Problem 3.166

The gap between the plates of a parallel-plate capacitor is filled up with two dielectric layers 1 and 2 with thicknesses $d_{1}$ and $d_{2},$ permittivities $\varepsilon_{1}$ and $\varepsilon_{2}$, and resistivities $\rho_{1}$ and $\rho_{2}$. A dc voltage $V$ is applied to the capacitor, with electric field directed from layer 1 to layer 2 . Find $\sigma$, the surface density of extraneous charges at the boundary between the dielectric layers, and the condition under which $\sigma=0$