\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 \${<