Boundary conditions for ${\bf E}$ and ${\bf D}$

When the space surrounding a set of charges contains dielectric material of non-uniform dielectric constant then the electric field no longer has the same functional form as in vacuum. Suppose, for example, that the space is occupied by two dielectric media whose uniform dielectric constants are $\epsilon_1$ and $\epsilon_2$. What are the boundary conditions on ${\bf E}$ and ${\bf D}$ at the interface between the two media?

Imagine a Gaussian pill-box enclosing part of the interface. The thickness of the pill-box is allowed to tend towards zero, so that the only contribution to the outward flux of ${\bf D}$ comes from the flat faces of the box, which are parallel to the interface. Assuming that there is no free charge inside the pill-box (which is reasonable in the limit in which the volume of the box tends to zero), then Eq. (809) yields

$$D_{\perp 2}-D_{\perp 1} = 0,$$ (815)

where $D_{\perp 1}$ is the component of the electric displacement in medium 1 which is normal to the interface, etc. If the fields and charges are non time-varying then the differential form of Faraday's law yield $\nabla{\times}{\bf E} ={\bf0}$, which gives the familiar boundary condition (obtained by integrating around a small loop which straddles the interface)

$$E_{\parallel 2}-E_{\parallel 1} = 0.$$ (816)

Generally, there is a bound charge sheet on the interface whose density follows from Gauss' law:

$$\sigma_b = \epsilon_0 (E_{\perp 2}-E_{\perp 1}).$$ (817)

In conclusion, the normal component of the electric displacement, and the tangential component of the electric field, are both continuous across any interface between two dielectric media.