### Joule’s law via Drude Model

exercise 3_7922

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 -  $$\langle P\rangle=\langle \frac{\Delta\varepsilon_{tot}}{\Delta t}\rangle = \sigma E^2$$  ,    $$[\varepsilon]=\frac{energy}{volume}$$