Physics307L:People/Le/Formal

The Search for Charge, Milikan Oil Drop Experiment
Linh Le, Cary Dougherty

Physics and Astronomy Department, Junior Lab 307, University of New Mexico, Albuquerque, New Mexico

Email: linhle@unm.edu



Abstract
The fundamental charge of an electron has stumped many scientists for many years. To find the charge of an electron, we charged oil droplets with excess electrons, then suspended them within an electric field. By measuring their rate of fall, we could find their mass, and by measuring the rate of rising in the field, we can measure the amount of charge stored within the oil droplets. We measured the charge to be about $$ 1.5x10^{-19}$$ Coulombs, very close to the accepeted value of $$1.60217733x10^{-19}$$[1]. 

Introduction
The concept of charge has been around long before a number was put to it. Finding a numerical number for the fundamental charge has eluded many scientists for many years (JJ Thomson found a number, but Milikan's apporach was much better) [1]. It was also uncertain if these charges came in discrete intervals (ie. 1q, 2q, 3q..., not 1.23q). This experiment answered both those questions.

The idea that Milikan came up with was to charge oil droplets, suspend them in an electric field, and balance the weight of the droplets and the force of the charges keeping the droplets up to find a value of the charge.[1]

Methods and Materials
We used "Millikan Oil Drop Apparatus" by PASCO Scientific Model AP-8210. The manual could be of use and can be found | here

Here is the basic design of the apparatus: It is a parralel plate capacitor that allows oil droplets to enter, attactched to an alpha particle generator, to force the oil droplets to become ionized. The capacitor itself can have it's plate polarities to be switched and has a thermistor hooked up to it, allowing the temperature inside to be determined. Mounted to the right of the plates is a light source, and in the front is a microscope for you to view and measure the droplets from.

The apparatus is almost built for "plug and play" and did not require any extensive setup. We attached a power supply to the capacitor to charge the plates and the ionizer. We attached a multimeter to the supply so we could accurately measure the voltage accross the plate. We also placed a second multimeter to the thermistor to allow us to measure its resistance, and from the, the temperature inside the capacitor. We leveled the whole apparatus, focused the light source and the microscope with a needle inside the capacitor, and filled a ionizer spray bottle with mineral oil.

Procedure
After calibrating the microscope, we cover the capacitor with its cylindrical housing. AFfter dimming the lights and plugging all the power supplies in, we place the nozzle of the oil filled sprayer into a small hole in the top of the housing. There is a lever to the left of the capacitor that we turned to open, to allow the oil to enter. One quick spray and a few slower ones, forces the oil into the capacitor. While looking through the microscope, you will see a cluster of "sparkles" come into view. When a good amount comes into view, we switch the lever to close.

With oil inside the apparatus, we now turn the lever to on. This starts the bombardment of the oil droplets by alpha particles. After a few seconds, reset the lever to close. Using the switch to change the polarity of the plates, we are able to move the droplets that were charged up and down. We tried to pick out one that rose and fell slowly. There is a grid in the eyepice of the microscope that gives us a scale for measurements. With our drop found, we measured how long it took for it to fall about .5mm due to gravity, and to rise the same amount due to the force of the electric field created by the plates.

After taking a few measurements, we turn the lever back to on for a few seconds, to re-bombard our droplet again. This allows us to remasure the same droplet with a new amount of charge. We now repeat the expieriment with the rising and fall times.

Results
We measured, and remeasured, the same droplet as many times as possible, gathering fall times and rise times.

Raw Data
Spacing of plates (as computed by the average of the data above): 8.10mm

Density of Squibb's Mineral Oil: 886kg/m^3

Atmospheric Pressure in Abq (as looked up on the internet): 8.33X10^4 Pa [2]

Set 1
Voltage: 501V Resistance: 1.9982 M(ohms) Distance: .5mm

In this set, we were able to change the charge on the droplet and remeasure

Voltage=501.9V Resistance=1.976M(ohms)

Set 2
Voltage: 500V Resistance: 1.923M(ohms)

We changed the charge of the oil droplet in this set as well, but the droplet moved so fast, that it was very hard to sync Cary's observations and oral commands with my data collecting. As that is the case, we only took 2 data points.

Set 3
Starting Voltage: 501.0 V Resistance: 2.07 M(ohms) Ending Voltage:501.8 V 2.041M(Ohms)

Set 4
Starting Voltage:501.4 V Resistance:1.945 M(Ohms)

Calculations
Using the fall and rise times, we were able to determine a few things about the droplets and the charge.

To find the fall and rise times, we said that $$V=\frac{d}{t}$$. We use this simple formula since the droplets reach terminal velocity very quickly and travel at a constant rate after that.



To find the charge stored in the electron:

$$q = {\frac{4}{3}}\pi \rho g[\sqrt{(\frac{b}{2p})^2 +\frac{9 \eta v_f}{2g\rho}}-\frac{b}{2p}]^3\frac{v_f + v_r}{Ev_f} $$

A nice detailed derivation of this formula can be found in the manual, linked above.

q- the charge of the electron

p-barometric pressure-$$8.33^4Pa$$

d-distance between capacitor plates

g- acceleration due to gravity- $$9.8 \frac{m}{s^2}$$

b- constant $$8.20E(-3)Pa * m$$

a- radius in drop measured in meters

$$ \rho $$-density of the oil which is $$886 \frac{kg}{m^3}$$

$$ \eta $$- viscosity of air (found by the comparing temp inside the capacitor with chart in manual appendix A)

V- potential difference across the plates in Volts

$$v_r$$- rise velocity (dividing .5mm by rise time)

$$v_f$$- falling velocity (dividing .5mm by rise time)

E electric field (found by $$\frac{V}{d})$$

After a good amount of number crunching, we came up with some final values for the charges in the oil droplets

Set 1A
Voltage: 501V

Resistance: 1.9982 M(ohms)

Temp=25C

$$\eta=1.8480\frac{Ns}{m^{2}}x10^{-5}$$

$$ v_favg=3.408E-5 \frac{m}{s}$$

$$ v_ravg=1.025E-4 \frac{m}{s}$$

E= V/d= 61851.85 V/m

Plug everything into the formula: q=3.39E-19 C

Set 1B
Voltage=501.9

Resistance=1.976M(ohms)

Temp=25.5C

$$\eta=1.8480\frac{Ns}{m^{2}}x10^{-5}$$

$$ v_favg=3.608E-5 \frac{m}{s}$$

$$ v_ravg=2.118E-4 \frac{m}{s}$$

E= V/d= 61886.56 V/m

Painstakenly plug and chug into the formula and get q= 6.38E-19C

Set 2
Voltage: 500V

Resistance: 1.923M(ohms)

Temp= 26C

$$\eta=1.8520\frac{Ns}{m^{2}}x10^{-5}$$

$$ v_favg=3.225E-5 \frac{m}{s}$$

$$ v_ravg=1.683E-4 \frac{m}{s}$$

E= V/d= 61652.28 V/m

Another round of math and I get q=4.84E-19

Set 3
Voltage: 501.0 V

Resistance: 2.07 M(ohms)

Temp: 24C

$$\eta=1.8440\frac{Ns}{m^{2}}x10^{-5}$$

$$ v_favg=2.855E-5 \frac{m}{s}$$

$$ v_ravg=1.106E-4 \frac{m}{s}$$

E= V/d= 61775.59 V/m

Insert values into the formula and get q=3.08E-19C

Set 4
Voltage:501.4 V

Resistance:1.945 M(Ohms)

Temp:26C

$$\eta=1.8520\frac{Ns}{m^{2}}x10^{-5}$$

$$ v_favg=3.243E-5 \frac{m}{s}$$

$$ v_ravg=2.304E-4 \frac{m}{s}$$

E= V/d= 61901.23 V/m

Last but not least, calculations yield q as 6.34E-19C

Final


By looking at the data, we can see that a faster rise time lead to a higher value for q. There looks to be some kind of trend, leading me to believe that charges are quantized. With that in mind, I will take the minimum value for q as my base. If we look at set 1A and set 3, their charges are about even, sets 1B and 4 are about twice as big, and set 2 is about 1.5times as big.

So now, we can take the avg of them and find some value for this "Q"

$$Q= \frac{1+1+2+2+1.5}{5} x10^{-19}$$

$$Q=1.5x10^{-19}C$$ 

Conclusion
This lab is not perfect, but I was very surprised with the results. There are systematic errors that, all added up, will affect the data.

Below, I estimated how far off errors skewed certain measurements we made:

$$ error.propigation =\sqrt{x^2 + y^2 +z^2...}$$
 * Human Error in timing: 10%
 * Barmoetric pressure off internet: 5%
 * Trying to find viscosity off a chart: 5%
 * Brownian motion of particles, skewing measurements: 5%

$$ep=\sqrt{10%^2 + 3x(5%^2)}$$

$$ep=13.2%$$

Since the data is not 100% accuate, I would surmise the answer lies within the limits set by the error propigation.

$$Q=(1.5 +/- .1984) x10^{-19} C $$

We can compare our results with that of the accepted value: $$1.602 x 10^{-19} C $$

$$%error= \frac{|Actual-Experimental|}{|Actual|}x100$$

$$%error= \frac{|1.60217646 x 10^{-19}-1.5 x 10^{-19}|}{|1.60217646 x 10^{-19}|}x100$$

$$%error=6.37$$ 

Sources of Error
There is plenty of room for error in this lab. 
 * I went on the internet to find the barometric pressure in Albuquerque, but that is probably an estimated quanity and changes with specific altitude of your area and the weather at the time
 * The viscosity of the air inside the capacitor is measured by the temperature inside the capacitor. That value is determined by measuring the resistance in a thermistor and then finding values on a chart.
 * The temperature is estimated off a chart, but when you get a value for the resistance that falls between values, you have to round
 * Once the temperature is found, you look at a graph to find the value, and as above, you have to round
 * While measuring the fall and rise times of the droplets, it is uncertain how close or far the droplets are, so adjusting the microscope might change the distance that they travel (although it is assumed the apparatus is calibrated to prevent this)
 * There is also "lag time" between measurements as one of us stared into the scope and the other was running the stopwatch
 * In the calculations, I rounded the values we found as 2 times greater than or 3 times greater than ect. instead of using an acutal ratio

Acknowledgements
I would like to thank the following people: 
 * My lab partner Cary, for helping take measurements
 * The lab professor, Dr. Koch, for getting this all working
 * Milikan, for discovering this in the first place

Koch comments
Overall, a good start, looks like some good data. However, significant additions and improvements are needed. I recommend another iteration before your final version. Here are some main points now (not all-inclusive):
 * 1) Need bigger introduction, with many citations to original research articles
 * 2) Need to make most areas more formal (more like a real scientific paper; less like an individual lab summary).  Also spell check.
 * 3) Consolidated data tables
 * 4) More (see comments above)
 * 5) Retake data now that you've thought about how to do it better!  I know you spent a whole day blazing the trails for the experiment, and I think you can take some better data during the final week.
 * 6) Check out [Jesse's informal write-up] for data analysis ideas.  Also, I notice that he has a different value for the spacer than you do, I think, which would be important!