A rapidly convergent modified Green' function for Laplace' equation in a rectangular region

Abstract

A computationally useful expression for the modified Green' function in a rectangular region is derived. This can be used to obtain solutions of Laplace' equation or Poisson' equation for the electrical potential associated with any configuration of current sources that satisfies the condition of charge balance, and represents a generalization of a well–known result due to Newman applicable to an infinite channel. The starting point for the analysis is the very slowly converging Fourier–series expansion of the modified Green' function. This Fourier series can be separated into a Fourier series in one variable that can be summed in closed form, and a double Fourier series that can be partly summed with respect to one index and transformed by application of the Poisson summation formula and integration by residues. The result of applying the Poisson formula can be expressed either as a rapidly converging series of logarithmic functions, or as a rapidly converging Fourier series which can be integrated analytically with respect to the coordinates of the source point. The use of the modified Green' function is demonstrated first by determining potential distributions that arise from specification of uniform current density on two or more of the boundary surfaces. The potential in a cell with two electrodes embedded in opposite sides of the rectangle is then calculated, and circumstances in which this potential approaches the potential in an infinite channel are determined.