there:

Already in equilibrium as there are no other charges.

Case 2 - Two charges:

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

Case 3 - Three charges:

Three charges make a shape of an equilateral triangle.

Case 4 - Four charges:

A regular tetrahedron comes up when they reach the state of equilibrium.

While extrapolating these cases to higher number of charges, one roadblock comes up: how to adapt a more-than-three-dimensional figure to the three-dimensional sphere?

Any hint taking to the correct answer or better approach to the problem is welcome.

Explanation / Answer

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.

References:

Wikipedia

Mathworld.wolfram

Mathpages

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