Low Frequency Behavior

Near $\omega = 0$ the qualitative behavior depends upon whether or not there is a ``resonance'' there. If there is, then $\epsilon(\omega\approx 0)$ can begin with a complex component that attenuates the propagation of EM energy in a (nearly static) applied electric field. This (as we shall see) accurately describes conduction and resistance. If there isn't, then $\epsilon$ is nearly all real and the material is a dielectric insulator.

Suppose there are both ``free'' electrons (counted by $f_f$) that are ``resonant'' at zero frequency, and ``bound'' electrons (counted by $f_b$). Then if we start out with:

$$\epsilon(\omega) = \epsilon_0\left (1 + \frac{N e^2}{m} \sum_i \frac{f_i}{(\omega_i^2 - \omega^2 - i \omega \gamma_i)}\right)$$

$$= \epsilon_0\left (1 + \frac{N e^2}{m} \sum_b \frac{f_b}{(\omega_b^2 - \omega^2 - i \omega \gamma_b)}\right)$$

$$\hspace{1cm} + \frac{N e^2}{m} \sum_f \frac{f_f}{(- \omega^2 - i \omega \gamma_f)}$$

$$= \epsilon_b + i \epsilon_0 \frac{N e^2 f_f}{m \omega (\gamma_0 - i \omega)}$$ (9.125)

where $\epsilon_b$ is now only the contribution from all the ``bound'' dipoles.

We can understand this from

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$H$} = \mbox{\boldmath$J$} + \frac{d\mbox{\boldmath$D$}}{dt}$$ (9.126)

(Maxwell/Ampere's Law). Let's first of all think of this in terms of a plain old static current, sustained according to Ohm's Law:

$$\mbox{\boldmath$J$} = \sigma \mbox{\boldmath$E$}.$$ (9.127)

If we assume a harmonic time dependence and a ``normal'' dielectric constant $\epsilon_b$, we get:

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$H$} = \left(\sigma - i\omega \epsilon_b \right)\mbox{\boldmath$E$}$$

$$= -i \omega\left ( \epsilon_b + i \frac {\sigma}{\omega} \right ) \mbox{\boldmath$E$}.$$ (9.128)

On the other hand, we can instead set the static current to zero and consider all ``currents'' present to be the result of the polarization response $\mbox{\boldmath$D$}$ to the field $\mbox{\boldmath$E$}$. In this case:

$$\mbox{\boldmath$\nabla$}\times \mbox{\boldmath$H$} = - i\omega \epsilon \mbox{\boldmath$E$}$$

$$= -i \omega\left ( \epsilon_b + i \epsilon_0 \frac{N e^2}{m} \frac{f_f}{(\gamma_0 - i \omega)} \right) \mbox{\boldmath$E$}$$ (9.129)

Equating the two latter terms in the brackets and simplifying, we obtain the following relation for the conductivity:

$$\sigma = \epsilon_0 \ \frac{n_f e^2}{m}\frac{1}{(\gamma_0 - i \omega)}.$$ (9.130)

This is the Drude Model with $n_f = f_f N$ the number of ``free'' electrons per unit volume. It is primarily useful for the insight that it gives us concerning the ``conductivity'' being closely related to the zero-frequency complex part of the permittivity. Note that at $\omega = 0$ it is purely real, as it should be, recovering the usual Ohm's Law.

We conclude that the distinction between dielectrics and conductors is a matter of perspective away from the purely static case. Away from the static case, ``conductivity'' is simply a feature of resonant amplitudes. It is a matter of taste whether a description is better made in terms of dielectric constants and conductivity or complex dielectric.