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\begin{document}
\heading{E8483: Millikan type experiment} \auname{Arthur Shulkin}

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Question

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{enumerate}
\item Write a Langevin equation for the velocity $v$ with a friction coefficient
$\gamma$ describing the particle dynamics. 
\item For $E=E^{*}$ find the time $T_{D}$ (assuming $T_{D}\gg1$) after
which the diffusion is observed. 
\item For $E\neq E^{*}$ the equation has a steady state solution $<v_{z}>=v_{d}$.
Find the drift velocity $v_{d}$. 
\item Rewrite the equation in terms of $v_{d}$ and find the long time limit
of $<z^{2}>$. From the condition that the observation time is $t\ll T_{D}$
deduce a limit on the accuracy in measuring $E^{*}$. 
\item If the vacuum is improved (i.e. air density is lowered) but $T$ is
maintained, will the accuracy be improved.
\end{enumerate}
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Solution

1.

\begin{equation}
\dot{v}=-\frac{\gamma}{m}v+A(t)+\frac{eE}{m}-g\end{equation}


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2. For time $T_{D}\gg\tau_{col}$ takes place the relation

\begin{equation}
<r^{2}>=\frac{6T}{\gamma}T_{D}\end{equation}


we will assume isotropic media and use the fact $<z^{2}>=d^{2}$

\begin{equation}
<r^{2}>=<x^{2}>+<y^{2}>+<z^{2}>=3<z^{2}>=3d^{3}\end{equation}


from the last one can extract an expretion for $T_{D}$

\begin{equation}
T_{D}=\frac{\gamma d^{2}}{2T}\end{equation}


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3. For a steady state $\dot{v}=0$, $E\neq E^{*}$ and remembering
that $<A(t)>=0$ we get

\begin{equation}
\frac{\gamma}{m}<v>\equiv\frac{\gamma}{m}v_{d}=\frac{eE}{m}-g\rightarrow v_{d}=\frac{e}{\gamma}(E-E^{*})\end{equation}


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4. According to the last definition of $v_{d}$ the Langevin equation
can be written as follow

\begin{equation}
\dot{v}=-\gamma(v-v_{d})+A(t)\end{equation}


from the draft term we can recognize the effective velocity as $v-v_{d}$,
therefor

\begin{equation}
<z_{effective}^{2}>=<(z-v_{d}t)^{2}>=<z^{2}>+v_{d}^{2}t^{2}=\frac{2T}{\gamma}t++v_{d}^{2}t^{2}\end{equation}


where in the final expression the linear term in $t$ is due to diffusion
and the quadratic term is due to the drift velocity.

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5. Using $t=\frac{d}{v_{d}}$ and the expression for $T_{D}$ from
(2) $t\ll T_{D}$ leads to

\begin{equation}
\frac{d}{v_{d}}=\frac{\gamma d}{e(E-E^{*})}\ll\frac{\gamma d^{2}}{2T}\end{equation}


from here we deduce the limit accuracy in measuring $E^{*}$

\begin{equation}
\frac{2T}{ed}\ll E-E^{*}\end{equation}


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6. By improving the vacuum quality we change the parameter $\gamma$,
but from (5) we see that the accuracy don't depend on $\gamma$ therefor
no improve will be achived.

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\Dn
\end{document}
