The circular disk parallel plate capacitor

One of the more familiar systems in electrostatics is the parallel plate capacitor (PPC). While this system has received considerable attention in the close plate approximation, little is known about the exact solution for arbitrary plate separations. Although the solution was first given, in cylindrical coordinates by Sneddon, it was part of a more general treatise on mixed boundary value problems and appears to be unknown to much of the physics community. We present here a dedicated derivation of the solution to the boundary value problem for parallel disks, in cylindrical coordinates. The resulting expressions for potential and capacitance are in closed form, but depend on a function f(t) which is determined from an integral equation of the Fredholm type, known as Love’s equation. By adopting an orthogonal series approach to the solution of Love’s equation, we have calculated the capacitance for a number of plate separation to plate radius ratios. A quantitative measure of the close plate approximation is then presented by comparing these values to those one would obtain using the elementary capacitance equation ε0A/d.