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

If the region in the vicinity of a collection of free charges contains dielectric material of non-uniform dielectric constant then the electric field no longer has the same form as in a vacuum. Suppose, for example, that space is occupied by two dielectric media whose uniform dielectric constants are $\epsilon_1$ and $\epsilon_2$ . What are the matching 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 its two flat faces. These faces are parallel to the interface, and lie in each of the two media. Their outward normals are $d{\bf S}_1$ (in medium 1) and $d{\bf S}_2$ , where $d{\bf S}_1= - d{\bf S}_2$ . Assuming that there is no free charge inside the pill-box (which is reasonable in the limit that the volume of the box tends towards zero), Equation (503) yields

$${\bf D}_1 \cdot d{\bf S}_1 + {\bf D}_2 \cdot d{\bf S}_2 = 0,$$ (508)

where ${\bf D}_1$ is the electric displacement in medium 1 at the interface with medium 2, et cetera. The above equation can be rewritten

$$({\bf D}_2 - {\bf D}_1) \cdot {\bf n}_{21} = 0,$$ (509)

where ${\bf n}_{21}$ is the normal to the interface, directed from medium 1 to medium 2. If the fields and charges are non-time-varying then Maxwell's equations yield

$$\nabla\times{\bf E} = {\bf0},$$ (510)

which gives the familiar boundary condition (obtained by integrating around a small loop that straddles the interface)

$$({\bf E}_2 - {\bf E}_1)\times {\bf n}_{21} = 0.$$ (511)

In other word, 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.