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

When the space near a set of charges contains dielectric material of non-uniform dielectric constant then the electric field no longer has the same 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 matching conditions on ${\bfm E}$ and ${\bfm D}$ at the boundary between the two media?

Imagine a Gaussian pill-box enclosing part of the boundary surface between the two media. The thickness of the pill-box is allowed to tend towards zero, so that the only contribution to the outward flux of ${\bfm D}$ comes from its flat faces. These faces are parallel to the bounding surface and lie in each of the two media. Their outward normals are $d{\bfm S}_1$ (in medium 1) and $d{\bfm S}_2$, where $d{\bfm S}_1= - d{\bfm S}_2$. Assuming that there is no free charge inside the disk (which is reasonable in the limit where the volume of the disk tends towards zero), then Eq. (3.7) yields

$${\bfm D}_1\!\cdot\!d{\bfm S}_1 + {\bfm D}_2\!\cdot\!d{\bfm S}_2 = 0,$$ (429)

where ${\bfm D}_1$ is the electric displacement in medium 1 at the boundary with medium 2, ${\em etc}$. The above equation can be rewritten

$$({\bfm D}_2 - {\bfm D}_1)\!\cdot\!{\bfm n}_{21} = 0,$$ (430)

where ${\bfm n}_{21}$ is the normal to the boundary surface, directed from medium 1 to medium 2. If the fields and charges are non time varying then Maxwell's equations yield $\nabla{\wedge}{\bfm E} = 0$, which give the familiar boundary condition (obtained by integrating around a small loop which straddles the boundary surface)

$$({\bfm E}_2 - {\bfm E}_1)\wedge {\bfm n}_{21} = 0.$$ (431)

In other word, the normal component of the electric displacement and the tangential component of the electric field are both continuous across any boundary between two dielectric materials.