The capacitance of the circular parallel plate capacitor obtained by solving the Love integral equation using an analytic expansion of the kernel

TL;DR

This paper presents an analytical expansion method using Fourier cosine series and integral calculations to accurately compute the capacitance of circular parallel plate capacitors, especially at small separations, improving upon previous numerical approaches.

Contribution

The authors develop an analytical approach to evaluate the Love integral equation, enabling larger matrices and enhanced accuracy for small plate separations.

Findings

Improved numerical accuracy at small separations

Analytical calculation of expansion integrals using Sine and Cosine integrals

Error estimates demonstrating substantial improvement over previous methods

Abstract

The capacitance of the circular parallel plate capacitor is calculated by expanding the solution to the Love integral equation into a Fourier cosine series. Previously, this kind of expansion has been carried out numerically, resulting in accuracy problems at small plate separations. We show that this bottleneck can be alleviated, by calculating all expansion integrals analytically in terms of the Sine and Cosine integrals. Hence, we can, in the approximation of the kernel, use considerably larger matrices, resulting in improved numerical accuracy for the capacitance. In order to improve the accuracy at the smallest separations, we develop a heuristic extrapolation scheme that takes into account the convergence properties of the algorithm. Our results are compared with other numerical results from the literature and with the Kirchhoff result. Error estimates are presented, from which we conclude that our results is a substantial improvement compared with earlier numerical results.