The conductivity of checkerboards treated by conjugate functions

Abstract

The flow of electrical current in a conducting plate is defined by an electrical potentialu(x, y) and a stream functionv(x, y) whose level lines are the lines of flow. If the plate has unit conductivity, thenu andv satisfy the Cauchy-Riemann equations and (u, v) is termed a conjugate pair. Then (v, -u) is also a conjugate pair and sou andv play dual roles. However, suppose that the conductivity of the plate is not constant but periodic, such as an infinite checkerboard of tiny black and white squares. This poses the problem of determining the effective conductivity of the checkerboard in the large. By using generalized Cauchy-Riemann equations and duality the effective conductivity of the checkerboard is shown to be the geometric mean of the conductivities of black and white.