Physics307L:People/Knockel/formal

= Millikan's oil drop experiment (charge of the electron) =

Author: Bradley Knockel

Experimentalists: Nikolai Joseph and Bradley Knockel

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

Date of experiment: September 19, 2007

Abstract
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
I want to measure the charge of an electron by 1) measuring the charge on a bunch of oil droplets and 2) seeing if I can find that my calculated charges are integer multiples of some fundamental charge. 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 fundamental charge is $$e$$=1.60x10-19 C. The charge of the electron is $$-e$$.

Millikan was the first to attempt this calculation, and he used oil droplets, just like us. 

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.

Setup


Our main piece of equipment was the Millikan device (Model AP-8210 by PASCO scientific), which includes scope, viewing chamber, light, level, plate charging switch, focusing wire, thermistor, etc.

To setup this experiment, we plugged in a high-voltage direct-current power source (turned off) to wall and then to the Millikan device using banana plug patch cords. We used these 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).

We also made sure we had mineral oil and an atomizer on hand to spray droplets into the viewing chamber of the Millikan device. We also needed to find a stopwatch and micrometer.

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 moves slowly in a field (we want drops that have little charge). Perhaps, in hindsight, we should have selected some drops that were moving a little bit faster 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.

We measured the speed at which it falls, $$v_f$$. Having a partner to hold the stopwatch and write data while the other person watches the droplet is very helpful. We then created an electric field that caused the droplet to rise and measured the speed, $$v_r$$. We took many measurements of both of these speeds 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. 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. 

List of Values Needed
Since there are so many values required to do the calculations, I feel that listing before providing the results will prevent confusion when reading the results.

Known (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)
 * $$p=8.5\times10^4 Pa$$ (air pressure in Albuquerque)
 * $$b=8.20\times10^{-3} Pa\cdot m$$ (some stupid constant)
 * $$l=1.0\times10^{-3} m$$ (length droplet will be timed over)

To be found when taking data:

 * $$T$$ (temperature from thermistor in °C)
 * $$V$$ (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)

To be calculated later:

 * $$\eta\,$$ (viscosity of air as a function of T found in a table in 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)

Comment on derivation of radius (a):
Using Stokes equation and Newton's 2nd law for a falling droplet in no field, one gets:

$$mg=9\pi\eta_{eff}a v_f\,$$,

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

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

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

Comment on derivation of charge (q):
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 get a little tricky 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.

For a person who enjoys doing things differently, one can 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 the experimentalist cannot get a power supply powerful enough to actually have the droplet of smallest mass and charge you can find rise, this is another instance where $$v_r$$ would need to be negative.

Results
My initial observations are recorded in "Data," and the subsequent calculations are in "Calculations."

Data
In all the following measurements, I use the number of significant figures that I recorded when doing the experiment.

Droplet 1, Charge A: 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
 * $$T$$=23°C

Droplet 2, Charge A:
 * $$V$$=-503V
 * $$T$$=26°C

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 lazy.
 * $$V$$=-504V
 * $$T$$=26°C

Droplet 3, Charge A:
 * $$V$$=-504V
 * $$T$$=27°C

Droplet 6, Charge A: Droplets 3B, 4 and 5 acquired either one or two data points before going out of focus and becoming lost.
 * $$V$$=+505V
 * $$T$$=27°C

Calculations
For the velocity values, the number in parenthesis is the uncertainty due to random error of the last digit(s). 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.



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.

I notice that we always chose the droplet with the smallest charge. Taking into account there being three of five droplets all having the same and low charge and that all five of the droplets are multiples of this charge, I conclude that the three droplets with smallest charge have one unit of a fundamental charge called $$e$$. To calculate $$e$$, I will set the sum of the charges equal to the sum of the suspected multiples of $$e$$.

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

I now can solve for $$e$$.

$$e=1.90\times10^{-19} C$$

Let's see how good of an experimentalist I am by comparing my $$e$$ with the actual value of 1.60x10-19 C...

$$Relative\ Error=\frac{\left|1.60\times10^{-19}C-e\right|}{\left|1.60\times10^{-19}C\right|}=0.19=19%$$

The average uncertainty due to random error in my charge calculations is not enough to explain this error. The $$e$$ 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, $$\eta$$. 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
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 $$e$$ was too large, but there was a good bit of systematic error to explain this!

General Koch comments

 * 1) See my email for some more comments
 * 2) Overall, the writing style needs to be changed to be more appropriate for formal report.
 * 3) Some specific things that need to be fixed (not complete yet):
 * 4) You need an estimate of your random uncertainty in your final best estimate for fundamental charge!!!
 * 5) Expanded introduction (see above comment)
 * 6) Expanded conclusions (see above comment)
 * 7) Diagram / photo for methods (see above comment)
 * 8) 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.
 * 9) Calibration of microscope grid?  Calibration of other instruments...how do you know they are working?
 * 10) 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.
 * 11) Other?