W9_parallel_resonance.eps

144 Week 4: Capacitance

in a ball of radiusais (see above or better yet, use Gauss’s Law to derive it again for yourself):

Eatom=−ke(Ze)z

a^3

(240)

(down). Thus the forces balance when:

+ZeE 0 −

ke(Ze)^2 z 0

a^3 = 0 (241)

We can then solve for the dipole moment of the polarized atom:

pz= (Ze)z 0 =

a^3

ke

E 0 = 4πǫ 0 a^3 E 0 (242)

There are two very important things to note about this. One is thatthe polarization of the model
atom isdirectly proportional to the applied field. Second, sinceeachatom has a dipole moment of this
magnitude, one can compute theaveragedipole moment per unit volume by dividing this estimate
by the approximate volume occupied by each polarized atom in a solid orliquid or gas. We call this
“dipole moment per unit volume thepolarizationof the material and give it the (vector) symbolP~.
If (for example) we imagine a simple cubic lattice of spherical atoms, there is one atom per cube of
side 2a, with volume 8a^3. Thus:
P=

pz

8 a^3

=

π

2

ǫ 0 E 0 (243)

whereE 0 is the field in the immediate vicinity of the atom (which in general will be the fieldinside
the material, not necessarily the applied external field).

There was nothing special about our guestimate of a volume of 8a^3 per atom, and of course the
actual field will probably not be exactly what we compute above in themodel – we might well expect
it to depend on the kind of atom and its quantum structure, on the time dependence of the field
(if any) and perhaps on still other things – but we neverthelessexpectthat the restoring force will
be linear in the charge displacement for weak fields because of the usual argument, a Taylor series
expansion of the energy about the equilibrium position gets a leading possible contribution from the
quadratic piece, corresponding to a linear restoring force.

Overall, we expect quite generally that an insulating material will polarize, that the polarization
for weak to moderate field strengths will be linear in the field, and that the order of the polarization
density will be some pure number timesǫ 0 E. We give thatdimensionlessnumber a special name
and its own symbol – we call it theelectric susceptibilityχesuch that:

P~=χeǫ 0 E~ (244)

Note well that the units of polarization arecoulombs per square meter– those ofsurface charge
density. It remains to find a surface for which the polarization tells us a surface charge density.

To continue our observations above,χewill, in general, be characteristic of the material; it will
depend on whether the material is solid or liquid or gas (gases usually have a very weak polarization
response because of the large volume occupied per atom) and of course upon the neglected details
of the material in our model – the quantum structure and/or molecular structure of the material.
For solids and liquids it will generally be of the order of unity – in our example,χe=π/ 2 ≈= 1.5 –
where for gases it will usually be “small” as there simply aren’t a lot of atoms or molecules per unit
volume, so no matter how well they polarize individually you won’t build upmuch of a polarization
density.

We are only interested in the static limit of the susceptibility in thisintrocourse, but it really
depends on thetime dependent behavior of the electric field, on temperature, and much more. It
takes the charge in a real materialtimeto respond to changes in the applied field and response times
depend on the natural frequencies and damping times of the charges that are responding. Many
physicists have spent their entire careers studying quantities that amount to general susceptibilities
for various materials (which can have very odd properties indeed!)