Analytical solution for two-dimensional Laplace's equation in a shallow domain containing coplanar interdigitated boundaries

TL;DR

This paper derives an analytical solution for the potential and flux in a two-dimensional Laplace's equation scenario with interdigitated boundaries in a shallow domain, useful for miniaturized device design.

Contribution

It introduces a conformal mapping approach using Jacobian elliptic functions to solve Laplace's equation in complex interdigitated geometries with arbitrary parameters.

Findings

Potential and flux expressions depend only on relative dimensions.

Shallow domain behavior approaches semi-infinite domain when height exceeds band separation.

Equal-width bands minimize total surface for fixed flux.

Abstract

Laplace's equation appears frequently in physical applications involving conservable quantities. Among these applications, miniaturized devices have been of interest, in particular those using interdigitated arrays. Therefore, we solved the two-dimensional Laplace's equation for a shallow or finite domain consisting of interdigitated boundaries. We achieved this by using Jacobian elliptic functions to conformally transform the interdigitated domain into a parallel plates domain. The obtained expressions for potential distribution, flux density and flux allow for arbitrary domain height, different band widths and asymmetric potentials at the interdigitated array, besides considering fringing effects at both ends of the array. All these expressions depend only on relative dimensions, instead of absolute ones. With these results we showed that the behavior in shallow or finite domains approaches that of a semi-infinite domain, when its height is greater than the separation between the centers of consecutive bands. We also found that, for any desired but fixed flux, bands of equal width minimize the total surface of the interdigitated array. Finally, we present approximate expressions for the flux, based on elementary functions, which can be applied to ease the calculation of currents (faradaic or non-faradaic), capacitances and resistances among other possible applications.