\sect{Millikan experiment}
Consider a Millikan type experiment to measure the charge $e$ of
a particle with mass $m$. The particle is in an electric field $E$
in the $z$ direction, produced by a capacitor whose plates are distance $d$ apart.
The experiment is at temperature $T$ and in a poor vacuum:
the average time $\tau_{col}$ between collisions of the air molecules
and the charged particle is short.
The field is opposite to the force of gravity,
and the experiment attempts to find the exact field $E^{*}$, for which $eE^{*}=mg$,
by monitoring the current.
\begin{enumerate}
\item Write a Langevin equation for the velocity $v$ of the particle,
with a friction coefficient $\gamma$.
\item For $E=E^{*}$ find the time $T_{D}$
after which the diffusion is observed.
\item For $E\neq E^{*}$ the equation has
a steady state solution $\langle v_{z} \rangle =v_{d}$.
Find the drift velocity $v_{d}$.
\item Rewrite the equation in terms of $v_{d}$ and find the long time limit of $\langle z^{2}\rangle$.
From the condition that the observation time is $t \ll T_{D}$,
deduce a limit on the accuracy in measuring $E^{*}$.
\item If the air density is lowered such that the vacuum is improved,
maintaing the same temperature $T$, would the accuracy be improved?
\end{enumerate}