\sect{Baruch's  D24.}

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, i.e. $\tau_{col}$ is short. ( $\tau_{col}$
is the average time between collisions of the air molecules and
the charged particle). The field is opposite to the gravity force
and the experiment attempts to find the exact field ${E^*}$ where
${eE^*=mg}$ by monitoring the charge arriving at the plates.

\begin{itemize}
\item [(a)]
Write a Langevin equation for the velocity ${\bf v}$ with a
friction coefficient $\gamma$  describing the particle dynamics.
If ${E=E^*}$ find the time ${T_{D}}$ (assuming  ${\gamma
T_{D}>>1)}$ after which a current noise due to diffusion is
observed. What is the condition on $\tau_{col}$ for the validity
of this equation?

\item [(b)]
When ${E\neq E^*}$ the equation has a steady state solution
${\langle v_{z}\rangle=v_{d}}$. Find the drift velocity ${v_{d}}$.
Rewrite the equation in terms of ${\tilde{v} _{z}=v_{z}- v_{d}}$
and find the long time limit of ${\langle z^{2}\rangle}$. From the
condition that the observation time is ${<<T_{D}}$ deduce a limit
on the accuracy in measuring ${E^*}$.

\item [(d)]
If the vacuum is improved (i.e. air density is lowered) but T is
maintained, will the accuracy be improved?\\
\end{itemize}

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