 Steady fields in two leaky dielectric layers

For two planar homogeneous layers in series, charge conservation makes the steady normal current density the same in both layers. Combining Ohm's law with $E_1{d}_1+E_2{d}_2=V$ determines the fields when both conductivities are positive. The free interfacial surface charge density is ${ϵ}_2{E}_2-{ϵ}_1{E}_1$, whereas total free-plus-bound charge is ${ϵ}_0(E_2-E_1)$. The two notions of charge are distinct. This DC state can differ from the immediate response to switching the voltage; interfacial free charge accumulates during the transient.