Things to Note

Before we go on, we should understand a few things:

1. $\epsilon$ is now complex! The imaginary part is explicitly connected to the damping constant.
2. Consequently we can now see how the index of refraction
   

   $$n = \frac{c}{v} = \frac{\sqrt{\mu\epsilon}}{\sqrt{\mu_0\epsilon_0}},$$ (9.107)

   
   
   can be also be complex. A complex index of refraction describes absorption (or amplification!) and arises from the damping term in the electrons' EOM (or non-linear, non-equilibrium effects in lasers, which we will not consider here). This makes energy conservation kind of sense. Energy absorbed by the electrons and dissipated via the ``frictional'' damping force is removed from the EM field as it propagates through the medium. This (complex dispersion of incident waves) is the basis for the ``optical'' description of scattering which is useful to nuclear physicists.
3. The term
   
   $$\frac{1}{\omega_i^2 - \omega^2 - i\omega \gamma}$$
   
   
   has a form that you will see again and again and again in your studies. It should be meditated upon, studied, dreamed about, mentally masticated and enfolded into your beings until you understand it. It is a complex equation with poles in the imaginary/real plane. It describes (very generally speaking) resonances.

   It is useful to convert this into a form which has manifest real and imaginary parts, since we will have occasion to compute them in real problems one day. A bit of algebra gives us:
   

   $$\frac{1}{\omega_i^2 - \omega^2 - i\omega \gamma} = \frac{(\o... ...i\omega \gamma}{(\omega_i^2 - \omega^2)^2 + \omega^2 \gamma^2}$$
   
   

4. If $N$ is ``small'' ($\sim 10^{19}$ molecules/cc for a gas) $\chi_e$ is small (just like in the static case) and the medium is nearly transparent at most frequencies.
5. if $N$ is ``large'' ($\sim 10^{23}$ molecules/cc for a liquid or solid) $\chi_e$ can be quite large in principle, and near a resonance can be quite large and complex!

These points and more require a new language for their convenient description. We will now pause a moment to develop one.