## 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 $\sigma $/$\sigma _{0}$ = 1-$\alpha _{n}$f + o(f$^{2}$), where f is the area fraction of the inclusions and the coefficient $\alpha _{n}$ = $\frac{\tan \, (\pi /n)}{2\pi n}\Gamma ^{4}\left(\frac{1}{n}\right)$/$\Gamma ^{2}\left(\frac{2}{n}\right)$, which is 2.5811, 2.1884, 2.0878 for triangles, squares and pentagons, and tends to the circle limit of 2 as n $\rightarrow \infty $. The coefficient $\alpha _{n}$ 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 $\sigma $/$\sigma _{0}$ = 1 - f, while for the perpendicular conductivity, I find that $\sigma $/$\sigma _{0}$ = 1 - n$\pi $a$^{2}$, 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.

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