Week 4: Capacitance 147
Now let’s imagine this figure redrawn on a length scale where atoms aretiny– too small to be
seen in the figure (as they are in any macroscopic chunk of matter large enough to be seen with
the naked eye). When we consider the field between the surface charge layers, the block of matter
starts to look like, and behave like, acapacitorinternally, with a reaction fieldErthat flows from
the positive to the negative charge layers in theopposite direction to the applied external field. This
situation is portrayed in figure 47.
Applying Gauss’s Law to the induced surface charge layers in this simple rectangular geometry,
we expect:
Er=
σb
ǫ 0 (247)
The total field is then:
E=E 0 −σb
ǫ 0=E 0 −
P
ǫ 0=E 0 −χeE (248)We can rearrange this into:
E(1 +χe) =E 0 (249)
and solve forE, the field inside the material, in terms ofE 0 , the applied external field:
E= E^0
1 +χe=E^0
ǫr(250)
where we have introduced therelative permittivity
ǫr= (1 +χe) (251)as a dimensionless constant characteristic of the material. Note thatE≤E 0 becauseχe≥0. This
also means thatǫr≥1! The electric field isreducedinside a dielectric – this is what the “di-” in
“dielectric” means!
Note Well:Most introductory physics books written for college or high schoolphysics courses
omit any explicit mention of the susceptibility (leaving students with quite a chore later if they go
on in physics and have never seen it the next time they take electricity and magnetism) and use the
symbolκto represent 1 +χeand call it thedielectric constantfor the material, as in:
κ= (1 +χe) =ǫr (252)This usage is deprecatedeven in introductory treatments because in general neitherǫrnor
κareconstant (doh!) and because it encourages confusion with the sensible definition of the
permittivity of the material. We therefore useǫrexclusively in this textbook.
This may seem very confusing to you, so let me review. ǫ 0 is functionally equivalent toke, a
constant of nature that connects the units of charge and lengthto those of field and force at the
microscopic scale of elementary particles (or in a vacuum), where ofcourseke= 1/(4πǫ 0 ). The
presence of bulk neutral mattermodifiesthe electric fieldE~ 0 produced by bare/isolated/free charges
Qfthatwouldbe there in a vacuum; the fieldpolarizesthe material, which creates a reaction field
that strictly reduces the applied field inside the material. The polarization density (dipole moment
per unit volume) of the medium is related to thenetfield in the mediumE~byP~=χǫ 0 E~. The net
field itself is related to the applied field byE~=E~ 0 /ǫrwhereǫr= 1 +χ.
There is one more thing we can do with therelativepermittivity, the thing that gives it its
name. We can use it to define thepermittivity of any medium:
ǫ=ǫrǫ 0 (253)