Capacitance and charging of metallic objects

TL;DR

This paper presents a new formulation for calculating the capacitance of arbitrarily shaped objects using the Neumann-Poincaré operator, enabling more accurate and efficient computations for complex geometries.

Contribution

It introduces a reformulation of capacitance in terms of the Neumann-Poincaré eigenfunctions and extends the approach to capacitors, with numerical validation for axisymmetric shapes.

Findings

Eigenfunction norm varies slowly with shape changes

Numerical method matches finite element results for axisymmetric geometries

Shape factors and capacitance of nanowires and biological membranes are discussed

Abstract

The capacitance of arbitrarily shaped objects is reformulated in terms of the Neumann-Poincar\'{e} operator. Capacitance is simply the dielectric permittivity of the surrounding medium multiplied by the area of the object and divided by the squared norm of the Neumann-Poincar\'{e} eigenfunction that corresponds to its largest eigenvalue. The norm of this eigenfunction varies slowly with shape changes and allows perturbative calculations. This result is also extended to capacitors. For axisymmetric geometries a numerical method provides excellent results against finite element method results. Two scale-invariant shape factors and the capacitance of nanowires and of membrane in biological cells are discussed.