Conduction in inhomogeneous materials: Hot and high-field spots

Consider electrical conduction in a macroscopically inhomogeneous material with two randomly distributed phases of different conductivity. There will be regions with a particularly high generation of Joule heat (hot spots) and regions with a particularly high electric field (high-field spots). These spots are determined both by the conductivity at the spots themselves (local effects) and by the phase composition in the region surrounding the spots (long-range effects). We have studied the local and long-range effects in two and three dimensions, using analytical models with one and two inclusions in a uniform matrix and numerical calculations on random resistor networks. Hot and high-field spots are favored by long-range compositional fluctuations with borderlines similar to an hourglasslike double cone and with the cone axes along the applied field and the hot or high-field spot at the point where the apexes of the cones meet. The well-conducting phase dominates inside the cones, and the poorly conducting phase dominates outside. The opening angle of the cones is 90^∘{} in two dimensions and 2 arccos(1/ {}3 )=109^∘{} in three dimensions.