%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \sect{Thermionic emission of electrons from a metal} A piece of metal ("cathode") is placed inside a vacuum metal tube ("anode"). The cathode has a work function $W$ and surface area $A$, while the anode has work function $W'$. The cathode by itself can be regarded as a potential-well: the depth of the potential floor close to the surface is zero, while deeper inside the metal it is $V_0 (\gg W)$. The system is held at temperature $T$ . \begin{itemize} \item[(1)] Write an integral expression for the saturation current $I_s$ that would be measured if the bias voltage is very large. \item[(1a)] Show that $V_0$ does not appear in the final result: the outcome of the calculation is the same for sections that are close to the surface or deep in the metal. \item[(1b)] Calculate the integral in the Boltzmann approximation, and explain whether the validity condition corresponds to low temperatures (${ T \ll W }$) or high temperatures. \item[(2)] Using the result of the previous section write an estimate for the current if a reverse (stopping) voltage $V_{\text{battery}}$ is applied. Explain whether $W$ or $W'$ is relevant. \item[(2a)] Explain the relation to the analysis of the stopping voltage in the photoelectric effect. \item[(3)] Assume that the cathode is detached and left alone in free space. Calculate the charge $Q(t)$ of the cathode as a function of time assuming that $Q(0)=0$. \item[(3a)] Explain the limitations of the result that you have obtained. \end{itemize} %%begin{figure} \putgraph{Ex603} %%\caption{}\label{} %%end{figure}