Uniform distribution on ball/Positively loaded/Thought experiment/Exercise

Consider the following thought experiment: given a sphere made of metal, and ${}n$ identical particles with the same positive electric charge. Therefore, these particles repel each other. We put these particles on the sphere, they still repel each other, but they stay on the sphere. What is the (final) configuration of these particles? Don't we expect "due to physical laws“ that there is a "uniform“ configuration where all particles are on an equal footing? Don't we expect that for any two particles ${}P,Q$, there exists a symmetry of the sphere and of the configuration that transforms ${}P$ into ${}Q$?

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