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\documentclass{article}
\usepackage{tabularx,amsmath,boxedminipage,epsfig}
\begin{document}
\title{Millikan ``Oil Drop'' Experiment}
\author{} \date{}
\maketitle
\section*{Introduction and Theory}
Consider Fig. \ref{moplates}
\begin{figure}[h]
\centerline{\epsfig{width=3in, file=moplates.eps}}
\caption{\label{moplates}Very schematic Millikan Oil Drop System}
\end{figure}
It turns out that very small droplets fall very slowly. Clouds, for
example, are very small water droplets, trying to fall, but held aloft
by very slight air currents.
An electric field can take the place of the air current and
even cause the oil drop
to rise. Thus, for a rising oil drop:
\begin{equation}\label{rise}
Eq=mg+kv_r
\end{equation}
where $E$ is the electric field, $q$ is the charge on the droplet, $m$
is the mass of the droplet, $g$ is the acceleration due to gravity, $v_r$
is the velocity rising, and $k$ is a drag coefficient which will be
related to the viscosity of air and the radius of the droplet.
If the field is off and the droplet is just falling, then:
\begin{equation}\label{fall}
mg=kv_f
\end{equation}
Combining Eqs. \ref{rise} and \ref{fall} we can find the charge $q$:
\begin{equation}\label{q}
q=\frac{mg(v_f+v_r)}{Ev_f}
\end{equation}
The drag coefficient, $k$, can be determined from the viscosity, $\eta$, and
the radius of the droplet, $a$:
using Stokes law:
\begin{equation}
k=6\pi a\eta
\end{equation}
Since
\begin{equation}\label{m}
m=\frac{4}{3}\pi a^3 \rho
\end{equation}
one may solve for $a$:
\begin{equation}\label{simple}
a=\sqrt{\frac{9\eta v_f}{2g\rho}}
\end{equation}
Here $rho=.886\cdot 10^3 kg/m^3$ is the density of the oil. (We ignore the
density of air, which is roughly 1/1000 less.)
There is a
small correction because the oil drop radius is not so different
from the mean free path of air. This leads to an effective viscosity:
\begin{equation}\label{etaeff}
\eta_{eff}=\eta\frac{1}{1+\frac{b}{pa}}
\end{equation}
where $b\approx 8.20 \times 10^{-3}$ (Pa m) and $p$ is atmospheric
pressure (1.01 $10^5$ Pa). The idea here is that the effect should be
related to the
ratio of the mean free path to the drop radius. This is the form
here since the mean free path is inversely proportional to pressure.
The particular numerical constant can be obtained experimentally if
the experiment were performed at several different pressures. A feature
Milikan's apparatus had, but ours does not.
There are two approaches at this point that one could take.
\begin{enumerate}
\item One could use Eq. \ref{simple} to determine $a$ using an uncorrected
$\eta$, then use this to determine $\eta_{eff}$ then use this viscosity
in Eq. \ref{simple} again to find a somewhat better $a$, and then proceed
around the loop again until convergence is achieved. If the correction
is large, this can get tedious.
\item Put $\eta_{eff}$ of Eq. \ref{etaeff} into Eq. \ref{simple} and
then solve this more complex equation for $a$.
\end{enumerate}
This second approach leads to:
\begin{equation}\label{complex}
a=\sqrt{\frac{9\eta v_f}{2g\rho}+\left(\frac{b}{2p}\right)^2}-\frac{b}{2p}
\end{equation}
Having found $a$ one can then find $m$ using Eq. \ref{m} and then find
$q$ from Eq. \ref{q}. I would use this approach rather than the Pasco one.
Try to find drops that do not rise too quickly for they will likely have
a large number of electrons on them and, further, it will be difficult to
determine the $V_r$. If you can't find slow risers, then lower the voltage
so as to get better precision. Be sure to measure the space thickness in
order to determine the field from the voltage.
\newpage\
\subsection*{Required for your report}:
Make a table of your measurements. Identify the drop and charge.
Determine the charge in each case. Make a table of charge differences.
Determine the smallest charge for which all the charges could be multiples
of this smallest charge. Estimate the error in your determination of e.
Answer these questions somewhere in your report:
\begin{enumerate}
\item You will notice that some drops travel upward and others downward
in the applied field. Why is this so? Why do some drops travel
very fast, and others slow?
\item Is the particle motion in a straight line? Or, do you notice that
the particle "dances" around ever so slightly? This is due to
Brownian motion: the random motion of a small particle in a gas or
fluid.
\item Do you notice distinct steps in the terminal velocity in applied
field? That is, do the terminal velocities appear to clump around
similar values? What does this say about the discrete nature of
charge?
\item We made three assumptions in determining the charge from Equation 1
above. What are they? Hint: They are related to Stoke's Law.
\item How does the average particle diameter you extracted from the
terminal velocity without the field on compare to the value given
on the bottle? Try to explain any discrepancies.
\item Would you, like Millikan, spend 10 years on this experiment?
\end{enumerate}
Extra credit: Millikan and his contemporaries were only able to
measure integer values of electron charge (as you are). Has anyone
measured free charges of other than integer multiples of e?
\end{document}
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