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When the space surrounding a set of charges contains dielectric material
of nonuniform 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 and . What are the boundary
conditions on and at the interface between the
two media?
Imagine a Gaussian pillbox enclosing part of the interface. The thickness of the pillbox
is allowed to tend towards zero, so that the only contribution to
the outward flux of comes from the flat faces of the box, which are
parallel to the interface. Assuming that there
is no free charge inside the pillbox (which is reasonable in the limit
in which the volume of the box tends to zero), then Eq. (809)
yields

(815) 
where is the component of the electric displacement in medium 1 which is normal to the interface, etc. If the fields and charges are non timevarying then the differential
form of Faraday's law yield
, which gives the
familiar boundary condition (obtained by integrating around a
small loop which straddles the interface)

(816) 
Generally, there is a bound charge sheet on the interface whose
density follows from Gauss' law:

(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.
Next: Boundary value problems with
Up: Dielectric and magnetic media
Previous: Polarization
Richard Fitzpatrick
20060202