Rules for the distribution of point charges on a conducting disk

Abstract

The minimum energy configurations of N equal point charges interacting via the Coulomb potential on an infinitely thin conducting disk are determined and the rules for the distribution of charges on the disk are deduced.

Minimum Energy Configurations of Classical Charges: Large N Asymptotics

“…[Fr03] or see the original Russian article [Zh60]), but requires the slightly stronger hypothesis N < Z + 1 which excludes singly-negative ions. The "no shells" result of Proposition 1.1b) is false for true atoms (see [BB55] for experimental data showing multiple maxima of the radial electron density in Argon), but interestingly, it is also false, e.g., for classical Coulomb particles confined to a disc in two dimensions, in which case minimizers would extend into the radial direction [EO00]; but it would become true again if the interaction was replaced by the Green's function of the two-dimensional Laplacian. For a result related to Theorem 1.2 for repulsive classical charges confined to a compact set see [Lan72], where it is proved that every sequence of empirical measures of minimizers of the particle system contains a subsequence converging to a minimizer of the relevant continuum limit.…”

Comment on “Thomson rings in a disk”

“…This problem has been previously studied by several authors in a series of papers, refs. [2][3][4][5][6][7][8], and it can be regarded as a generalization of the well-known Thomson problem [9] (finding the configurations of minimum energy of N equal charges on the surface of a sphere). Despite the apparent simplicity, both problems provide a serious computational challenge, of increasing difficulty with N : in particular, the number of local minima of the total electrostatic energy grows very fast with N (for the case of the Thomson problem see for example the discussion in Ref.…”

“…For the case of the disk, Erkoc and Oymak [5,6] have observed the tendency, for systems with modest number of charges (N ≤ 109), to accomodate the charges on concentric rings, empirically deducing the rules for the distribution of charges on the disk (incidentally, most of the energies reported by these authors in Tables [6] do not correspond to global minima). The analysis performed by Cerkaski et al in Ref.…”

Thomson rings in a disk

“…In particular, the particle number, which corresponds to the opening of a new shell (starting from one particle in the center), can be calculated exactly up to n = 90 with the aid of the formula n = (2p + 1)(2p + 2) (see also [20]). It gives n = 132 at p = 5, while the MD results provide the opening of the sixth shell at n = 134.…”