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Suppose we take 'n' electrons and put them on the surface of a conducting cube. How can we calculate the charge distrubution and position of these electrons once the static situation has been arrived at.

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The same problem on a sphere is called the Thomson problem and is unsolved (analytically, at least) for arbitrary $n$. I am virtually certain that the cubical version is as well.

The solution for $n=2$ is obvious. The solution for $n=8$ seems obvious but is probably nontrivial to prove.

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  • $\begingroup$ Not so sure about n=8. One square could be rotated 45 degrees. $\endgroup$ – Ben Crowell Aug 10 at 19:29
  • $\begingroup$ @BenCrowell I’m not following you. Can you clarify what you mean? I meant put one electron at each vertex. $\endgroup$ – G. Smith Aug 10 at 22:10
  • $\begingroup$ @G.Smith - Thank you very much for that information. It seems that for 8 electrons, each electron would occupy one of the eight corners of the cube, as the surface charge density comes out to be zero on the plane surfaces of the cube because the radius of curvature is infinity there. So now the question is where will the 9th electron go? Can this be questioned just with classical physics and Gauss's theorem or we have to use uncertainty principle and probability theories? $\endgroup$ – Samarth Gupta Aug 12 at 11:21
  • $\begingroup$ It can be answered with classical physics. It’s just electrostatics, but some electrostatics problems can only be solved numerically, on a computer, rather than analytically. You want to minimize the electrostatic potential, which is a function of $3n$ variables, so it is a high-dimensional minimization problem. $\endgroup$ – G. Smith Aug 12 at 15:40

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