W9_parallel_resonance.eps

Week 4: Capacitance 153

The last important relation involving bound charge is well beyond the scope of this course to
discuss, but note well that onecanin principle generate a dielectric material with nonzerobulk
bound charge, that is, with a bound chargedensityρbdistributed throughout the material itself and
not just confined to the surface. In this case, the polarization density becomes a function of this
bound charge that is given by solving:
∇~·P~=−ρb

or equivalently: ∮

S

P~·ˆndA=−

∫

V/S

ρbdV

(the two are equivalent due to the divergence theorem).

This expression looks a lot like Gauss’s Law, but for the polarization density, which in turn is
related to the local field, which is in turn related to the total chargeinside Gaussian surfaces, and
in fact one derives this expression in the next course up from this one by considering all of these
things and working out how the total field is modified by the presenceof a (e.g. linear response)
dielectric materialandextra bound charge distributed through the dielectric.

That is:

E~=E~ 0 −

P~

ǫ 0

(see above) and if we take the divergence of both sides:

∇~·E~=ρtot

ǫ 0

=∇~·E~ 0 −

∇~·P~

ǫ 0

=ρf+ρb

ǫ 0

where we used∇~·E~ 0 =ρf/ǫ 0 from Gauss’s Law for the free charge only. It all works out just asit
should!

So much to look forward to, if you are going on in physics!

As a last remark, consider field energy density inside a dielectric. If we recapitulate the argument
for field energy density for a parallel plate capacitor filled with a dielectric, we get:

U=^1

2

CV^2 =^1

2

ǫrǫ 0 A

d

(Ed)^2 (271)

whereEis still the field between the plates, in this case the field inside the dielectric. Hence

ηe=

dU

dV

=

1

2

ǫE^2 (272)

whereǫ=ǫrǫ 0 is the dielectric permittivity of the material. This is the correct form of the energy
density to use inside a linear dielectric material.

This is all we need to know about dielectrics, although the problems below will challenge you
with half-filled capacitors and the like to make sure you understand itwell enough to be able to use
it.