Application of Schwarz–Christoffel mapping to the analysis of conduction through a slot

We consider the generic problem of steady conduction through a slot traversing a non-conducting plate that separates two semi-infinite conducting regions. The current-density field is conservative; the dimensionless problem governing its potential depends upon a single geometric parameter, h , the ratio of the slot length (i.e. the plate thickness) to its width. We construct a Schwarz–Christoffel transformation to handle this two-dimensional transport problem. The transformation is expressed in terms of two parameters which are related to h through two implicit equations; in the limit h →0, it becomes explicit. Because of the slow decay of the current density at large distances from the slot, the integral representing the slot resistance diverges. The excess resistance of a finite-length slot relative to that of a zero-length slot is, however, finite. This excess resistance depends only upon the asymptotic behaviour of the potential far from the slot; it may therefore be directly obtained as a function of the two transformation parameters. Asymptotic approximations are found for the excess resistance at small and large h , respectively, scaling as h ln ⁡ h and h . The single-slot solution is used to analyse conduction through a periodic array of widely spaced slots.