Briefly explain the main assumptions leading to Drude's theory of conductivity. Show that these assumptions lead to the following equation for the average drift velocity $\langle\mathbf{v}(t)\rangle$ of the conducting electrons:

$$\frac{d\langle\mathbf{v}\rangle}{d t}=-\tau^{-1}\langle\mathbf{v}\rangle+(e / m) \mathbf{E}$$

where $m$ and $e$ are the mass and charge of each conducting electron, $\tau^{-1}$ is the probability that a given electron collides with an ion in unit time, and $\mathbf{E}$ is the applied electric field.

Given that $\langle\mathbf{v}\rangle=\mathbf{v}_{0} e^{-i \omega t}$ and $\mathbf{E}=\mathbf{E}_{0} e^{-i \omega t}$, where $\mathbf{v}_{0}$ and $\mathbf{E}_{0}$ are independent of $t$, show that

$$\mathbf{J}=\sigma \mathbf{E}$$

Here, $\sigma=\sigma_{s} /(1-i \omega \tau), \sigma_{s}=n e^{2} \tau / m$ and $n$ is the number of conducting electrons per unit volume.

Now let $\mathbf{v}_{0}=\widetilde{\mathbf{v}}_{0} e^{i \mathbf{k} \cdot \mathbf{x}}$ and $\mathbf{E}_{0}=\widetilde{\mathbf{E}}_{0} e^{i \mathbf{k} \cdot \mathbf{x}}$, where $\widetilde{\mathbf{v}}_{0}$ and $\widetilde{\mathbf{E}}_{0}$ are constant. Assuming that $(*)$ remains valid, use Maxwell's equations (taking the charge density to be everywhere zero but allowing for a non-zero current density) to show that

$$k^{2}=\frac{\omega^{2}}{c^{2}} \epsilon_{r}$$

where the relative permittivity $\epsilon_{r}=1+i \sigma /\left(\omega \epsilon_{0}\right)$ and $k=|\mathbf{k}|$.

In the case $\omega \tau \gg 1$ and $\omega<\omega_{p}$, where $\omega_{p}^{2}=\sigma_{s} / \tau \epsilon_{0}$, show that the wave decays exponentially with distance inside the conductor.