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

^{SJK 00:33, 7 November 2007 (CST)}

In an attempt to measure the charge of an electron, we sent microscopic droplets of oil plummeting to their soon-to-occur eternal doom at the bottom of a viewing chamber due to gravity's evil iron grip. But before their demise, we, being saviors of the oil droplets, activated an electric field hoping that oil droplets would have enough faith in us (charge) to rise to eternal bliss. It turns out that the oil droplets all the droplets had integer multiples of a fundamental unit of charge, and that this unit was about 1.90x10^-19 C, which is not the accepted value, but which can be reconciled with the accepted value by understanding that we are not perfect experimentalists.

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 fundamental 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, so 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 of measuring the charge of the electron. In 1913, he published that the charge was -1.59x10^-19 C. 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 fundamental 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 [math]\displaystyle{ e }[/math]=1.60x10^-19 C. The charge of the electron is [math]\displaystyle{ -e }[/math] and the charge of the proton is [math]\displaystyle{ e }[/math]. To as many significant digits that are relatively certain, the world knows that [math]\displaystyle{ e }[/math]=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 affects 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.

**Figure 1:** A basic schematic of the Millikan device. (Source: Instruction manual for the device written by PASCO scientific.)

To setup this experiment, we plugged in a high-voltage (500 V max) 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 filament 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, 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 to measure resistance, 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.

**Figure 2:** The primary circuit diagram.

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 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, [math]\displaystyle{ t_f }[/math]. 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, [math]\displaystyle{ t_r }[/math]. 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.

[math]\displaystyle{ d=7.59\times 10^{-3} m }[/math] (distance between charged plates using micrometer)
[math]\displaystyle{ \rho=8.86\times 10^2 \frac{kg}{m^3} }[/math] (density of mineral oil given on bottle)
[math]\displaystyle{ g=9.8 \frac{m}{s^2} }[/math] (gravitational acceleration)
[math]\displaystyle{ p=8.4\times10^4 Pa }[/math] (air pressure in Albuquerque)
[math]\displaystyle{ b=8.20\times10^{-3} Pa\cdot m }[/math] (some stupid constant)
[math]\displaystyle{ l=1.0\times10^{-3} m }[/math] (length droplet will be timed over)

Values to be found when taking data

[math]\displaystyle{ T }[/math] (temperature from thermistor in K)
[math]\displaystyle{ V }[/math] (Voltage between plates in viewing chamber in volts)
[math]\displaystyle{ t_f }[/math] (time droplet takes to fall in no field in seconds)
[math]\displaystyle{ t_r }[/math] (time droplet takes to rise in field in seconds)

Values to be calculated later

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

[math]\displaystyle{ mg=9\pi\eta_{eff}a v_f\, }[/math],

where [math]\displaystyle{ \eta_{eff} }[/math] is a correction to [math]\displaystyle{ \eta }[/math] for small [math]\displaystyle{ a }[/math]. Substituting

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

into this equation and solving for [math]\displaystyle{ a }[/math] should get you the correct equation.

Derivation of charge equation

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

[math]\displaystyle{ mg=k v_f\, }[/math] and [math]\displaystyle{ Eq=mg+k v_r\, }[/math],

where [math]\displaystyle{ k }[/math] is how much the air effects the drag force and [math]\displaystyle{ E }[/math] is the electric field strength where up is positive. Eliminating [math]\displaystyle{ k }[/math] and then solving for [math]\displaystyle{ q }[/math] produces

[math]\displaystyle{ q=\frac{mg\left(v_r+v_f\right)}{E v_f} }[/math].

If you substitute

[math]\displaystyle{ m=\frac{4}{3}\pi a^3\rho }[/math] and [math]\displaystyle{ E=\frac{V}{d} }[/math]

into this [math]\displaystyle{ q }[/math] equation, you should get the correct final equation.

The sign [math]\displaystyle{ V }[/math] can get a little tricky when calculating [math]\displaystyle{ q }[/math] (all other values used to find [math]\displaystyle{ q }[/math] are positive). When the plate charging switch is set to negative, this means that the top plate is negative so the value for [math]\displaystyle{ V }[/math] should be positive. To get the droplet to rise, [math]\displaystyle{ V }[/math] will sometimes need to be positive and sometimes negative, which means the charge [math]\displaystyle{ q }[/math] 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 [math]\displaystyle{ v_r }[/math] would be negative when finding [math]\displaystyle{ q }[/math] since the droplet is falling instead of rising. The equation for [math]\displaystyle{ q }[/math] is very flexible and can handle a negative [math]\displaystyle{ v_r }[/math]. 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 [math]\displaystyle{ v_r }[/math] 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.

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 greatly, 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 [math]\displaystyle{ t_r }[/math] was very different and we suspect a change in charge, so we are discarding it, even though I am displaying it below.

[math]\displaystyle{ V }[/math]=+503V
[math]\displaystyle{ T }[/math]=296K

[math]\displaystyle{ t_f }[/math] (s)	41.3	47.0	49.0	51.3	45.5	43.9
[math]\displaystyle{ t_r }[/math] (s)	10.9	4.5	4.6	4.8	4.9	4.8

Droplet 2, Charge A:

[math]\displaystyle{ V }[/math]=-503V
[math]\displaystyle{ T }[/math]=299K

[math]\displaystyle{ t_f }[/math] (s)	59.2	60.1	69.9	62.6
[math]\displaystyle{ t_r }[/math] (s)	9.6	9.3	9.3	9.1

Droplet 2, Charge B: Our first observation for [math]\displaystyle{ t_r }[/math] 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.

[math]\displaystyle{ V }[/math]=-504V
[math]\displaystyle{ T }[/math]=299K

[math]\displaystyle{ t_f }[/math] (s)	85.0	87.1
[math]\displaystyle{ t_r }[/math] (s)	2.0	1.43	1.53	1.43	1.52	1.51

Droplet 3:

[math]\displaystyle{ V }[/math]=-504V
[math]\displaystyle{ T }[/math]=300K

[math]\displaystyle{ t_f }[/math] (s)	42.3	47.2	50.8	47.1
[math]\displaystyle{ t_r }[/math] (s)	12.1	12.1	12.9	13.5

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.

[math]\displaystyle{ V }[/math]=+505V
[math]\displaystyle{ T }[/math]=300K

[math]\displaystyle{ t_f }[/math] (s)	57.5	63.5	63.0
[math]\displaystyle{ t_r }[/math] (s)	10.0	9.7	9.1

After taking this data, I performed the appropriate calculations to find the radius and charge of each droplet. By observing that 1) we always 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 fundamental units of charge [math]\displaystyle{ e }[/math] 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.

Droplet	[math]\displaystyle{ \eta\, }[/math] (x10^-5 Pa*s)	[math]\displaystyle{ v_f\, }[/math] (x10^-5 m/s)	[math]\displaystyle{ v_r\, }[/math] (x10^-4 m/s)	[math]\displaystyle{ a\, }[/math] (x10^-7 m)	[math]\displaystyle{ \left\| q\right\|\, }[/math] (x10^-19 C)	Suspected Multiple of [math]\displaystyle{ e }[/math]
1	1.83	2.17(7)	2.11(3)	4.08	4.00(13)	2
2A	1.85	1.60(6)	1.07(1)	3.46	1.75(6)	1
2B	1.85	1.16(1)	6.74(9)	2.89	7.81(14)	4
3	1.86	2.14(8)	0.79(2)	4.09	1.76(6)	1
6	1.86	1.63(5)	1.04(3)	3.52	1.75(6)	1

To calculate [math]\displaystyle{ e }[/math], I set the sum of the charges equal to the sum of the suspected multiples of [math]\displaystyle{ e }[/math].

[math]\displaystyle{ \sum \left|q\right|=\left(4.0044+1.7533+7.8087+1.7582+1.7539\right)\times10^{-19} C =\left(2+1+4+1+1\right)e }[/math]

Solving for [math]\displaystyle{ e }[/math] gives

[math]\displaystyle{ e=1.90\times10^{-19} C }[/math] .

Discussion

My first observation is that making the charge more positive in 2B significantly decreases the size of the droplet. This is interesting because it shows that the collisions between the droplets and alpha particles are violent.^{SJK 00:59, 7 November 2007 (CST)}

00:59, 7 November 2007 (CST)
How sure are you that the radius changed? Is the difference *significant?* ... you need to discuss the uncertainty if you're going to make observations like this!

I want you to take more data using your ideas for how to do it better.

We should have selected some drops that were moving a faster in a field since most of our selected drops had plus or minus one electron, but we thought we were dealing with larger charges with our selected drops. Also, it probably would have been better if we would have waited for the power supply and thermistor to warm up to reduce fluctuations in voltage and temperature.

Let's see how good of an experimentalist I am by comparing my [math]\displaystyle{ e }[/math] with the actual value of 1.60x10^-19 C...

[math]\displaystyle{ Relative\ Error=\frac{\left|1.60\times10^{-19}C-e\right|}{\left|1.60\times10^{-19}C\right|}=0.19=19% }[/math]

The average uncertainty due to random error in my charge calculations is not enough to explain this error. The [math]\displaystyle{ e }[/math] I calculated is above the accepted value either because the accepted value is wrong (unlikely) or because of systematic error. A large part of this error may be due to not taking Albuquerque's high altitude into account when calculating viscosity, [math]\displaystyle{ \eta }[/math]. Also, many of the values, such as air pressure, may be incorrect. I did not include these uncertainties in my calculations because I was only finding the uncertainty due to the random error of the velocity measurements.

Conclusion

^{SJK 01:00, 7 November 2007 (CST)}

There definitely is a fundamental electric charge that both the electron and proton have, and the currently accepted value is most likely right on! My random error was not enough to explain why my value for [math]\displaystyle{ e }[/math] was too large, but there was a good bit of systematic error to explain this!

References

^{SJK 01:02, 7 November 2007 (CST)}

This experiment is based on the instruction manual for the Millikan device (Model AP-8210 by PASCO scientific).

General Koch comments

See my email for some more comments
Overall, the writing style needs to be changed to be more appropriate for formal report.
Some specific things that need to be fixed (not complete yet):
1. You need an estimate of your random uncertainty in your final best estimate for fundamental charge!!!
2. Expanded introduction (see above comment)
3. Expanded conclusions (see above comment)
4. Diagram / photo for methods (see above comment)
5. Analysis of sensitivity of answer to uncertainties in the parameters. E.g., given the data you have, make a plot of "estimated fundamental charge versus assumed air pressure." You can make similar plots for all parameters you think are important. If making a plot is too difficult, you can calculate it for +delta and -delta to obtain the slope of the curve near your assumed value. This may also be a method that you use to estimate your uncertainty in your final value.
6. Calibration of microscope grid? Calibration of other instruments...how do you know they are working?
7. To get an excellent grade, I will want you to take some more data to try to make up for the deficiencies you noticed while writing up the report.
8. Other?

Physics307L:People/Knockel/formal2

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