Physics307L:People/Knockel/formal2

= Measuring charge of single electrons via Millikan's oil drop experiment = Author: Bradley Knockel

Experimentalists: Nikolai Joseph and Bradley Knockel

Location: UNM Department of Physics, Albuquerque, New Mexico, United States

Date: December 9, 2007

Abstract
In an attempt to measure the charge of an electron, we used oil droplets of radii smaller than a micrometer with a small net charge that results from a deficit or excess of several electrons. By analyzing how these droplets fell and rose when both under electric fields and when under no field, the charge of the droplets could be calculated. Once these charges were obtained, I found an integer multiple of an elementary charge (the charge of an electron) that composed the charges on the droplets. I this calculated the magnitude of this charge to be $$e$$=1.917(25)x10-19 C. This is 20% larger than the accepted value of $$e$$=1.602x10-19 C due to systematic error that can be corrected in later experiments that can use Millikan's basic procedure that I used.

Introduction
The magnitude of the charge of every electron and every proton is the same. Knowing the value of this charge allows the human race to build many things including cathode ray tubes and the first televisions and computer monitors. In fact, something so elementary as this charge has limitless application and importance in understanding the physical world. The charge of the quark is more fundamental, but the magnitude of this value is 1/3 or 2/3 of the charge of the electron, and knowing the charge of the electron allows one to know the charge of the quark.

Robert Millikan was the first person to devise a method to definitively measure the charge of the electron. In 1913, he published that the charge was -1.592x10-19 C [1]. This result won him the Nobel Prize 10 years later. His method was to give very small oil droplets a very small charge of only several electrons and then record the velocities of the droplets when an electric field is introduced. A charge can be calculated only if the mass of the droplet is known, and the mass can be known by measuring the velocity that the droplet falls in no electric field. He hoped to find and found that, after calculating charges, he could notice integer multiples of some elementary charge.

In my experiment, I want to copy Millikan's method and measure the charge of an electron. There is no good reason why the charge of the electron gets all the attention with this experiment since the charge of a proton is also measured. The currently accepted value for this charge is $$e$$=1.602x10-19 C. The charge of the electron is $$-e$$ and the charge of the proton is $$e$$. To as many significant digits that are relatively certain, the accepted value is $$e$$=1.60217646x10-19 C.

Methods and Materials
We preformed the setup a week before doing the procedure, but we performed the entire procedure within one day to have more precise data since atmospheric pressure affected our results. Our data was taken between 3:00 and 5:00 pm MST on September 19, 2007.

Setup
Our main piece of equipment was the Millikan device (Model AP-8210 by PASCO scientific), which includes the following (as shown in Figure 1): a viewing chamber that will contain the droplets, a scope for viewing the droplets inside the viewing chamber, a light to shine in the viewing chamber to see the droplets, a DC transformer for the light, a level for making the platform horizontal, a plate charging switch for changing the electric field in the chamber, a focusing wire for focusing the scope, mineral oil and atomizer for creating oil droplets, a thorium-232 source for altering the charge of a droplet, and a thermistor for determining the temperature of the chamber.



To setup this experiment, we plugged in a high-voltage (500 V) direct-current power source into the Millikan device using banana plug patch cords. We used patch cords so we could attach a multimeter in parallel to measure the precise voltage from the power supply. Before turning on the power supply, we leveled the Millikan device, plugged in a DC transformer to the light that will be used to view the droplets, focused the viewing scope using the focusing wire, and aimed the light on the focusing wire. We then checked to make sure our multimeter was measuring the voltage correctly before connecting another multimeter to the built-in thermistor (a thermistor uses a measure of resistance to find the temperature). I am not providing the model numbers of the multimeters since this information does not help in predicting how well they work (only testing them can do this), but I can say that the multimeters were in great agreement with the power source and the approximate temperature of the room.

There is one important circuit in this experiment. The thermistor requires an extremely small current created when the multimeter sends current through a small resistor, but this is a trivial circuit. The power for the light source and multimeters are also irrelevant circuits. The main circuit is shown in Figure 2 and involves a somewhat complicated switch (the plate charging switch) that has three settings: positive voltage, no voltage, and negative voltage. The capacitor in the diagram is what creates the electric field in the viewing chamber, and the voltmeter is a simple multimeter.



Our setup included some other minor equipment. We had mineral oil and an atomizer to spray droplets into the viewing chamber of the Millikan device, a stopwatch was needed to measure rise and fall times of the droplets, and a micrometer was needed to measure the distance between the plates that create the voltage.

Procedure
After turning off the external lights, we sprayed oil droplets into the viewing chamber using the atomizer by pumping droplet rich air into it. There was a small hole for the droplets could enter the chamber, and there was another small hole to allow air flow so the oil droplet filled air could make its way into the chamber (the latter hole could be closed when the droplets entered the chamber). There is no science to this; we just kept trying over and over until droplets appeared in the center of the screen. We then selected drops that were barely falling through the viewing chamber in no electric field (we want drops that have little mass). From those drops, we selected one that moved slowly in a field (we want drops that have little charge).

Once we had singled out a desirable droplet, we measured the time the droplet took to fall a millimeter, $$t_f$$. We could do this because the scope had a mesh that measures distance in millimeters. Having a partner to hold the stopwatch and write data while the other person watches the droplet was very helpful. We then created an electric field that caused the droplet to rise and measured the rise time, $$t_r$$. We took many measurements of both of these times over and over on the same droplet. We then tried to introduce alpha particles using the thorium-232 source to change the charge of the oil droplet (to be either more positive or negative depending on how the collision between the oil and alpha particles occurred), but the droplet would often become lost in the viewing chamber before we could do this. This process took practice, and it was hard to be sure that the droplet was not changing its charge unexpectedly, which happened a few times.

We also recorded the temperature as given by the thermistor and the voltage across the capacitor plates for each droplet. Since the fluctuations in these readings were so small compared to the output of the multimeters providing them, we did not need to take this data very often.

Known values
The following values are needed for calculations are are given to as many significant figures as are reasonably certain.


 * $$d=7.59\times 10^{-3} m$$ (distance between charged plates using micrometer)
 * $$\rho=8.86\times 10^2 \frac{kg}{m^3}$$ (density of mineral oil given on bottle)
 * $$g=9.8 \frac{m}{s^2}$$ (gravitational acceleration)
 * $$p=8.4\times10^4 Pa$$ (air pressure in Albuquerque)
 * $$b=8.20\times10^{-3} Pa\cdot m$$ (a constant used when finding $$\eta_{eff}$$ in the derivation of the radius formula)
 * $$l=1.0\times10^{-3} m$$ (length droplet will be timed over)

Values to be found when taking data

 * $$T$$ (temperature from thermistor in K)
 * $$V$$ (Voltage between plates in viewing chamber in volts)
 * $$t_f$$ (time droplet takes to fall in no field in seconds)
 * $$t_r$$ (time droplet takes to rise in field in seconds)

Values to be calculated later
(Sutherland's formula gives viscosity of air in Pa*s as a function of temperature in Kelvins)
 * $$\eta=18.27\times10^{-6}\frac{291.15K+120K}{T+120K} \left(\frac{T}{291.15K}\right)^\frac{3}{2}$$ Pa*s
 * $$v_f=\frac{l}{t_f}$$ (average velocity of oil droplet falling in no field in m/s)
 * $$v_r=\frac{l}{t_r}$$ (average velocity of oil droplet rising in a field in m/s)
 * $$a=\sqrt{\left(\frac{b}{2p}\right)^2+\frac{9\eta v_f}{2g\rho}}-\frac{b}{2p}$$ (radius of droplet in meters)
 * $$q=\frac{4}{3}\pi\rho g d\frac{a^3}{V}\frac{\left(v_r+v_f\right)}{v_f}$$ (charge of oil droplet in Coulombs)

Derivation of radius equation
Using Stokes equation and Newton's 2nd law for a falling droplet in no field, one gets:

$$mg=9\pi\eta_{eff}a v_f\,$$,
 * Steve Koch 16:54, 11 December 2007 (CST):I'm not checking all these formulas, but should it be six pi?

where $$\eta_{eff}$$ is a correction to $$\eta$$ for small $$a$$. Substituting

$$m=\frac{4}{3}\pi a^3\rho$$ and $$\eta_{eff}=\eta\left(\frac{1}{1+\frac{b}{pa}}\right)$$ [1]

into this equation and solving for $$a$$ should get you the correct equation.

Derivation of charge equation
Newton's laws for a falling (in no field) and rising droplet create

$$mg=k v_f\,$$ and $$Eq=mg+k v_r\,$$,

where $$k$$ is how much the air effects the drag force and $$E$$ is the electric field strength where up is positive. Eliminating $$k$$ and then solving for $$q$$ produces

$$q=\frac{mg\left(v_r+v_f\right)}{E v_f}$$.

If you substitute

$$m=\frac{4}{3}\pi a^3\rho$$ and $$E=\frac{V}{d}$$

into this $$q$$ equation, you should get the correct final equation.

The sign $$V$$ can be confusing when calculating $$q$$ (all other values used to find $$q$$ are positive). When the plate charging switch is set to negative, this means that the top plate is negative so the value for $$V$$ should be positive. To get the droplet to rise, $$V$$ will sometimes need to be positive and sometimes negative, which means the charge $$q$$ will sometimes be positive or negative.

An alternate method of doing this experiment is to take velocity measurements with the field pushing the droplet down, in which case $$v_r$$ would be negative when finding $$q$$ since the droplet is falling instead of rising. The equation for $$q$$ is very flexible and can handle a negative $$v_r$$. However, this is a bad idea since slower velocities are easier to time. If a power supply powerful enough to actually have the droplet of smallest mass and charge you can find rise cannot be found, this is another instance where $$v_r$$ would need to be negative.

Calculating the charge of an electron
After I calculated all of the charges, I found a value that all the charges are an integer multiple of. By doing this, I had a guess for how many electrons were on each droplet. I then calculated the charge of the electron to be the sum of all the electrons on all of the droplets divided by the sum of all the charge on all the droplets.

Results
Our first matter of business was to take data, and I am providing that data below. In all the following measurements, I use the number of significant figures that I recorded when doing the experiment, which is equivalent to the number of useful significant digits.

The voltage and temperature varied extremely little, so I only took one value for each. Although the values of the voltage and temperature changed from droplet to droplet making it appear that these values fluctuated, much of the time that elapsed between these droplets was trying to get droplets into the viewing chamber and choosing a suitable droplet.

Droplet 1: Our first observation for $$t_r$$ was very different and we suspect a change in charge, so we are discarding it, even though I am displaying it below.
 * $$V$$=+503V
 * $$T$$=296K

Droplet 2, Charge A:
 * $$V$$=-503V
 * $$T$$=299K

Droplet 2, Charge B: Our first observation for $$t_r$$ was very different and we suspect a recording error, so we are discarding it, and I am displaying it below. We only took two falling times because these took much longer than the rising times and we were afraid that we would lose the droplet if we took too much time.
 * $$V$$=-504V
 * $$T$$=299K

Droplet 3:
 * $$V$$=-504V
 * $$T$$=300K

Droplet 6: Droplets 4 and 5 provided either one or two data points before going out of focus and becoming lost, so I will not provide them.
 * $$V$$=+505V
 * $$T$$=300K

After taking this data, I performed the appropriate calculations to find the radius and charge of each droplet. By observing that 1) we usually chose the droplet with the smallest charge, 2) three of five droplets all had practically the same and low charge, and 3) all five of the droplets are multiples of this charge, I could guess the number of elementary units of charge $$e$$ that were on each droplet.

In the table below, the number in parenthesis following the velocities is the uncertainty due to random error. I am using the standard error of the mean to represent this uncertainty. For the radius values, my uncertainty is due to the propagation of the uncertainty from the velocity values, and this happens to be very small. For the charge values, my uncertainty is the propagation of the uncertainties from the radius and the velocity values. I am providing the number of significant figures that are reasonably well-known while using the full-length (double precision) numbers in my calculations.

To calculate $$e$$, I set the sum of the charges equal to the sum of the suspected multiples of $$e$$.

$$\sum \left|q\right|=\left(4.0656+1.7708+7.8829+1.7683+1.7632\right)\times10^{-19} C =\left(2+1+4+1+1\right)e$$

Solving for $$e$$ gives

$$e=1.917(25)\times10^{-19} C$$.

Discussion
My first observation is that making the charge more positive in 2B significantly decreases the size of the droplet. This is the only conclusion I can reach since the uncertainty in my radius calculations were so low that they do not even appear on my table. However, I may have confused two particles since this experiment was very difficult on the eyes. This conclusion, if correct, is interesting because it shows that the collisions between the droplets and alpha particles are violent.

My result of $$e$$=1.917(25)x10-19 C is 20% larger than the actual charge of an electron, which is far outside of the uncertainty due to random error. This random error was due to imperfect operations of the stopwatch and Brownian motion of the droplets. I can think of many causes for the systematic error: faulty multimeter or stopwatch, the mesh on the scope being incorrectly calibrated, air viscosity ($$\eta$$) being affected by altitude, etc. To determine which values that may have caused the most systematic error, I increased the value of each by 10% and have recorded the relative change in $$q$$ for my droplet 1 calculations.

From these results, I am fairly certain that much of the systematic error is due to $$l$$, the length each droplet was timed over. I am fairly certain because I was already suspicious of this variable because I had no way to calibrate it and because there seemed to be many ways for the calibration to be off (for instance, if some droplets are farther from the lens than others). $$\eta$$ worries me greatly because I could not find a way to take into account Albuquerque's altitude into its calculation (in the equation I used, the only variable $$\eta$$ depends on is $$T$$). $$T$$ does not worry me since I the results were no more than 5 K off since the results matched the perceived temperature of the room. The variable $$d$$ also does not worry me because we used a very accurate micrometer.

Much systematic error may be do to the lack of an in depth understanding of how the droplets move through the air. I say this because the three droplets with one electron produce charges that are very close. Taking the value of $$e$$ to be 1.77x10-19 C, which is the average charge of these three droplets, I find that the two-electron droplet is 2.30 times this amount and the four- electron droplet is 4.45 times this amount. This is very odd considering the high precision seen with one droplet.
 * Steve Koch 17:03, 11 December 2007 (CST):This discussion is excellent, and I love the sensitivity table. I believe from Millikan's paper that the "l" uncertainty was his major problem too.  In your case, it is supposed to be accounted for by the focusing wire...that is, the grid is supposed to be calibrated when the center of the wire is in focus.  So, if you bump or change the focus during an experiment, l will surely be systematically wrong.  Also note the question about your Stokes drag equation above.  Finally, that is interesting that you see 2.30 and 4.45x the n=1 value...that probably is a great indicator of the systematic problem, though I'm not sure what it means yet.

Conclusion
In the footsteps of Millikan, I wanted to measure the magnitude of the charge of the electron, $$e$$, by measuring how oil droplets with a magnitude of net charge no larger than $$5e$$ responded to an electric field. To do this, I needed to know the mass of the droplets, which I accomplished by analyzing the speed at which the droplets fell in no electric field. In the end, I calculated the magnitude of the charge of the electron to be $$e$$=1.917(25)x10-19 C, which is 20% larger than the accepted value.

My random error is not enough to explain why my value for $$e$$ is too large, and there are a few things that can be done in future experiments to reduce systematic error. Not having to depend on equipment from a junior lab would be a great start. Not depending on PASCO to design the scope's mesh or simply calibrating the mesh would also be very beneficial. Using a barometer to calculate the actual air pressure in the room and experimentally determining the viscosity of air at the time of the experiment would greatly improve accuracy. Also, understanding how air's viscosity affects very small object would be useful.