Solutions of Laplace's equation with simple boundary conditions, and their applications for capacitors with multiple symmetries

TL;DR

This paper introduces a geometrical method using Basic Harmonic Functions to solve Laplace's equation with specific boundary conditions, simplifying calculations of electrostatic potentials and capacitances in symmetric configurations.

Contribution

The paper presents a novel approach employing Basic Harmonic Functions in curvilinear coordinates to efficiently solve Laplace's equation and determine capacitances for complex symmetric electrostatic problems.

Findings

Simplified solutions for Laplace's equation with boundary conditions taking values zero or one.

Explicit formulas for capacitance and electric fields in symmetric configurations.

Application to complex geometries and multiple conductor arrangements.

Abstract

We find solutions of Laplace's equation with specific boundary conditions (in which such solutions take either the value zero or unity in each surface) using a generic curvilinear system of coordinates. Such purely geometrical solutions (that we shall call Basic Harmonic Functions BHF's) are utilized to obtain a more general class of solutions for Laplace's equation, in which the functions take arbitrary constant values on the boundaries. On the other hand, the BHF's are also used to obtain the capacitance of many electrostatic configurations of conductors. This method of finding solutions of Laplace's equation and capacitances with multiple symmetries is particularly simple, owing to the fact that the method of separation of variables becomes much simpler under the boundary conditions that lead to the BHF's. Examples of application in complex symmetries are given. Then, configurations of succesive embedding of conductors are also examined. In addition, expressions for electric fields between two conductors and charge densities on their surfaces are obtained in terms of generalized curvilinear coordinates. It worths remarking that it is plausible to extrapolate the present method to other linear homogeneous differential equations.