A unified solution of Laplace's equation in an arbitrary region

Abstract

An exact solution for the Laplace's equation with Dirichlet boundary conditions in an arbitrary region is presented in this work. The solution is in the form of an integral in terms of the potential on the boundary, and depends on the geometry of the region. The method is valid for bounded as well as unbounded regions. Poisson's integral formula and Schwarz's integral rep resentation for the half-plane potential problem are obtained as a verifi cation of the approach. The method also presents a generalized mean-value theorem for harmonic functions at any interior point of a circular disk in terms of weighted values at the boundary of the region.

A modified methodology for dielectric constant measurement of an arbitrary-shaped sheet by using capacitance technique

“…Therefore, in order to measure dielectric constant, the size of the dielectric is chosen to be less than 0.6r o [11]. Reference [10] provides a unified method for solving Laplace's equation in an arbitrary region.…”

“…From the theoretical point of view, the edge effect can be considered in an accurate full wave modeling of the capacitors. Many numerical and analytical methods of solving Laplace's equation [10] of a parallel plate have been developed in order to accurately calculate the exact capacitance of a parallel plate capacitor. However, regardless of their complexity, one has to inversely solve a very complicated problem in order to obtain the dielectric constant of an arbitrary-shaped sheet material.…”