Derivation of the Love equation for the charge density of a circular plate condenser

TL;DR

This paper derives the Love equation for the charge density of a circular plate condenser by solving the electrostatic problem of two coaxial disks with different radii using complex potential expansion.

Contribution

It introduces a method to derive coupled integral equations for the charge densities of two disks, generalizing Love's equation for equal radii and opposite charges.

Findings

Derived coupled integral equations for two coaxial disks.

Reduced equations to Love's integral equation for equal radii and opposite charges.

Provided a framework for analyzing charge distributions in circular plate condensers.

Abstract

The electrostatic problem of two coaxial parallel charged disks of different radii in infinite space is solved by expansion in terms of complex potentials. For perfectly conducting disks a pair of coupled integral equations is derived for the two weight functions of the expansion. The weight function on each disk is linearly related to its radially symmetric charge density. If the radii are equal and the disks are oppositely charged the pair of equations reduce to Love's integral equation for the circular plate condenser.