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Recall, (from sections 4.5 and 4.6) 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:

(9.94) 
acting to pull a charge from a neutral equilibrium in the presence
of an electric field:

(9.95) 
where
is the applied external field. The dipole moment of
this (presumed) system is
. 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:

(9.96) 
From the linear approximation you obtained an equation for the total
polarization (dipole moment per unit volume) of the material:

(9.97) 
(equation 4.68). This can be put in many forms. For example, using the
definition of the (dimensionless) electric susceptibility:

(9.98) 
we find that:

(9.99) 
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, . In term of , this is:

(9.100) 
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:

(9.101) 
or

(9.102) 
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
semiquantitative 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.
Next: Dynamic Case
Up: Dispersion
Previous: Dispersion
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Robert G. Brown
20071228