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diff --git a/chapters/millikan.tex b/chapters/millikan.tex new file mode 100644 index 0000000..085c22f --- /dev/null +++ b/chapters/millikan.tex @@ -0,0 +1,254 @@ +%\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} |