Some comments on the electrostatic forces between circular electrodes

TL;DR

This paper analyzes electrostatic forces between circular electrodes, deriving formulas for various configurations, revealing force behaviors depending on contact zone dimensionality, supported by numerical computations.

Contribution

It provides new formulas and a classification of electrostatic force behaviors between circular electrodes, including short-distance force expressions and the influence of contact zone dimensionality.

Findings

Force can be logarithmic and repulsive even for opposite charges.

Force depends strongly on the contact zone dimensionality.

Numerical computations support the theoretical results.

Abstract

We study the force between two circular electrodes in different configurations. A formula analogous to Kelvin's formula for the spheres is given in the case of equal disks held at the same potential and when one plate is earthed. An expression for the force at short distance between two arbitrarily charged disks is found: the generic case shows a logarithmic repulsive force, also for disks carrying charges of opposite sign. Some numerical computations support the results. A classification for the possible behaviors of the force is proposed on the basis of a decomposition of the capacitance matrix. It is shown that the forces depend strongly on the dimensionality of the contact zone between the conductors. The analysis is supported by a numerical computation carried for the case of two disks of different radii.