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\sect{Thermionic emission of electrons from a metal}

A spherical piece of metal ("cathode"), 
that has radius $R$ and temperature $T$,  
is placed inside a vacuum tube. 
A second metallic piece ("anode") is used to collect the electrons 
that are emitted from the cathode.
The effective temperature of the anode is zero.
%
The cathode has a work function $W$, while the anode has work function $W'$. 
The depth of the potential that holds the electrons 
inside the cathode, aka the potential floor, is $V_0$. 

\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 using the Boltzmann approximation. 
Specify the range of temperatures for which the 
approximation is valid.  

\item[(2)]
Using the result of the previous item 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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