The gap between the plates of a prallel-plate capacitor is filled up with two dielectric layers 1 and 2 thickness `d_(1)` and `d_(2)` permittivities `epsilon_(1)` and `epsilon_(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 with `sigma = 0`.

Explanation

We have, `E_(1) d_(1) + E_(2) d_(2) V`
and by current conservation
`(1)/(rho_(1)) E_(1) = (1)/(rho_(2)) E_(2)`
Thus, `E_(1) = (rho_(1) V)/(rho_(1) d_(1) + rho_(2) d_(2)`,
`E_(2) = (rho_(2) V)/(rho_(1) d_(1) + rho_(2) d_(2))`
At the boundary between the two dielectrics,
`sigma = D_(2) - D_(1) = epsilon_(0) epsilon_(2) E_(2) - epsilon_(0) epsilon_(1) E_(1)`
`(epsilon_(0) V)/(rho_(1) d_(1) + rho_(2) d_(2)) (epsilon_(2) rho_(2) - epsilon_(1) rho_(1))`