# Physics307L:People/Joseph/Formal Lab Report2

**Determination of the Charge of a Single Electron**

Nikolai Joseph University of New Mexico Albuquerque, NM njoseph@unm.edu

## Contents

## Abstract

The Millikan Oil Drop Experiment was a landmark experiment in physics because it was the first time the charge of the electron was experimentally found. The experiment consists of measuring the time it takes for one drop of oil to fall some distance under the influence of gravity and then the time it takes to rise when subjected to an electric potential. With that data and measurements of the environment around the experiment (temperature, pressure, viscosity) , the charge of the electron can be found. We took the data and measured the charge of the electron to be [math]e=1.56(5)\times10^{-19} C[/math], which differs slightly from the accepted value of [math]1.602\times10^{-19}C[/math]^{1}, possibly due to inexact environmental considerations and delays in the recording equipment.

## Introduction

The electron sits high in the world of particles and is fundamental to understanding the physical universe. Understanding the charge gives an explanation of how particles interact and the structure of matter. Determining the charge allows accurate calculations of fundamental interactions between matter and electro-magnetic fields. Determining the charge can be done by allowing droplets of oil to fall through an electric potential and measuring how much the potential moves the droplet.^{2} This is not the only method of determining the charge of the electron^{3}This ought give an integer value as the measured charges change. There is has been strong evidence for the existence of fractional charges as well^{4}, but the fundamental charge is what we are interested in here. Even though the currently accepted value was not determined this way, when Millikan first performed this experiment he brought the world closer than anyone before.
It has been said that Millikan's mistake was the values he used for air viscosity^{5}. I hope that using more refined values for the viscosity and the air pressure I'll find the fundamental charge with a good degree of accuracy. I plan on doing this by using a computer applet to time the drops as they fall, and then record the times as they rise in the presence of an electric field. Those measurements, along with several determined values will allow the fundamental charge to be determined.

## Method

To measure the charge of the electron the influence on the electron by a known electric potential was used. We measured the times that it took for the droplet of oil to fall a known distance ([math]l=1.0\times10^{-3}m[/math]) solely under the influence of gravity. We then applied the electric potential ([math]V=500 volts[/math]) and measured the time it took for the droplet to rise over that same distance. Doing this multiple times generated the data necessary to compute the charge of the electron.

### Equipment

The experiment centered around the use of the Millikan Oil Drop Apparatus (Pasco AP-8210). See fig 1 or look at the [manual]). There was also:

- TEL-Atomic 50V-500V power source: connected to the Millikan Apparatus via BNC cables.
- Fluke III True RMS multimeter #1: connected to the Millikan Apparatus via BNC cables to show what voltage was being applied to the plates that when a acurrent was applied, would cause the electron to rise
- Fluke III True RMS multimeter #2: : connected to the Millikan Apparatus via BNC cables indicated the voltage through the thermistor
- stopwatch or stopwatch applet: for recording the rise and fall times
- atomizer: for spraying the oil in a fine mist into the oil chamber
- mineral oil: the oil that would be divided into the actual drops that were being watched

### Setup

The apparatus needs to be on a level surface to remove as much uneven gravitational influence as possible. The apparatus was placed atop several textbooks, to provide a more suitable viewing height, and was leveled using the adjustable screw legs and a bubble level on the device. The power supply was connected to the ports labeled "Plate voltage" on the apparatus and multimeter was plugged into the power supply to allow the voltage through the plates to be known. The light source was plugged into the wall, and it had no switch; the light was on only if the cable was plugged in. BNC cables connected the port labeled "Chamber Temperature" to a multimeter that displayed the voltage through the thermistor. A table on the apparatus gave corresponding temperatures to voltages. The chamber could be disassembled by removing the outer housing, the droplet hole cover, the capacitor plates, and the spacer within it. The plates were cleaned and reassembled. The viewing reticle had to be calibrated and to do required the chamber be opened up and and the viewing pin needed to be inserted into the hole where the droplet hole cover was. With the light on, someone looked through the reticle and focus the reticle, using the focusing knobs, until the pin was as clearly visible as possible. Once this was done, the pin was removed and droplet hole cover was replaced, and the chamber was reassembled.

This picture show the main components of the AP8210 Oil Drop Apparatus.

## Procedure

The first data to take was recording the thickness of the spacer. This was done with a micrometer and the attained value was [math]d=7.59\times 10^-3 m[/math]. That value has use later on. The next step was to reassemble the chamber setup for data taking. Allow the power supply, light source, and thermistor all to warm up for a few minutes before beginning. The atomizer was filled with mineral oil of density [math]\rho=8.86\times 10^2 \frac{kg}{m^3}[/math] (given by the manufacturer on the bottle). The ionization control was set to **Spray Droplet** position and the initial temperature of the thermistor was recorded. All overhead lighting was killed and one person looked through the viewing reticle. Before the oil was introduced all that was visible was the grid of the viewing reticle against a dim yellow-orange background from the light source. With a person looking, the other person put the atomizer so its tip was in the hole in the top center of the out housing. The person with the atomizer squeezer the bulb very quickly one time and allowed it to refill with air. Then another, much slower squeeze on the atomizer bulb, not allowing it to refill with air while it was still in the hole on the housing. During the second squeeze, the person viewing should see tiny, golden droplets fill the viewing area beyond the grid. This technique requires a degree of finesse and may take a few tries! During all of this the charging plate should be in the **Plates Grounded** position
Once the droplets were all visible, the viewer would change the ionization source to **Off**. The viewer selects a droplet that is falling slowly, and is located so that the grid does not interfere with the visibility of the drop. The viewer starts the timer as the drop passes one grid line and will stop timing when it passes the next lower one. Following the same drop, the viewer applies a voltage from the charging plates, and records the time it takes for the droplet to rise. The ionization source can be turned to the **On** position and the thorium-232 source within the chamber will bombard the droplet with alpha particles, charging the charge on the droplet. This will causes the droplet to rise much quicker when a voltage is applied. It is very important that **the same drop be followed**. Collecting data for various droplets doing various things produces inconsistent and useless data.
For the final calculations we need the air pressure. Unfortunately, published air pressures are always adjusted to sea level, whereas we are about 5300 feet above sea level. Making the conversion equation as:
[math]\frac{P_1}{P_2}=\frac{P_x}{P_3}[/math]
With:
[math]P_1[/math]=standard pressure at 5300 ft=[math]833.756 hPA[/math]
[math]P_2[/math]=standard pressure at sea level=[math]1013.56 hPA[/math]
[math]P_3[/math]=sea level pressure in Albuquerque on 12/5=[math]1014.8 hPa[/math]
[math]P_3[/math]=actual air pressure in Albuquerque on 12/5
Finding P_1 took a trick. I used the standard air pressure at 5000 feet, [math]843.08 hPA[/math], and the standard air pressure at 6000 feet, [math]812 hPA[/math]. I added them and multiplied by .3. I then subtracted that total value from the pressure at 5000 feet.
After everything I got the number [math]p=834.78 hPA[/math]

## Data

A total of 8 droplets were followed, we determined it only relevant to use the data from 2 of the droplets. This was not reckless abandonment of data though, There was a strong systematic tendency for the drops to drift behind the coordinate grid and they became very difficult to see. So difficult in fact, that it was determined that any measurements recorded whilst the droplet was obscured from sight was far too unreliable to be of any worthwhile contribution. All of the discarded data is listed in Appendix A, and won't be used in our calculations. Below is the listed useful data. The data is structured as follows:

- [math]V[/math]: The potential that was applied to create the repulsive electric field, volts ([math]V[/math])
- [math]T[/math]: The temperature within the viewing chamber, in degrees Celsius (°C)
- [math]t_f[/math] (s): The fall time of the droplet, in seconds (s)
- [math]t_r[/math] (s): The rise time of the droplet, in seconds (s)

** Here are a few data points, the rest are contained in Appendix B. **

#### Droplet 1

[math]V=500.2 +/- 1 [/math]V

[math]T=22.6[/math]°C

Fall time (s) | 11.5 | 12.1 | 12.8 | 12.1 | 12.25 |
---|---|---|---|---|---|

Rise time (s) | 2.7 | 2.8 | 2.9 | 2.8 | 2.9 |

#### Droplet 2

[math]V=500 +/- 1[/math]V

[math]T=25 +/-.1[/math]°C

Fall Time (s) | 9 | 9.8 | 9.7 | 9.8 | 9.7 |
---|---|---|---|---|---|

Rise Time (s) | 2.3 | 2.6 | 2.1 | 2.2 | 2.2 |

To test whether or not more charge could be accumulated the drops were ionized and data was taken afterwards. That data is in Appendix B, and the distribution of the how many times the droplets fell in given time brackets.

## Calculations and Results

The target value to be determined is the charge of the electron, [math]q[/math]. Here are the necessary equations and values used in doing the calculations for [math]q[/math]:

### Values

- [math]\rho=8.86\times 10^2 \frac{kg}{m^3}[/math] (density of the mineral oil)
- [math]g=9.8 \frac{m}{s^2}[/math] (gravitational acceleration)
- [math]p=8.3478\times10^{-4} PA[/math] (air pressure)
- [math]l=1.0\times10^{-3} m[/math] (length droplet will be measured over)
- [math]b=8.20\times10^{-3} Pa\cdot m[/math] (proportion constant)

#### Determined

- [math]T[/math] (temperature from thermistor in °C)
- [math]V[/math] (Voltage between plates in viewing chamber in volts)
- [math]t_f[/math] (fall time in seconds)
- [math]t_r[/math] (rise time in seconds)
- [math]d=7.59\times 10^{-3} m[/math] (width of the plastic spacer)

### Equations

- [math]\eta[/math] (viscosity of air as a function of T found in a table in Pa*s)
- [math]v_f=\frac{l}{t_f}[/math] (average velocity of oil droplet falling in no field in m/s)
- [math]v_r=\frac{l}{t_r}[/math] (average velocity of oil droplet rising in a field in m/s)
- [math]a=\sqrt{\left(\frac{b}{2p}\right)^2+\frac{9\eta v_f}{2g\rho}}-\frac{b}{2p}[/math] (radius of droplet in meters)

Which all contribute to: [math]q=\frac{4}{3}\pi\rho g d\frac{a^3}{V}\frac{\left(v_r+v_f\right)}{v_f}[/math] (charge on oil droplet in Coulombs)

Droplets are listed by number. An "A" signifies 1 second of ionization, a "B" signifies 2 seconds of ionization, and "C" signifies 3 seconds of ionization.

Droplet/Charge | [math]\eta\left(\times10^{-5} Pa\cdot s\right)[/math] | [math]v_f\left(\times10^{-5} \frac{m}{s}\right)[/math] | [math]v_r\left(\times10^{-4} \frac{m}{s}\right)[/math] | [math]a\left(\times10^{-7} m\right)[/math] | [math]\left| q\right|\left(\times10^{-19} C\right)[/math] |
---|---|---|---|---|---|

1 | 1.83 | 4.179(1) | 1.78(8) | 5.83 | 5.76 |

1A | 1.83 | 4.150(4) | 1.25(2) | 5.8 | 4.33 |

2 | 1.85 | 5.030(5) | 2.19(9) | 6.44 | 7.89 |

2B | 1.85 | 5.040(1) | 1.60(5) | 6.44 | 6.16 |

2C | 1.85 | 4.776(1) | 2.00(5) | 6.26 | 7.02 |

Where the uncertainty is in a value is in parenthesis.

From here I found the averages of the similar suspected Q values, and plotted those against the suspected multiples of Q. There would be a relationship of [math]Q=n*e[/math], with n being the suspected multiple and [math]e[/math] being the fundamental charge. Once I plotted this and found the uncertainty in the fitting, the linear slope was:

[math]e=1.56(5)\times 10^{-19} C[/math]

With a standard error of about 5.3%

All of these calculations can be found in the attached Excel file.

### Conclusion

This error is most likely from the lack of control in the environment. The exact air pressure in the room was not known, and the EXACT viscosity of the air was not known. There were also possible temperature fluctuations that weren't accounted for. The most reasonable was the timing. Regardless of how well someone moves, there is a delay between when someone sees an action and can hit a button to record it. This delay would influence the times recorded, which make a world of difference experiment.

Yet a value of [math]e=1.56(5)\times 10^{-19} C[/math] with a standard error of about 5.3% is quite reasonable, and I am confident that under the right conditions that 5% could be done away with. The value is actually quite close to the value Millikan himself got^{5}! Unfortunately, for now a 5% error is not enough for any reasonable person to bank on. A 5% error in current density might kill a man, and no one would want that. This experiment can be done better in the future, and I am sure that as time goes on a great many aspiring physicists will narrow this error down until it exists no longer.

### References

1. [1]

3. [3]

5. [4] [PASCO Scientific Model AP-8210] <reference />

### Acknowledgements

I'd like to thank the following:

- Bradley Knockel, my lab for the last 2 years and expert theorist who helped me sort out everything I needed to know from everything I already knew. Respectful and brilliant, he is not just a colleague but a good friends as well.
- Dr. Steven Koch, thank you for the help with the experiment and comments on my report. Thank you for putting our work into context and applying meaning to it on a weekly basis. Without you everything would be a whole lot worse.
- Kyle Martin, thank you for the great idea on data analysis.
- UNM, for providing us with all of the equipment.

## Appendices

### Appendix A

Old data that was not taken with enough information at the time, but nonetheless of some interest:

#### Droplet 1

- [math]V[/math]=+503V
- [math]T[/math]=23°C

[math]t_f[/math] | 47 | 49 | 51.3 | 45.5 | 43.9 |
---|---|---|---|---|---|

[math]t_r[/math] | 4.5 | 4.6 | 4.8 | 4.9 | 4.8 |

#### Droplet 2

- [math]V[/math]=-503V
- [math]T[/math]=26°C

[math]t_f[/math] | 59.2 | 60.1 | 69.9 | 62.6 |
---|---|---|---|---|

[math]t_r[/math] | 9.6 | 9.3 | 9.3 | 9.1 |

#### Droplet 3

- [math]V[/math]=-504V
- [math]T[/math]=27°C

[math]t_f[/math] | 42.3 | 47.2 | 50.8 | 47.1 |
---|---|---|---|---|

[math]t_r[/math] | 12.1 | 12.1 | 12.9 | 13.5 |

#### Droplet 4

- [math]V[/math]=+505V
- [math]T[/math]=27°C

[math]t_f[/math] | 57.5 | 63.5 | 63.0 |
---|---|---|---|

[math]t_r[/math] | 10.0 | 9.7 | 9.1 |