Recall, (from sections 4.5 and 4.6 in Jackson) that when the electric
field penetrates a medium made of bound charges, it *polarizes*
those charges. The charges themselves then produce a field that
opposes, and hence by superposition reduces, the applied field. The key
assumption in these sections was that the polarization of the medium was
a *linear function* of the total field in the vicinity of the atoms.

Linearity response was easily modelled by assuming a harmonic (linear)
*restoring force*:
= -m&omega#omega;_0^2
acting to pull a charge
into a new neutral equilibrium in the
presence of an electric field
acting on a presumed charge
.
The field exerts a force
, so:
e
- m&omega#omega;_0^2
= 0
is the condition for equilibrium. The dipole moment of this (presumed)
molecular system is
_mol = e
= e^2m&omega#omega;_0^2
=
(1&epsi#epsilon;_0 e^2m&omega#omega;_0^2) &epsi#epsilon;_0 =
&gamma#gamma;_mol &epsi#epsilon;_0
where
is the ``molecular polarizability'' in suitable
units.

Real molecules, of course, have *many* bound charges, each of which
at equilibrium has an approximately linear restoring force with its own
natural frequency, so a more general model of molecular polarizability
is:
&gamma#gamma;_mol = 1&epsi#epsilon;_0 &sum#sum;_i e_i^2m_i&omega#omega;_i^2.

This is for a *single* molecule. An actual medium consists of
molecules per unit volume. From the linear approximation you obtained
an equation for the total *polarization* (dipole moment per unit
volume) of the material:

(11.56) |

(equation 4.68) where the factor of 1/3 comes from averaging the linear response over a ``spherical'' molecule.

This can be put in many forms. For example, using the definition of the
(dimensionless) *electric susceptibility*:
= &epsi#epsilon;_0 &chi#chi;_e
we find that:

(11.57) |

The susceptibility is one of the most often measured or discussed quantities of physical media in many contexts of physics.

However, as we've just seen, in the context of waves we will most often
have occasion to use polarizability in terms of the *permittivity*
of the medium,
. Recall that:
= &epsi#epsilon;= &epsi#epsilon;_0 + = &epsi#epsilon;_0 (1 + &chi#chi;_e)

From this we can easily find in term of :

(11.58) |

From a knowledge of (in the regime of optical frequencies where for many materials of interest) we can easily obtain, e. g. the index of refraction: n = cv = &mu#mu;&epsi#epsilon;&mu#mu;_0&epsi#epsilon;_0 &ap#approx;&epsi#epsilon;&epsi#epsilon;_0 &ap#approx;1 + &chi#chi;_e or n = 1+2N&gamma#gamma;_mol31-N &gamma#gamma;_mol3 if and are known or at least approximately computable using the (surprisingly accurate) expression above.

So much for static polarizability of insulators - it is readily
understandable in terms of real physics of pushes and pulls, and the
semi-quantitative models one uses to understand it work quite well.
However, *real* fields aren't static, and *real materials*
aren't all insulators. So we gotta

- Modify the model to make it
*dynamic*. - Evaluate the model (more or less as above, but we'll have to work harder).
- Understand what's going on.

Let's get started.