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*:

(9.100) |

(9.101) |

(9.102) |

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:

(9.103) |

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:

(9.104) |

This can be put in many forms. For example, using the definition of the
(dimensionless) *electric susceptibility*:

(9.105) |

(9.106) |

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:

(9.107) |

From this we can easily find in term of :

(9.108) |

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:

(9.109) |

(9.110) |

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.