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Electric Current

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 \(\sigma=0\)

Reveal Answer
σ=ε0V(ε2ρ2−ε1ρ1)/(ρ1d1+ρ2d2),σ=0 if ε1ρ1=ε2ρ2\sigma=\varepsilon_{0} V\left(\varepsilon_{2} \rho_{2}-\varepsilon_{1} \rho_{1}\right) /\left(\rho_{1} d_{1}+\rho_{2} d_{2}\right), \quad \sigma=0 \quad \text { if } \quad \varepsilon_{1} \rho_{1}=\varepsilon_{2} \rho_{2}
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