W9_parallel_resonance.eps

Week 4: Capacitance 151

+ + + + + + + + + + + + + + + +

+ + + + + + + +

+σ

−σ

−σb

+σbf

f

E = E / 0 εr

Figure 48: Bound and free charge in a capacitor filled with a dielectric.

where recall thatEris the reaction field generated by the surface chargeσb, which is also equal to
the local polarization density at the surface. If we write out the fieldsE 0 andErin terms of the
charges that produce them (basically using Gauss’s law on the two surface charges), we get:

4 πkeσf

ǫr

= 4πkeσf− 4 πkeσb (264)

If we cancel out the common factor of 4πke= 1/ǫ 0 , we get:

σf

ǫr

=σf−σb (265)

or

σb =

(

1 −

1

ǫr

)

σf

=

(

ǫr− 1

ǫr

)

σf

=

(

−χ

1 +χ

)

σf (266)

where the last form is in terms of the material’s susceptibility instead of the more commonly used
ǫr.

Note that an alternate, perhaps simpler, route to this relation is through the observation that
themagnitudeof the bound surface charge densityσb=P=ǫ 0 χeE(from our previous discussion
of polarization density and the definition of the susceptibility).

σb = ǫ 0 χeE

= ǫ 0 χeE^0

ǫr

= ǫ 0 χe

σf

ǫ 0 ǫr

= σf

χe

1 +χe

(267)

where we once again usedǫr= 1 +χeby definition. In this case one must put in the sign relation
(the bound charge always has the opposite sign of the free chargethat it faces) by hand.

We see that the bound surface charge on the dielectricσbis closely related to the free surface
chargeσfon the actual plate of the conductor. Note well thatQf=σfAis the actual chargestored
on the conductor, but the presence of the bound charge layer reduces the field that charge produces
across the dielectric and therefore reduces the potential difference between the plates of the capacitor
for any given charge. This is, by definition, an increase in the capacitance of the arrangement –
more charge stored per volt of potential difference.

Although we’ve done all of our derivation and examples in the cases above in the context of a
parallel plate capacitor, they hold in thegeneralcase for fields in materials, even where the fields
vary. The electric field in a medium isalwaysgiven byE=E 0 /ǫr, even where the field is varying as