Millikan’s Oil Drop Experiment

Michael Toomey

Partner: Rob Shreiner

14 February 2017

Version 2

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Abstract

We conducted an experiment to prove the existence of charge quantization and determine the

value of the fundamental unit of charge. This value corresponds to the charge of a single

electron. The setup and methods used in the lab closely resembled those used by Robert Millikan

to determine e 1. In the experiment, droplets of oil were introduced into a chamber formed by

two capacitor plates that could be charged at a known voltage. Observations of the time for the

droplet to fall under the influence of gravity and rise under the presence of a known electric field

were recorded to determine the mass of the oil droplets, the force they experienced, and to

calculate e. Many drops were observed over the course of the experiment. The results show

evidence for the existence of charge quantization. In this experiment, the value of e was

determined experimentally to be (4.6 ± 0.6) × 10−10 e.s.u.

Introduction

At the beginning of the 20th century, physicists were tantalizingly close to accurately calculating,

for the first time, the mass of the electron. The subatomic particle was discovered, by J.J.

Thompson in 1897, when some of the first measurements of the charge-to-mass ratio of the

electron were being published2. As such, many attempts were made to estimate the value of e

following his discovery, but with marginal success. This changed when Millikan made several

improvements on methods used by his predecessors. One such insight he had was to observe the

droplets with a much higher applied voltage so as to measure the rate at which drops moved as

opposed to the point at which the droplets became suspended – when the gravitational force

balanced with the electric. This allowed Millikan to focus on a single drop, as opposed to

dealing with large clouds of oil in his field of view, which produced more reliable results. As a

result, he was able to calculate the charge of an electron and Avogadro’s number with great

accuracy2.

The methods developed by Millikan to manipulate oil droplets, in a controlled environment using

an electric field, to measure e, have been employed in this experiment to conduct an independent

calculation of the quantity. The point of the experiment is not just to calculate the value, but also

to see firsthand the quantized nature of charges. In other words, the experiment is interested with

how charge is distributed on the oil drops. Are they integer multiples of each other? Or the

contrary? Is there no common denominator between the measured charges of a collection of oil

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droplets? These form the main reason for pursuing this experiment. However, as previously

mentioned, values for the mass of the electron and Avogadro’s number can be derived from

knowing e.

Unfortunately, the simple answers to why one may do this experiment brings forth some

questions of its own. For example, why might it be important to calculate e? Knowing e is

necessary to understand electrons. For example, knowing e can predict the energy of an electron

that is accelerated across a potential difference or determine its movement through a magnetic

field. Having this insight into how electrons and other charges can move, because e is known,

has had direct implications for the technology in our day to day lives. The benefits from knowing

the mass of the electron and Avogadro’s number also have had far-reaching implications from

physics to chemistry.

Experimentally, in its most simple sense, the charge is calculated from observations of the

velocity of oil droplets when they fall under the influence of gravity and when they are pulled

back upward, opposing gravity, by a known electric field. Theory tells us that knowing these

quantities, in addition to a few other caveats, we can derive the charge of the oil droplets.

Theory

The derivation of the expression to calculate charge, q, is relatively straight forward. The

foundation of the derivation comes from analyzing two special situations the oil droplets will be

exposed to in the lab. The first situation is when the oil droplet is introduced into the chamber

only under the influence of gravity. In this situation, the oil droplets experience a gravitational

force downward and a corresponding drag force, proportional to its velocity downward,

antiparallel to gravity. The second situation is when the capacitor plates have a voltage applied to

them that causes some of the oil drops to rise due to an electric force opposing the force of

gravity with a drag force, again proportional to the velocity, but now parallel with gravity. These

two equations can be used to eliminate the drag coefficient from both, such that the charge of the

oil drop is expressed in terms of its mass, the potential of the capacitor, the two velocities, and

little g.

The above equations are only halfway done as the mass of the falling oil drops is not known. The

mass term needs to be substituted out by replacing it with the equation for the volume a sphere

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multiplied by the density of the oil. This has to be altered further as the radius of the oil droplets

is not precisely known. To address this, Stokes’ Law needs to be used to get an expression for

the radius of the droplet in terms of the viscosity, velocity, and a few other parameters. While the

derivation is fairly straightforward, the final equation itself is quite messy – in appearance only –

and is reproduced below in Equation (1) with a full derivation located in Appendix A.

(1)

Experimental Method

The steps to complete this experiment are deceptively easy as they require a significant amount

of patience and concentration. Perhaps the most important step, and one that is necessary before

beginning this experiment, is cleaning the entire housing assembly as seen in Figure 1. Oil tends

to build up in the hole of the top of the upper capacitor plate and the hole cover which can block

the flow of fresh oil into the gap between the capacitors. Therefore, it is prudent that the whole

housing assembly is removed from the stand and cleaned thoroughly using isopropyl alcohol and

Figure 1. This depicts the experimental set up of the Millikan oil drop experiment. The main

piece of equipment is PASCO Scientific’s apparatus, composing all components of the

housing, microscope, light, and attached stand. In addition is a digital mustimeter to measure

the temperature in the housing and the potential across the capacitor plates, a power supply to

provide a high voltage across the capacitor, and oil/oil dispensing devise. These form the

entirety of the equipment necessary to conduct this experiment.

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whipping all surfaces with laboratory quality tissue. The first time this is disassembled is an ideal

time to measure the thickness of the spacer which will be needed for calculation of the charge.

Before starting, it is also prudent to check a barometer for air pressure and record it for later

calculations. When this is complete, it is time to begin the experiment.

The first step using the Millikan apparatus requires that the microscope be focused. This can be

done using a focusing wire that is screwed into the platform that the microscope is on. By

placing this wire through the top of the upper capacitor, as seen is Figure 1, the microscope can

be focused by adjusting the focus knob. Additionally, while the focusing wire is installed, the

knobs of the light can be used to adjust where the beam is concentrated in the housing – the ideal

position is when the right edge of the focusing wire is brightest. When this is complete, the

focusing wire can be returned to its previous location and the entire housing assembly can be

reassembled and returned to the stand.

The next step involves turning on the power supply after making sure that it has been properly

plugged into the apparatus. Using the digital multimeter (DMM) the potential across the

capacitors should read roughly 500 V, otherwise, adjust it with the power supply. The DMM was

then plugged in so that it was reading the resistance of the circuit – this effectively measures the

temperature of the chamber. Over the course of the experiment, multiple measurements of the

temperature of the room were taken every 15 minutes due to a non-consistent temperature in the

housing.

With all of these steps prior being accounted for, the oil could then be introduced into the

chamber. After aligning the nozzle of the oil sprayer with the top of the housing and placing the

ionization lever in the spray droplet position, the bulb on the sprayer was given a firm squeeze

followed by a very long squeeze. Typically this results in an almost immediate sighting of the oil

droplets in the microscope. If after several tries no oil is sighted, the housing apparatus should be

disassembled and cleaned again – this seems always to work. Once oil droplets can be seen, the

ionization lever should be put to the “on” position for a few seconds and then placed on the “off”

position so that the Thorium 232 can ionize the oil in the chamber. Using the control switch and

changing the direction of the electric field, the oil droplets should change direction when the

field is swapped. If this is not sufficient, the droplets can be exposed to radiation for another few

seconds.

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Provided there is now a good group of oil droplets that fall when no field is applied and respond

well to the electric field, the velocity of the droplets can now be measured. This can be done by

using a stopwatch to time how long it takes for a droplet to go over a certain distance of the grid

when it is experiencing only gravity and when it is pushed up by the electric field. These

measurements should be taken several times, say 10, for the same drop to get good statistics.

Additionally, these observations were repeated many times with different drops to assure that e

could be calculated – in this experiment 25 times. When sufficient data has been collected, data

analysis could then begin. A table of all the data gathered in the experiment is included in

Appendix B.

Results

To analyze the data that had been collected in an efficient manner, all of the time measurements

were copied from the notebook used in the experiment to a text file. A Python program was then

written that automated the calculation of the charge for each drop in the experiment based off of

Equation (1). This was done by taking the average value of the time it took the drops to travel the

given distance for both cases and converting them to velocities, followed by using the

appropriate resistance, pressure, and voltage values depending on which day the data were taken.

Another short code was written to fit the data for the viscosity of air with respect to temperature

to a line and another to fit a quadratic to find an expression for temperature as a function of

resistance, using a least squares approach. This covered the basis of simply calculating the

charge of each oil drop.

The results of the calculation for all drops is displayed graphically in Figure 2. While there is

some extent to the 3rd and 4th grouping of charges, there is a very clear indication of the presence

of charge quantization in our data. The next obvious step would be to calculate the amount of

charge that separates the clusters of data points. While not the most statistically rigorous, it was

decided only to calculate the fundamental unit of charge based off of the difference in charge

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from the 1st to the 2nd cluster because the other points were too inconsistent. To calculate the

difference in charge, the error in charge for all oil droplets was calculated using propagation of

errors – this is detailed in the Experimental Uncertainties section. Once this was complete

inverse-variance weighting was used to calculate the mean of the 1st and 2nd clusters of charges.

The same method was also used to calculate the variance for both clusters. The 1st cluster and the

2nd cluster had charges of (9.3 ± 0.3) × 10−10 e.s.u. and (13.9 ± 0.5) × 10−10 e.s.u.

respectively. After subtracting the charge of the 1st from the 2nd cluster and adding the

uncertainties in quadrature a value for e of (4.6 ± 0.6) × 10−10 e.s.u. was determined. This

value for the fundamental unit of charge was divided into the charge of each drop to determine

how many electrons each had.

Another parameter that can be solved for is the mass of the electron. This can be done

using the currently accepted value of the charge-to-mass ratio for the particle, 1.76 × 1011 C/kg,

which gives a mass for the electron of (9.7 ± 0.2) × 10−28 g with the value for e calculated in

this experiment. The last parameter to be calculated is Avogadro’s number. This can be

calculated simply using Faraday’s constant, 2.89 × 1014 e.s.u./mol. Faraday’s constant

Figure 2. This histogram shows the distribution of charges for all oil drops from the

experiment. The plot shows clear signs of the existence of charge quantization.

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measures the amount of charge in a mole of electrons, so this constant divided by e gives a value

for the mole of (6.3 ± 0.1) × 1023 mol−1.

Experimental Uncertainties

In this experiment, there are a lot of experimental uncertainties to consider. This has to do with

the fact that a good portion of the parameters that are needed to calculate the charge of the oil

droplets need to be recorded by the experimenter. In this experiment, two types of relevant error

contribute to the final uncertainty in the charge of the drops.

One type of uncertainty is that caused by the limited measuring ability of an instrument such as

the micrometer used to measure the thickness of the spacer. In this situation, the uncertainty is

judged off of the smallest unit of measure the device provides. In addition to the micrometer, the

voltage measurements and the barometric pressure measurements exhibited the same type of

uncertainty.

The other type of uncertainty in this experiment resulted from data that fluctuated more than the

smallest measuring element of the device being used to measure. An example of this was

measuring the time it took the drops to traverse the grid while looking through the microscope.

Depending on the particular drop, a time measurement could vary anywhere from a tenth of a

second to two seconds. These types of data had to be treated differently. Specifically, the same

data had to be taken many times to determine the mean and standard deviation from the mean for

each parameter. For this experiment, that included all the thermistor readings and time

measurements.

Perhaps the most difficult part of the lab was determining the final uncertainty in the charge. To

accomplish this, propagation of errors needed to be done. The variance in the charge is equal to

the sum of the partials of Equation (1) with respect to each measured value, squared, times each

measured values own variance. This process is explained in detail in Appendix C. A Python

script was written to calculate the variance of the charge for each oil droplet in the experiment.

This data was used so that an inverse-variance weighted mean and variance could be calculated

for each group of like oil droplets. The uncertainty for e was calculated by summing the

uncertainties in quadrature for the charges used to calculate the difference from the 1st and 2nd

cluster of oil droplets.

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Discussion

The results indicate the presence of charge quantization. This can clearly be seen in Figure 2 as

there appears to be clustering near integer multiples of the value of e calculated in this

experiment. There is some slight cause for concern in between 4e and 5e as there doesn’t appear

to be a break. This could be the result for a few reasons. The most likely reason is that there are

some poor data points from a couple of oil drops. While taking data for the experiment,

sometimes the droplets would decide to drift left or right of center where they started after we

had already taken data. Sometimes we also got drops that seemingly floated at times, as opposed

to falling straight down. Perhaps another reason might be that our statistics are not that good.

Perhaps if we had enough time to do a couple of hundred drops, as opposed to 25, the

discreteness would be more consistent. Nonetheless, there is sufficient evidence to prove

quantization exists based on our data.

The actual value for the fundamental constant, e, turned out to be all right. The uncertainty is a

little larger than one would like, but considering the amount of data that was taken and the data

that could be used, the overall result was good. Our result is inclusive of the real constant value

for e when comparing to the widely accepted value. Naturally, due to the closeness of the

experimental value of e to the actual value, it was not surprising to see the mass of the electron

and Avogadro’s number close to their real values.

Conclusion

It has been shown in this experiment that the oil drop apparatus originally designed by Millikan

can do an excellent job of measuring the smallest of charges. Even using considerably less data

than Millikan did, it was possible to independently determine that charge is quantized and that

the fundamental charge is (4.6 ± 0.6) × 10−10 e.s.u. With this newly calculated constant, it

was possible to determine the mass of the electron, (9.7 ± 0.2) × 10−28 g, and Avogadro’s

number, (6.3 ± 0.1) × 1023 mol−1.

References

1 R.A. Millikan, Physical Review, 2, 109-143 (1913).

2 J.J. Thompson, Philosophical Magazine, 44, 293 (1897).

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Appendix A: Derivation of charge equation

The derivation of the equation of charge in the Millikan oil drop experiment begins with an

analysis of the forces that the oil drops undergo during the two different types of observation –

falling under the influence of gravity and rising against gravity due to a magnetic field.

When the oil droplets fall, they experience a gravitational force downward and a buoyant force

upward.

𝑚𝑔 = 𝑘𝑣𝑓

Here m is the mass, g is the gravitational constant, k is the coefficient of friction between the oil

and the air, and vf is the velocity of the particle falling.

When the droplet is rising under the influence of the electric field, it feels an upward force from

the capacitor and downward forces from gravity and the buoyant force.

𝐸𝑞 = 𝑚𝑔 + 𝑘𝑣𝑟

Here the new parameters are the electric field, E, electric charge, q, and the velocity of the drop

while rising, vr.

Both equations can be solved in terms of k to eliminate it and to equate the two expressions.

Simplifying gives an expression for q.

𝑚𝑔

𝑣𝑓= 𝐸𝑞 − 𝑚𝑔

𝑣𝑟

𝑚𝑔(𝑣𝑟 + 𝑣𝑓) = 𝐸𝑞𝑣𝑓

𝑞 = 𝑚𝑔(𝑣𝑟 + 𝑣𝑓)

𝐸𝑣𝑓

While this is an expression for the charge, the mass of the oil drops is not known. To get around

this, Stokes’ Law can be used if the density of the oil is known – in this case, it is known.

However, in this case, a special version of Stokes’ Law needs to be used that corrects for the

slow velocity that the drops fall at in this experiment.

𝑞 = 4𝜋𝜌𝑔(𝑣𝑟 + 𝑣𝑓)

3𝐸𝑣𝑓𝑎3

𝑞 = 4𝜋𝜌𝑔(𝑣𝑟 + 𝑣𝑓)

3𝐸𝑣𝑓(√(

𝑏

2𝑝)2

+ 9𝜂𝑣𝑓

2𝑔𝜌−𝑏

2𝑝)

3

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𝑞 = 6𝜋

√ 9𝜂3

2𝑔𝜌(1 + 𝑏𝑝 (

√(𝑏2𝑝)

2

+ 9𝜂𝑣𝑓2𝑔𝜌 −

𝑏2𝑝)

−1

)

3 (𝑣𝑟 + 𝑣𝑓)√𝑣𝑓

It can be shown, by converting to electrostatic units, that the above expression for the charge is

equivalent to Equation 1.

𝑞 = [400𝜋𝑑 (1

𝑔𝜌[9𝜂

2]3

)

1/2

]

[

(1

1 + 𝑏𝑝𝑎

)

3/2

]

[(𝑣𝑟 + 𝑣𝑓)√𝑣𝑓

𝑉] 𝑒. 𝑠. 𝑢

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Appendix B: Table of experimental data

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Appendix C: Propagation of error for charge

Propagation of errors was conducted to determine the uncertainty in the calculation of the charge

for this experiment. The general formula for the propagation of errors is as follows.

∆𝑓 = √∑(𝜕𝑓

𝜕𝑥𝑖 ∆𝑥𝑖)

2𝑛

𝑖=1

The following parameters from Equation 1 needed to be propagated: the separation of the plates,

d, the pressure in the room, p, the radius of the oil drops, a, the rising velocity, vr, the falling

velocity, vf, and the potential across the capacitor plates, V. Therefore the expression for the

propagation of error for q is as follows:

∆𝑞 = √(𝜕𝑓

𝜕𝑑 ∆𝑑)

2

+ (𝜕𝑓

𝜕𝑝 ∆𝑝)

2

+ (𝜕𝑓

𝜕𝑎 ∆𝑎)

2

+ (𝜕𝑓

𝜕𝑣𝑟 ∆𝑣𝑟)

2

+ (𝜕𝑓

𝜕𝑣𝑓 ∆𝑣𝑓)

2

+ (𝜕𝑓

𝜕𝑉 ∆𝑉)

2

Each of the uncertainty terms was known, so this left only the need to calculate the differentials.

The following is the result of each differential:

𝜕𝑓

𝜕𝑑= [400𝜋 (

1

𝑔𝜌[9𝜂

2]3

)

1/2

]

[

(1

1 + 𝑏𝑝𝑎

)

3/2

]

[(𝑣𝑟 + 𝑣𝑓)√𝑣𝑓

𝑉]

𝜕𝑓

𝜕𝑝= [400𝜋𝑑 (

1

𝑔𝜌[9𝜂

2]3

)

1/2

] [3𝑏

2(

𝑝𝑎3

(𝑎𝑝 + 𝑏)5)

1/2

] [(𝑣𝑟 + 𝑣𝑓)√𝑣𝑓

𝑉]

𝜕𝑓

𝜕𝑎= [400𝜋𝑑 (

1

𝑔𝜌[9𝜂

2]3

)

1/2

] [3𝑏

2(

𝑝3𝑎

(𝑎𝑝 + 𝑏)5)

1/2

] [(𝑣𝑟 + 𝑣𝑓)√𝑣𝑓

𝑉]

𝜕𝑓

𝜕𝑣𝑟= [400𝜋𝑑 (

1

𝑔𝜌[9𝜂

2]3

)

1/2

]

[

(1

1 + 𝑏𝑝𝑎

)

3/2

]

[𝑣𝑟𝑉]

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𝜕𝑓

𝜕𝑣𝑓= [400𝜋𝑑 (

1

𝑔𝜌[9𝜂

2]3

)

1/2

]

[

(1

1 + 𝑏𝑝𝑎

)

3/2

]

[𝑣𝑟 + 3𝑣𝑓

2𝑉√𝑣𝑓]

𝜕𝑓

𝜕𝑉= − [400𝜋𝑑 (

1

𝑔𝜌[9𝜂

2]3

)

1/2

]

[

(1

1 + 𝑏𝑝𝑎

)

3/2

]

[(𝑣𝑟 + 𝑣𝑓)√𝑣𝑓

𝑉2]

The error in the calculation of the charge was determined to be 6.0 × 10−11 e.s.u. using this

approach.