\sect{Millikan experiment}
Consider a Millikan-type experiment whose purpose is to measure the charge~$e$ of
a particle with mass ~$\mass$. The particle is located beteen plates of capacitor,
where the electric field~$\mathcal{E}$ is in the "up" direction,
while the gravitation~$g$ is in the "down" direction.
The distance between the plates is~$L$,
and the temperature of the system is~$T$.
Due to the poor vacuum the particle executes a Brownian motion
that is described by a Langevin equation with friction force~$-\eta v$.
The charge of the electron is estimated
via ${\delta F = e\mathcal{E}-\mass g = 0}$.
In item~(1) the system is prepared with a single particle
in the middle. In item~(3) assume a uniform gas of~$N$ particles.
In both cases the current is integrated during a time interval~$t$,
and the charge $Q=\int I(t')dt'$ is inspected as "readout".
\begin{enumerate}
\item[(1)]
Assuming that $\delta F =0$, determine the time $t_d$ such that
for $t\ll t_d$ it is not likely to get charge readout.
\item[(2)]
What is the $\delta F$ for which the condition $t \ll t_d$
is no longer valid. We shall regard this value, call it $\delta_1$,
as the resolution of the measurement.
\item[(3)]
Assuming that $\delta F =0$, determine the
power spectrum $C(\omega)$ of the current $I(t)$.
\item[(4)]
Assume that the time of the measurement is~$t$.
What is the $\delta F$ for which the condition ${\langle Q \rangle \ll \sqrt{\text{var}(Q)}}$
is no longer valid. We shall regard this value, call it $\delta_N$,
as the resolution of the measurement.
\item[(5)]
Express the ratio $\delta_N/\delta_1$ as a function
of $N$ and $t/t_d$.
\end{enumerate}
{\bf Tips:}
In the absence of fluctuations ${\delta F =0}$ is indicated by having zero readout.
In item (3) the ``readout" is a current versus voltage (``IV") measurement, and ${\delta F =0}$ is indicated
by zero current. Due to the fluctuations there is some blurring which determines
the resolution $\delta_N$. In order to calculate the fluctuations in item (3) define
the one-particle current as the velocity (up to a prefactor).