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3 added more details regarding Figure 5 and low dimensions

The behavior for continuous charge distributions amounts to classical potential theory; for discrete charges, you get this behavior in the continuum limit.

It is true that the distribution is uniform for a sphere. On other manifolds things can be more complicated. See http://www.ams.org/notices/200410/fea-saff.pdf for a very nice exposition and further references. (In particularFor example, Figure 5 from that paper (included here thanks to Joseph O'Rourke) shows some remarkable behavior the limiting distribution for particles on a torus.) torus under an inverse $s$-th power law:

In this example, for $s \ge 2$ you get a uniform distribution, which is the default behavior when the energy for a continuous charge distribution diverges. For $s < 2$ the particles converge to the continuous distribution that minimizes energy. When $s<1$, this distribution is not even supported on the entire torus.

You don't see these phenomena for the sphere, because of its symmetry, but they are typical for less symmetric manifolds.

Incidentally, the behavior of 1 and 2 dimensions is not so strange. The charges do indeed end up on the boundary, if one uses a harmonic potential function (for example, a logarithmic potential in $\mathbb{R}^2$). The difficulty is that the Coulomb potential is not harmonic in $\mathbb{R}^1$ or $\mathbb{R}^2$. One way of thinking about it is that if you view the needle or disk as sitting inside $\mathbb{R}^3$, then the charges do all end up on the boundary, because the boundary in $\mathbb{R}^3$ is the entire set.

More generally, in $\mathbb{R}^n$, if you use an inverse $s$-th power law for the potential function, then all the charge will be on the boundary if $s \le n-2$ (because the potential function is superharmonic and therefore satisfies the minimum principle). When $s > n-2$, that does not happen.

2 Inserted the cited Fig. 5.

The behavior for continuous charge distributions amounts to classical potential theory; for discrete charges, you get this behavior in the continuum limit.

It is true that the distribution is uniform for a sphere. On other manifolds things can be more complicated. See http://www.ams.org/notices/200410/fea-saff.pdf for a very nice exposition and further references. (In particular, Figure 5 shows some remarkable behavior on a torus.)

Incidentally, the behavior of 1 and 2 dimensions is not so strange. The charges do indeed end up on the boundary, if one uses a harmonic potential function (for example, a logarithmic potential in $\mathbb{R}^2$). The difficulty is that the Coulomb potential is not harmonic in $\mathbb{R}^1$ or $\mathbb{R}^2$. One way of thinking about it is that if you view the needle or disk as sitting inside $\mathbb{R}^3$, then the charges do all end up on the boundary, because the boundary in $\mathbb{R}^3$ is the entire set.

1

The behavior for continuous charge distributions amounts to classical potential theory; for discrete charges, you get this behavior in the continuum limit.

It is true that the distribution is uniform for a sphere. On other manifolds things can be more complicated. See http://www.ams.org/notices/200410/fea-saff.pdf for a very nice exposition and further references. (In particular, Figure 5 shows some remarkable behavior on a torus.)

Incidentally, the behavior of 1 and 2 dimensions is not so strange. The charges do indeed end up on the boundary, if one uses a harmonic potential function (for example, a logarithmic potential in $\mathbb{R}^2$). The difficulty is that the Coulomb potential is not harmonic in $\mathbb{R}^1$ or $\mathbb{R}^2$. One way of thinking about it is that if you view the needle or disk as sitting inside $\mathbb{R}^3$, then the charges do all end up on the boundary, because the boundary in $\mathbb{R}^3$ is the entire set.
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