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+%\chapter*{Millikan Oil Drop Experiment}
+%\addcontentsline{toc}{chapter}{Millikan Oil Drop Experiment}
+\documentclass{article}
+\usepackage{tabularx,amsmath,boxedminipage,epsfig}
+    \oddsidemargin  0.0in
+    \evensidemargin 0.0in
+    \textwidth      6.5in
+    \headheight     0.0in
+    \topmargin      0.0in
+    \textheight=9.0in
+
+\begin{document}
+\title{Millikan Oil Drop Experiment}
+\date {}
+\maketitle
+\noindent
+
+\textbf{Experiment objectives}: \\ \textbf{Week 1}: explore the experimental
+apparatus and data
+ acquisition procedure; develop the data analysis routing using a mock Millikan experiment. \\ \textbf{Week 2}: extract the value of a unit charge $e$ by observing the motion
+ of charged oil drops in gravitational and electric field.
+
+ \begin{boxedminipage}{\linewidth}
+\textbf{Warning: this is a hard experiment!} \\
+%
+You have two class sessions to complete this experiment - for a good reason:
+this experiment is very hard! After all, it took R. A. Millikan 10 years to
+collect and analyze enough data to make accurate measurement of the electron
+charge. It takes some (often considerable) time to learn how to use the
+apparatus and get reliable data with it, so make sure you take take good notes
+during the first session on what gives you good and bad results. Also prepare
+and debug all the data analysis routines (such as calculations of drop
+parameters from the velocity measurements). Then you will hopefully have enough
+time
+ to make reliable measurements during the second session.
+\end{boxedminipage}
+
+    \section*{Introduction and Theory}
+The electric charge carried by a particle may be calculated by measuring the
+force experienced by the particle in an electric field of known strength.
+Although it is relatively easy to produce a known electric field, the force
+exerted by such a field on a particle carrying only one or several excess
+electrons is very small. For example, a field of $1000$~Volts per cm would
+exert a force of only $1.6\cdot l0^{-14}$~N dyne on a particle bearing one
+excess electron. This is a force comparable to the gravitational force on a
+particle with a mass of $l0^{-l2}$~gram.
+
+The success of the Millikan Oil Drop experiment depends on the ability to
+measure forces this small. The behavior of small charged droplets of oil,
+having masses of only $l0^{-l2}$~gram or less, is observed in a gravitational
+and an electric field. Measuring the velocity of fall of the drop in air
+enables, with the use of Stokes’ Law, the calculation of the mass of the drop.
+The observation of the velocity of the drop rising in an electric field then
+permits a calculation of the force on, and hence, the charge carried by the oil
+drop.
+
+Consider the motion of a small drop of oil inside the apparatus shown in Fig.
+\ref{moplates}.
+\begin{figure}[h]
+\centerline{\epsfig{width=3in, file=modexp.eps}} \caption{\label{moplates}
+Schematic Millikan Oil Drop System with and without electric field.}
+\end{figure}
+
+
+Because of the air drag tiny droplets fall very slowly with some constant
+terminal velocity $v_f$:
+\begin{equation}\label{fall}
+mg=kv_f
+\end{equation}
+where  $q$ is the charge on the droplet, $m$ is the mass of the droplet, $g$ is
+the acceleration due to gravity, and $k$ is a drag coefficient which will be
+related to the viscosity of air and the radius of the droplet.
+
+Because of its small mass the motion of the droplets is sensitive to an
+external electric field $E$ even if they carry charges of only a few electrons.
+A sufficient electric field can cause the oil drop to rise with a constant
+velocity $v_r$, such that:
+\begin{equation}\label{rise}
+Eq=mg+kv_r
+\end{equation}
+Combining Eqs.~(\ref{rise},\ref{fall}) we can find the charge $q$:
+\begin{equation}\label{q}
+q=\frac{mg(v_f+v_r)}{Ev_f}
+\end{equation}
+
+Therefor, the charge of the droplet can be found by measuring its terminal
+velocity $v_t$ and rising velocity in the external magnetic field $v_r$.
+However, we also need to know the mass and the radius of a drop. These data has
+to be extracted from the same data. 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}
+The mass of a drop can be related to its radius:
+\begin{equation}\label{m}
+m=\frac{4}{3}\pi a^3 \rho,
+\end{equation}
+and one may solve for $a$ using Eq.~(\ref{fall}):
+\begin{equation}\label{simple}
+a=\sqrt{\frac{9\eta v_f}{2g\rho}}
+\end{equation}
+Here $\rho=.886\cdot 10^3 \mathrm{kg/m}^3$ is the density of the oil.
+
+The air viscosity at room temperature is $\eta=1.832\cdot 10^{-5}$Ns/m$^2$ for
+relatively large drops. However, there is a small correction for this
+experiment for a small drops 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$\cdot$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.
+
+To take into the account the correction to the air viscosity, one has to
+substitute the expression for $\eta_{eff}$ of Eq.~(\ref{etaeff})  into Eq.~(
+\ref{simple}) and then solve this more complex equation for $a$:
+\begin{equation}\label{complex}
+a=\sqrt{\frac{9\eta v_f}{2g\rho}+\left(\frac{b}{2P}\right)^2}-\frac{b}{2P}
+\end{equation}
+
+Therefor, the calculation of a charge carried by an oil drop will consists of
+several steps:
+\begin{enumerate}
+\item Measure the terminal velocities for a particular drop with and without
+electric field.
+\item Using the falling terminal velocity with no electric field, calculate
+the radius of a droplet using Eq.~(\ref{complex}), and then find the mass of
+the droplet using Eq.~(\ref{m}).
+\item Substitute the calculated parameters of a droplet into Eq.~(\ref{q}) to
+find the charge of the droplet $q$.
+
+\end{enumerate}
+%This second approach leads to:
+%
+%
+%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.
+
+
+
+
+\section*{Experimental procedure}
+
+\subsection*{Mock Millikan experiment - practice of the data analysis}
+\textit{The original idea of this experiment is described here:
+http://phys.csuchico.edu/ayars/300B/handouts/Millikan.pdf}
+
+The goal of this section is to develop an efficient data analysis routine for
+the electron charge measurements. You will be given a number of envelopes with
+a random number of Unidentified Small Objects (USOs), and your goal is to find
+a mass of a single USO (with its uncertainty!) without knowing how many USOs
+each envelope has. This exercise is also designed to put you in Robert Millikan
+shoes (minus the pain of data taking).
+
+Each person working on this experiment will be given a number of envelopes to
+weight. Each envelope contain unknown number of USO plus some packing material.
+To save time, all the data will be then shared between the lab partners.
+
+Then analyze these data to extract the mass of a single USO and its uncertainty
+in whatever way you’d like. For example, graphs are generally useful for
+extracting the data - is there any way to make a meaningful graph for those
+measurements? If yes, will you be able to extract the mean value of USO mass
+and its uncertainty from the graph? \textit{Feel free to discuss your ideas
+with the laboratory instructor!}
+
+After finding the mass of a USO, work with your data to determine how the size
+of the data set affects the accuracy of the measurements. That will give you a
+better idea how many successful measurements one needs to make to determing $e$
+in a real Millikan experiment.
+
+This part of the experiment must be a part of the lab report, including the
+results of your measurements and the description of the data and error analysis
+routine.
+
+
+
+\subsection*{Pasco Millikan oil drop setup}
+
+Follow the attached pages from Pasco manual to turn on, align and control the
+experimental apparatus. Take time to become familiar with the experimental
+apparatus and the measurement procedures. Also, it is highly recommended that
+you develop an intuition about ``acceptable'' drops to work with (see Pasco
+manual, ``Selection of the Drop'' section).
+
+\subsection*{Data acquisition and analysis}
+
+\begin{itemize}
+
+\item Choose a ``good'' drop and make about 10 measurements for its fall and rise
+velocities $v_t$ and $v_r$ by turning the high voltage on and off. Try to find
+a drop that does not rise too quickly for it will likely have a large number of
+electrons 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.
+
+
+\item Calculate the charge on the droplet. If the result of this first
+determination for the charge on the drop is greater than 5 excess electron, you
+should use slower moving droplets in subsequent determinations. Accepted value
+of the electron charge is $e=1.6\times10^{-19}$~C.
+
+\item If the drop is still within viewing range, try to change its charge. To
+do that bring the droplet to the top of the field of view and move the
+ionization lever to the ON position for a few seconds as the droplet falls. If
+the rising velocity of the droplet changes, make as many measurements of the
+new rising velocity as you can (10 to 20 measurements). If the droplet is still
+in view, attempt to change the charge on the droplet by introducing more alpha
+particles, as described previously, and measure the new rising velocity 10–20
+times, if possible. Since making measurements with the same drop with changing
+charge allows does not require repeating calculations for the drop mass and
+radius, try ``recharging'' the same drop as many times as you can.
+
+\item Be sure to measure the separation $d$ between the electrodes and the voltage potential in order to
+determine the field from the voltage.
+
+\end{itemize}
+
+Each lab partner should conduct measurements for at least one drop, and the
+overall number of measurements should be sufficient to make a reliable
+measurement for the unit electron charge. Make a table of all measurements,
+identify each drop and its calculated charge(s). 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 We made three assumptions in determining the charge from Equation 1
+%   above. What are they? Hint: They are related to Stoke's Law.
+%
+%
+%\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}