In this problem you will attempt to derive Joule’s law through the estimation of energy losses in Drude’s model. Joule's Law connect linearly between the __energy loss per unit time per unit volume__ as a consequence of the scatters to the __intensity of the field__ (the square field).

(a) Estimate the energy an electron loses each time it scatters of an ion. Assume it scattered in time \(t=0\) and again after time \(t\) - how much energy was lost? meaning what's the energy the electron gained between two scatters?

(b) For a given energy lost per scattering event - \(\epsilon(t)\), and a given density probability to scatter - \(p(t)\) get an integral expression for the total energy the electron loses. \([\epsilon]=energy\)

The probabilty density function for an electron to scatter is given by - \(p(t) = \frac{e^{-\frac{t}{\tau}}}{\tau}\)

(d) Get Joule’s law: the average of the energy loss per unit time per unit volume is proportional to the square filed - \(\langle P\rangle=\langle \frac{\Delta\varepsilon_{tot}}{\Delta t}\rangle = \sigma E^2 \) , \([\varepsilon]=\frac{energy}{volume}\)