The conductivity of a sheet containing a few polygonal holes and/or superconducting inclusions

Abstract

I use conformal mapping techniques to determine the change in the conductivity of a sheet containing a few well-separated holes. The hole shapes studied are the equilateral triangle, square, pentagon and regular n -gons. I show that the conductivity can be written as σ / σ 0 = 1 – α n f + o ( f 2 ), where f is the area fraction of the inclusions and the coefficient α n = (tan (π/ n )/2π n ) Г 4 (1/ n )/Г 2 (2/ n ), which is 2.5811, 2.1884, 2.0878 for triangles, squares and pentagons, and tends to the circle limit of 2 as n →∞ . The coefficient α n is proportional to the induced dipole moment around the polygonal hole which can be found using an appropriate conformal mapping. I have also examined and compared the results for long thin needle-like holes in the shape of diamonds, rectangles and ellipses. In all cases the conductivity parallel to the needles has the limiting form σ / σ 0 = 1 – f , while for the perpendicular conductivity, I find that σ / σ 0 = 1 – n π a 2 , where 2 a is the length of the needle, and n is the number of needles per unit area. For thicker needles, the shape becomes important and I compare the results with recent analog experiments and computer simulations. Because of the reciprocity theorem, all the results found here apply equally well to superconducting inclusions.