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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.
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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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