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28. Dielectrics II: Response to External Fields; Boundary Conditions
Introduction: Constitutive Relation
Experimentally, it is very well established that
nonconducting solids respond to an applied electric field by becoming to some
degree polarized, which on a macroscopic level is the same as generating a
dipole density. Obviously, this phenomenon is relevant in many areas of science
there are a lot of names describing the same thing.
In the following, we assume that the response (the
polarization ) of the dielectric medium to the applied
electric field is linear, a very good approximation in
practice for the field strengths normally encountered. For now, we’ll also assume
the response is isotropic, there are important exceptions (single crystal
materials) we’ll return to later.
The electric susceptibility
is defined by
so is dimensionless. It measures, obviously, how susceptible the
medium is to being polarized by an electric field. (The subscript e is for electric, as opposed to magnetic, susceptibility. We’ll get to the magnetic response later.)
But we could have asked: how does the displacement respond to the external field?
(Wait, isn’t that really the same question? right? Yes,
but bear with me.)
This is called the constitutive
and is the electric permittivity. So is the vacuum
There’s still more notation!
The dimensionless dielectric
constant is defined as the ratio of the permittivity to
the vacuum permittivity:
is also called the relative permittivity.
If the permittivity has no spatial variation, then
note the free
That is to say, in a uniform
medium, the fields, and therefore potentials, produced by a given charge
distribution are the same as in a vacuum, but
scaled down by a factor
We’ll find that always, the charge within the molecules moves
to partially shield external charge.
Hence for a given charge a capacitor with a dielectric medium rather
than nothing between the plates will have that much less potential difference
its capacitance is increased by a factor of great practical importance.
Recall from the previous lecture that
so in the absence of free charge, Therefore, for a thin pillbox disc at the
interface, Gauss’ theorem gives
The right-hand side will be nonzero if there really is free charge qt the interface, which there usually
The normal component of the electric field is
discontinuous at the interface, from the excess local (bound) charge inherent
in the polarization discontinuity.
The other boundary condition is established by using Stokes’
theorem for integrating around a loop contour half in the dielectric, half out,
Since the surface integral vanishes, and the line
In other words, the component of the electric field parallel
to the surface, is continuous across the surface.
Dielectric Boundary Value Problems
Image Method for Point Charge
First, we examine the standard problem of a point charge in
a semi-infinite dielectric having permittivity the charge being distance from the plane interface with a second
dielectric, permittivity filling the other half of space .
The point charge is the only unbound charge
in the system, there will of course be unbalanced bound charge on the interface
since the permittivity changes there.
(In the special case ,
the “bound charge” is no longer boundthis is
our old problem of a charge above a grounded conducting plane. But for finite the field penetrates the boundary.)
The equations are
Of course so for some potential, and are continuous at the border, has a step discontinuity from the sheet of
unbalanced bound charge.
This problem is a lot like the charge above a conducting
plane, except that this time we need to find the electric field on the other
side too. It turns out that there is a
simple image solution.
we take there to be an image charge at the point . Then
and for we assume (admittedly with hindsight)
since there is no real charge for Notice this means that in the region where
the charge isn’t, the lines of electric field are straight, and radiating from the charge (which itself is in the
Labeling the radial coordinate in the plane as at we can see that
It follows that the boundary conditions are satisfied by the
charge and the images in the appropriate regions provided (matching
the normal components of )
and from the continuity of the tangential components of
Solving for the image charges,
the field lines in the two regions. Draw two plots, for and
Finding the (bound) surface charge density at the interface
is straightforward: it’s equal to ,
and The jump in susceptibility generates a delta function in charge density a surface
and putting in the
values from the image solution,