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

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/σ0 = 1 – αnf+ o(f2), where f is the area fraction of the inclusions and the coefficient αn = (tan (π/n)/2πn) Г4(1/n)/Г2(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 αn 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/σ0 = 1 – f, while for the perpendicular conductivity, I find that σ/σ0 = 1 – nπa2, where 2a 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.