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When the space near a set of charges contains dielectric material
of nonuniform 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 and . What are the matching
conditions on and at the boundary between the
two media?
Imagine a Gaussian pillbox enclosing part of the boundary
surface between the two media. The thickness of the pillbox
is allowed to tend towards zero, so that the only contribution to
the outward flux of 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 (in medium 1) and
, where
. 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

(429) 
where is the electric displacement in medium 1 at the
boundary with medium 2, . The above equation can
be rewritten

(430) 
where 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
, which give the
familiar boundary condition (obtained by integrating around a
small loop which straddles the boundary surface)

(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.
Next: Boundary value problems with
Up: The effect of dielectric
Previous: Polarization
Richard Fitzpatrick
20020518