The positivity and other properties of the matrix of capacitance: physical and mathematical implications

TL;DR

This paper investigates the mathematical properties of the capacitance matrix in electrostatics, revealing its positive-singular nature, and explores the physical implications, potential space visualization, and energy minimization problems.

Contribution

It establishes the positive-singular property of the capacitance matrix and analyzes its physical and mathematical implications, including eigenvalue problems and energy minimization.

Findings

Capacitance matrix is positive-singular with a non-degenerate null eigenvalue.

Potential space can be constructed to visualize properties.

An equivalent capacitance for multiple conductors is derived.

Abstract

We prove that the matrix of capacitance in electrostatics is a positive-singular matrix with a non-degenerate null eigenvalue. We explore the physical implications of this fact, and study the physical meaning of the eigenvalue problem for such a matrix. Many properties are easily visualized by constructing a "potential space" isomorphic to the euclidean space. The problem of minimizing the internal energy of a system of conductors under constraints is considered, and an equivalent capacitance for an arbitrary number of conductors is obtained. Moreover, some properties of systems of conductors in successive embedding are examined. Finally, we discuss some issues concerning the gauge invariance of the formulation.