User:John Callow/Notebook/Junior Lab/Formal Report

=Computational Methods for The Millikan Oil Drop Experiment=

Contact Information
Author: John J. Callow

Experimentalists: John J. Callow & Johnny Gonzalez

Location: University of New Mexico Physics and Astronomy Department, Albuquerque, NM

e-mail: jcallow@unm.edu

Abstract
In this experiment we will observe the behavior of several oil droplets in a uniform electric field to find each droplets individual charge, and then using statistical methods calculate the charge of a single electron. The oil droplets will be produced by an atomizer, allowed to settle between two parallel plates acting as a capacitor, and then viewed with a telescope. By comparing the time for a droplet to fall a specified distance with no field to the time that same droplet takes to rise up that distance with an electric field applied we are able to calculate the mass of the drop and then find the charge. Using the assumption that electrons are identical and there is no fractional charge, each charge observed must be some integer n times the value of e, and then using a developed algorithm for the problem a value for e is found that best matches these assumptions.

Our value for the charge of a single electron is $$e = 4.839(20)*10^{-10} e.s.u.$$

The value found from Millikan

$$e = 4.774(9)*10^{-10} e.s.u.$$ [2, pg 140]

The currently accepted value is

$$e = 4.803*10^{-10} e.s.u.$$

Both our and Mullikan's are very close to the currently accepted value and thus showing that this experiment is very effective even compared to modern methods such as (fill this in later) for finding the charge of a single electron.

Introduction
From the results of J J Thomson's charge to mass ratio experiment concrete proof was finally available that the electron existed.[read this article] That different materials all had the same result in Thomson's experiment gave good reason to believe that all electrons carried the same charge. Soon Thomson would begin experiments to determine e, one of which was observing a cloud of identical water droplets carrying a single charge and the effects of gravity or an electric field on said cloud where observed.[1]. But because of the uncertain rate of evaporation and that there was no way to verify each drop was identical and carried a single charge, this method had high uncertainty in its experimental values.

Soon Millikan came up with his oil drop experiment, which eliminated most of the uncertainty of previous experiments. Also unlike the water cloud, Millikan's experiment allowed for the direct observation of a small number of charges on a single oil drop, giving further evidence that electrons are all identical. One issue with Millikan's experiment is that its accuracy greatly depended on Stokes's law and the assumption that oil droplets would behave as solid spheres. After a correction was introduced to Stokes's law for drops approaching the size of the mean free path of a gas molecule, and performing the experiment with liquids of varying viscosity, Millikan found they all produced the same result within error tolerance. Thus the assumption that the drops behaved as solids was shown to introduce little error and so was valid [2, pg 111]. Since the major issues with the experiment had been resolved, Millikan eventually found that e = 4.774(9)*10^-10 e.s.u. [2, pg 140] which is very close to the modern accepted value of 4.803*10^-10 e.s.u.[source]. That an experiment about one hundred years old holds up so well to modern methods demonstrates just how ingenious it is.

In our experiment we reproduced Millikan's original, observing individual charged oil drops in gravity alone and in an electric field. The reasoning for doing the report is to not only verify Millikan's findings being how important they are to atomic physics, but to also experimentally test an algorithm designed to solve under-determined systems of equations by restricting variables to a specified set of numbers. In our case, we are restricting each $$ {n_i} $$ to solve the system of equations $$ {n_i}*e = {q_i} $$ for the charge of an electron e where each q is the total charge found on each drop. Though methods have already been developed to solve for e using data from this type of experiment, it is hard to pass up the chance to explore others as understanding in physics or any science is commonly tied to mathematical concepts.

=Methods and Materials=

Taking Data
We used a commercial Millikan oil drop apparatus kit (pasco something) which consisted of a telescope, capacitor, oil droplet housing, light source, thermistor, and atomizer. Oil sprayed from the atomizer into the housing was viewed from the telescope. A switch allowed for us to turn off and on an electric field caused by the capacitor plates and an external power supply(tel-atomic 500V DC supply). A grid built into the telescope allowed us to accurately observe the rise and fall times over .5mm of an oil drop.

To correctly apply Stokes's law, we needed the temperature in the chamber along with the barometric pressure. To keep track of the temperature a multimeter was connected to the apperatus's thermistor. With this reading, the temperature can be found that corresponds measured resistance to temperature. Due to lack of time we used the initial reading of the thermistor finding the temperature to be 27 degrees centigrade [4]. We estimated the barometric pressure to be 76.0 cm of mercury using a reading done at the time at a nearby airport [3]. Because we didn't have exact values for either temperature or barometric pressure, this introduced minor error.

To calculate the charge of an electron from these times we needed to observe several drops of different charge. The apparatus had the ability to ionize droplets using a radioactive thorium 240somethins source exposed by a lever. This allowed for us to view a single sized drop with different charges. Because electrons have negligible mass compared to a drop of oil, this proved to be a great method as the fall times remained the same. It also insures that the data collected is that of drops with different charge which is required by our algorithm to find e. 

Analysis
With the data collected we were able to calculate the charge on each individual drop by using the formula

$$ q = \left[400{\pi}d\left(\frac{1}{g{\rho}}{\left[\frac{9*{\eta}}{2}\right]^3}\right)^{\frac{1}{2}}\right]*\left[\left(\frac{1}{1+\frac{b}{pa}}\right)^{\frac{3}{2}}\right]*\left[\frac{V_f+V_r\sqrt {V_f}}{V}\right] e.s.u.$$

in which q is the charge of the drop, d the separation of the capacitor plates, $${\rho}$$ the density of oil, g the acceleration of gravity, $${\eta}$$-viscosity of air in poise, b a constant equal to $$6.17*10^{-4} $$ cm of Hg, p the barometric pressure, a the radius of a drop, and V the potential difference across the plates. Then plugging the values for each charge into an algorithm written in Matlab 7.1.0 we were able to find the charge of an electron. This algorithm may be found in the appendix along with info on the above formula and constants. 

Tables and Figures


=Results and Discussion=

Figure 2 shows a plot of our algorithm trying to find the best value for the charge of an electron numerically using values from .1 to 5*10^-9 e.s.u. The method is based on finding a minimum euclidean distance between two vectors, where each value is q/e in one vector and this value rounded to the nearest whole number in another. As shown in the plot, the minimum appears at just about 4.8*10^-10 e.s.u. This function begins to oscillate wildly as e->0 simply because q/e is being evaluated. Figure 3 shows a window from 4.8-5*10^-10 e.s.u. giving a better view of how the function is behaving on a smaller scale.

In Figure 1 I plotted the charge divided by our best estimate of e for all the drops in blue with error bars and the closest integer multiple of our estimated e as red dots. From the plot it is clear that not only does each red dot lie well within the error bars but that they lie nearly on top of the calculated charges of the drops. This gives a lot of confidence in our method for finding e, and also tells us that the data we took is likely very accurate. All except one drop had very small standard error, and even our worst drop had standard error that is less than our best estimate for e. 

=Conclusions= Because we were unable to aquire exact values for the air pressure and forgot to record the thermistor reading for each individual measurement a bit of error is introduced here, but likely very little. Though accuracy in measuring rise and fall times alone had appriciable error, because of the power of the developed algorithm we still found a very good approximation for the charge of an electron. Our value for e was $$e = 4.839(4)*10^{-10} e.s.u.$$ using 1,000,000 steps in the algorithm. Even with only 1000 steps though it estimates e at $$e = 4.871(4)*10^{-10} e.s.u.$$, and this is with a very generous interval over which the algorithm searches. The uncertainty was calculated using a similar method to that of linear regression. Compared to the accepted value of $$e = 4.803*10^{-10} e.s.u.$$ we differ by less than .5%. Though with current data our standard error doesn't overlap the accepted value, with improved measurements and better values for air pressure and temperature it is very likely they will.

With both or results and that of Millikan's being so close to modern methods of experimentally finding an electron's charge, it really shows how powerful this method is for being fairly simple to set up. Being able to observe the effect of only a few charge also gives us very strong visual evidence that electrons are all equivalent.

=Acknowledgments= I would like to thank my lab partner Johnny Gonzalez for helping with setting up and taking data in this experiment. Also I thank my professor Dr. Koch and assistant Pranav Rathi for answering questions and assisting with some setup.

=References= [1] Modern Physics fifth edition, Paul A. Tipler and Ralph A. llewellyn

[2] Millikan R.A (1913), On the Elementary Electrical Charge and the Avogadro Constant. Physical Review, 2 pp. 109-143

[3] http://www.widespread.com/daily.aspx?id=2580&d=10%2f21%2f2009 (it seems to have broken since originally doing the lab)

[4] Pasco Millikan Apparatus lab manual http://openwetware.org/images/e/ea/Pasco_millikan_manual.pdf

=To be added=

I'd like to have a section under analysis on experimenting with the algorithm if I'm unable to actually prove anything. This would be things such as using simulated data and having it attempt to find some value.

=Appendix=

Here is where I plan on putting more info on how the algorithm works, and how I calculate it's uncertainty. I derived a method for it's uncertainty based off that of linear regression but not sure if this is anywhere near what should be done. Also analysis of my function has me believing that the value it puts out is provable to be the best estimate on a reasonable interval. I'll see if I can manage proof in time to turn in the final draft but it might be a bit beyond my current abilities given the time and that I don't really know much statistics other than what I've learned in this class. If I can't do some formal proofs though the experiments would probably be convincing enough. 

Matlab Code
Algorithm used to find an electron's charge

Error propagation code

Maple Worksheets
Worksheet used to calculate error for each droplet's charge

List of Used Constants and Their Values
$${\rho}$$ - density of oil $$ \frac{.866g}{cm^3}$$

g-acceleration of gravity $$ \frac{981cm}{s^2}$$

b-constant equal to $$6.17*10^{-4} $$ cm of Hg

V-potential difference across the plates 500 volts

p-barometric pressure 76.0cm of mercury found at http://www.widespread.com/daily.aspx?id=2580&d=10%2f21%2f2009. Used value at time 15:56 as that is around when measurements took place. Unfortunately the pressure inside the lab is probably different, but we had no equipment to measure it.

Formulas
$$ q = \left[400{\pi}d\left(\frac{1}{g{\rho}}{\left[\frac{9*{\eta}}{2}\right]^3}\right)^{\frac{1}{2}}\right]*\left[\left(\frac{1}{1+\frac{b}{pa}}\right)^{\frac{3}{2}}\right]*\left[\frac{V_f+V_r\sqrt {V_f}}{V}\right] e.s.u.$$

$$ a = \sqrt {\left(\frac{b}{2p}\right)^2 + \frac{9{\eta}*V_f}{2g{\rho}}}- \left(\frac{b}{2p}\right)$$