• A. Worley, Minimal energy configurations for charged particles on a thin conducting disc: the perimeter particles.

]]>Their Figure 5 (N=520) is similar to the figure at the top of your post.

]]>for a collection of points on the disk:

• A. Worley, Minimal energy configurations for charged particles on a thin conducting disc: the perimeter particles.

One interesting thing I hadn’t guessed is that the particles often lie on concentric shells.

Abstract.The lowest energy configurations for N equal charged particles confined to a thin conducting disc have been investigated in detail up to N = 160 and in outline for further values up to N = 500. For all values of N up to 160 the particle configurations can be described in terms of concentric shells. The number of perimeter particles p appears to be simply related to N and to the mean radius of the outermost internal shell. Justification for these relations is obtained from a simple model based on the well-known distribution of continuous charge on a conducting disc.

In the N → ∞ limit the density of particles in the unit disk is proportional to

Other interesting things:

In the range N ≤ 80, the configuration for N + 1 is, with one exception, the same as that for N with the addition of one particle to an existing shell or the creation of a new shell with a particle placed at the centre. The single exception is the pair N = 55 (with shell structure 5-13-37) and N = 56 (with shell structure 1-6-12-37). For the range 81 ≤ N ≤ 160 there are four exceptions to the usual pattern: N = 97–98, 117–118, 150–151 and 152–153.

The first case where the particles don’t lie on concentric shells is N = 185.

]]>in *three* dimensions. If we minimized potential energy for the potential that solved Poisson’s equation in *two* dimensions, the charge particles would indeed cluster on the boundary of the disk…

… at least in the limit where the number of particles approaches infinity.

In the 3-dimensional case, for finite numbers of particles confined to a ball minimizing the potential energy

the particles don’t always lie on the surface of the ball! In the 2-dimensional case, with the potential energy appropriate to 2 dimensions

I think the case of 7 particles would be a good one to try, to see this effect.

]]>• Ann Davis, Scott Malloy, Michael Neubauer, Mark Schilling, William Watkins, Joel Zeitlin, Potential minimizing configurations of points on the sphere.

The pictures go up to 75 points, with some gaps. Here is the case of 72 points, where apparently there are some squares:

]]>http://research.iaun.ac.ir/pd/ebrahimzadeh/pdfs/UploadFile_6472.pdf

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