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\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}

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