Conductivity of a “colored” plane
The conductivity of a “colored” plane, i.e., a plane divided into domains differing in conductivity, is calculated. The exact relation between the effective conductivities of the cited and dual (with inverse conductivities) systems is derived for the isotropic case (i.e., the effective conductivity tensor is proportional to the unit matrix). The conductivity of two-colored systems such as a “chessboard” or triangular lattice is exactly calculated to give σ=(σ1σ2)1/2. The particular case of a “hexagon, ” as well as the duality relations for anisotropic systems and for a system in a magnetic field are discussed.
PACS numbers73.50.Bk 75.70.Ak
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