The Millikan Oil-Drop Experiment

In this experiment, tiny charged oil drops drifting slowly through the air are viewed through a microscope. Their rate of fall is used to calculate, via Stokes law, the size of the drop. Then the drift velocity with an applied field is used to deduce the charge on the droplet. The drops are so tiny that the effect of a single electronic charge on a drop can be observed.

I. References

*Tipler, Paul, *Foundations of Modern Physics*, 2nd edition, Worth, New York, 197 Chapter 3. . Tipler includes a table reproducing Millikan’s original notebook.*

*Taylor, John, and Christopher Zafiratos,* Modern Physics for Scientists and Engineers*, Prentice Hall 1991, Section 4 and especially Problem 4.23, pg. 105.*

*Serway, Moses, and Moyer, *Modern Physics,* Saunders, San Francisco, 1989, pp. 83-87. *

II. Theory

The oil drops used in this experiment fall through air under the influence of gravity, an electric force, and air resistance. By measuring the drop's terminal velocity with and without an electric field, the drop radius and charge can be measured. The velocity is determined by measuring the time of free-fall.

v_{0} = L/T_{g}. (1)

First consider the forces acting on the drop when there is no electric field. Since the drop is not accelerating, the forces can be set equal to zero:

6p h av_{0}=mg (2)

Using

m=4/3p a³r **(3)**

gives

6p h av_{0}=4/3p a³r g **(4)**

Solving for the radius gives

a=[(9h v_{0})/2r g]^{½} (5)

Here h is the viscosity of air, and r is the density of the oil.

The correct density to use, however, is really the difference between the density of oil and the density of air:

r = r _{0 }–s

the density of air, s **, **depends on the barometric pressure and the temperature, and should be looked up in the Handbook of Chemistry and Physics. You will find a table and a formula. I suggest you also program the formula into the spreadsheet so that you can observe the sensitivity (or lack thereof) of your results upon the variation of T and P.

The correct coefficient of viscosity is, in fact, not a constant for drops as small as we use. It should be given by

h =h _{0} (1+b/aP)^{–1}** (6)**

where **a** is the drop radius in meters, **P** is the air pressure in centimeters of Hg (76 cm is a good enough approximation in this formula for our purposes), and b = 6.17 x 10^{-6} m-[cm Hg]. The value of h _{0} at room temperature is 1.83 x 10^{-5} in mks units (or 1.83 x 10^{-4} in cgs units).

The corrected value for h 5 is a function of drop size, and is best determined as follows: first calculate the drop size using the constant value h _{0} for the coefficient of viscosity. Then calculate h, and recalculate a corrected value for the drop size. You will want to make columns for this calculation.

In an electric field resulting in an upwards force, one has

6p h av_{down} = mg+qE_{down} **(7)**

(Since the charge is negative, a downwards electric field results in an upwards force).

By adjusting the electric field, one can hold v_{down} = 0, so q=mg/E, and m is found from the fall time in the absence of an electric field.

Procedures

Experimentally, the greatest difficulty in obtaining reasonable results in this experiment is the lack of patience found in today’s students (just kidding!). Actually, it has to do with the difficulty of the timing measurements, especially in terms of getting droplets of the right size and charge so that they will make transits in a time long enough to allow for some precision of measurement. Our procedure in the past has been to measure times up and down in a field, then use the analysis below to find q. The alternative we are now is to measure the charge on the droplet by balancing the force of gravity with the force of the electric field applied to the droplet charge.

From Eqn. 2, we see that the droplet weight can be determined from measuring the free-fall terminal velocity. Then Eqn. 7 tells us that at zero velocity, the electric force is exactly equal to the droplet weight. So all we do is measure the electric field needed to levitate the droplet.

III. Procedures, Also, be sure to read the Appendix on adjusting the optics.

- Level the apparatus and set up the illumination system to focus on the region where the drops will fall.
- Set up the optics and calibrate the system as follows:

- The tele-microscope must be calibrated sometime during the experiment. This involves observing the accurately ruled scale on the glass slide provided, and figuring out what distance in space each division on the graticule of the tele-microscope corresponds to. The distance between the plates must also be measured with a caliper, or, preferably, a micrometer. This distance is used to determine the electric field.
- Wire up the high voltage (about 400 volts). Try to arrange the wires so as to avoid accidental shock
**!** - Spray in some drops and observe qualitatively the effect of the field on them. Choose a drop which you like, and try to get it alone in the chamber, as follows: Run your drop up and down in the chamber repeatedly, by switching the sign of the field back and forth. Soon most of the other drops will be gone. Many of the ones that remain will have no charge. Eventually you should be able to eliminate all but your chosen drop.

While you are practicing maneuvering drops, you should also try steering the light source and moving the telescope from side to side, without losing the drop.

Some times while you are moving this drop around, its charge may become zero. Then you can't prevent it from dropping to the bottom of the chamber. Often, however, you can save the drop by leaving the field off for several seconds to permit the drop to change its charge. You may want to rotate the radioactive source into its active position while the drop is in free fall. Note that most charge changes take place when the field is off, as any ions produced in the chamber when the field is on will be rapidly swept to the plates. Even with the field off, however, the sources are weak enough that the likelihood of changing the charge is not large. - While you are taking measurements, you will have to keep the field on all the time, either up or down, to avoid having your measurements ruined by charge changes. To do this you have to toggle quickly from one field polarity to the other, with as short a field-off interval as possible. (This is not to say you never want the charge to change - see below.)
- ** Now choose a drop for charge measurements. An ideal drop takes about 10 to 12 seconds to fall one grid distance. Do not use drops having fall times less than 6 seconds or so per grid: these are too large, and will have too many charges to see the quantized charge. I suggest measuring free-fall times for one grid distance, and drift times with the electric field on for four grid distances. When you have a drop you like, make three or four measurements of t
_{0}, the free-fall time. This time will be used to determine the size of the drop. - ** Now make measurements of T
_{up}and T_{down}, the times to go up four grids and to go down four grids, respectively, with the field on. Note that the references to up and down mean up and down in real space. You will see the drop's motion inverted in the telescope; get the habit of translating this back to real-space directions. Note that it is**important**to obtain repeated measurements of T_{up}and T_{down}with the**same value**of the charge. To do this you must reverse the field instantly at top and bottom of the chamber. If the drop drifts without field for even an instant, it is likely to change its charge.

Take as many measurements as possible with a single drop. Try to get three or four (t_{up}, t_{down}) pairs for a single value of the charge. The times should reproduce rather accurately. Then, if the charge doesn't change spontaneously, make it change by letting the drop drift without field; you can profit from the occasion to take another value of t_{0}, to make sure that you still have the same drop. At this point, you can try to change the charge with the ionizing source – but don’t expect a dramatic effect. Continue for as many charge changes as possible. - **To analyze your data, use a spreadsheet, such as EXCEL. You will want to make columns and calculations for some or all of the following: T
_{down}, T_{up}, T_{down}’, T_{up}’, (T’ ‘s are the times recorded when the charge on the droplet has changed), T_{g}(the time in free fall), a_{0}, a_{0, corrected}, (1/T_{down }+1/T_{up}), q, n, [(1/T_{down }+1/T_{up}) – ( 1/T’_{down }+1/T’_{up})], 1/n(1/T_{down }+1/T_{up}), 1/n(1/T_{down }+1/T_{g}), and e. Part of the reason for so many columns is to allow different approaches, and to bring home the point about how easy it is to set different calculated columns on the spread-sheet.

New technique:

- ** Now choose a drop for charge measurements. An ideal drop takes about 10 to 12 seconds to fall one grid distance. Do not use drops having fall times less than 6 seconds or so per grid: these are too large, and will have too many charges to see the quantized charge. When you have a drop you like, make three or four measurements of t
_{0}, the free-fall time. After a period of balancing the field, return to take additional measurements of the fall time. This time will be used to determine the size of the drop. - ** Now balance the drop by adjusting the electric field. Take a half-dozen or so readings of the field needed to balanced, every 10 seconds or so. Be sure to think about the determining sources of error, and adjust your procedures accordingly. As indicated in
**7.**make additional measurements of the free-fall time. After several such cycles, if the charge has not changed spontaneously (you will see the difference in the balancing field value), insert the radioactive source during free-fall - but don’t expect a dramatic effect..

Take as many measurements as possible with a single drop. Continue for as many charge changes as possible. - Analyze your data, using a spreadsheet. You may have a different spreadsheet on a computer accessible to you, but at least initially, I want you to put your data into EXCEL on the machine in The 231. Save it onto your on disk, and into the EXCEL directory labeled mil_data. We’ll combine the data from the class to come up with a larger data set for you to work with, in addition to your own.

**Using EXCEL, you will want to make columns and calculations for some or all of the following: T_{down}, T_{up}, T_{down}’, T_{up}’, (T’ ‘s are the times recorded when the charge on the droplet has changed), T_{g} (the time in free fall), a_{0}, a_{0, corrected}, (1/T_{down }+1/T_{up}), q, n, [(1/T_{down }+1/T_{up}) – ( 1/T’_{down }+1/T’_{up})], 1/n(1/T_{down }+1/T_{up}), 1/n(1/T_{down }+1/T_{g}), the applied voltage, the value of **E **(the electric field), q (=ne), n, and e. Some of the columns, of course, refer to the old technique, others refer to the new, and some will be shared. Part of the reason for so many columns is to allow different approaches, and to bring home the point about how easy it is to set different calculated columns on the spread-sheet.

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Following is how to get started for those unfamiliar with spreadsheets:

- Start up one of the PC's and enter

win

This will put you into Windows - Click on the EXCEL icon to enter EXCEL. It will come up with a blank spreadsheet. Do<alt-File,Save> and enter your name and a 1; E.g., smith1, but you have to keep it to less than 8 characters. The 1 at the end is to keep track of what version or draft you are working on.

Here are a few EXCEL commands which may help you (I’ve enclosed (<>) the key to be struck.):

<Alt> or </> gets the menu of commands at the top of the screen

<F1> help

<esc> gets you out of something

<F2 edit the cell where you are

<.> anchors a point, in moving a block of data

<..> indicates a range of reference values which are omitted between the first and last values of a series.

<F4> toggles "absolute referencing"

<=> begins a formula in a cell (e.g.,, <= B2+B3> adds the contents of B2 and B3, and inserts the result in the current cell. In LOTUS or QUATTRO, the same operation is written as <+B2+B3>. Also, in LOTUS or QUATTRO library functions are prefaced with @, thus you might have <@avg(B2..B10)>and the corresponding EXCEL: <= avg(B2..B10)>, to calculate the average value of the column B2, B3,..., B10.)

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- Label and enter the values of the constants you will be using: viscosity and density of the oil, atmospheric pressure, g, spacing of the plates, spacing of the grids, anything else I may have forgotten. Be sure to use absolute referencing in the body of your spreadsheet, so that you can then modify the values of the constants to see the sensitivity of the results to error in these values.
- Enter and label columns indicated above (under 9.), including at least your free-fall time, [times up and down, the sum of the inverse of the up and down times,]** calculated value of the radius, the potential difference and the electric field, the value of q, the estimated value n, and calculated value of the charge.
- Now put in your own numbers for the free-fall times, for one single drop -

Note: to edit the contents of a cell, move to it - you will see the contents in the title bar –then push F2 to edit it.. To get rid of stuff you don't want, you can simply highlight and "delete" (press the delete key). You can cut and paste (or copy and paste) (from the edit menu) to move items around to different locations.. - Set up a column to calculate the value of a
_{0}. Calculate the value from the value of 1/T_{g}. And by combining the appropriate constants, and also calculate the corrected value of h , and corrected values for the radius. - Set up columns for the potential difference and the electric field. Set up your column for q, n, and e. (You will be determining n from a display of your data for q.) Enter your data! You will be able to guess the right order of magnitude for n, since e should be near 10
^{-19}coulombs.

When you have a lot of charge values (next week?) here's one way to make a histogram:

Make a set of lower bin edges in a series of squares. For example, put -<1.0E-18> in cell B52, put <+$B52+.1E-18> in cell B53, and copy B53 to B54...B72. This makes a histogram of 20 bins with values from 1.0E-18 to 21.0 E-18.. Then use the Frequency function to fill the bins. Then use Graph to graph them (as a bar graph). You should see peaks at the quantized charge values.

**NOTE:** when you save this spreadsheet, save it under a different name! I suggest that you save it to drive a:, on a floppy. __YOU NEED TO BRING ONE.__

- After thinking over the whole procedure and making any changes which suggest themselves, go back and take a final, more complete set of data.
- Calculate the charge for every measurement of voltage and freefall time [or for every value of t
_{up}or t_{down}]that you have recorded. Group together successive charge values all corresponding to the same value, and average them. Make a histogram of these values, using a finely divided scale on the x-axis so that you can see the spread of values around each peak. You should see accumulations of points near a series of peaks. How do you interpret these peaks? What do they tell you about quantization of charge? - An elegant way to calculate your "best" value for
**e**is to do a straight-line fit to all of your data. If the x-axis is**n**, the number of charges on the drop, and the y-axis is**q**, the slope is your value for**e**. Finding the slope is readily accomplished using the spreadsheet program, and it will give you both the plot and as well as calculating the best linear fit, and give you the error coefficients. Be sure you understand the errors you are quoting. The spreadsheet gives more error coefficients than you want. - At this point you will want to review your determination of uncertainty. As I suggested above, you will want to determine the sensitivity of your results to variation of your measured parameters. See if you can determine your uncertainty by observing the sensitivity of your fitted results to your measured and given parameters.
- Give your final result for the charge on the electron, and its error. This is definitely a time where you should separate statistical and systematic errors, and to discuss each.
- Finally, discuss the interpretation of these results. Take the point of view of someone living in 1900, when very little was known about the charges on elementary particles.

IV. Equipment

- Hoag-Millikan Oil-Drop Apparatus (Sargent-Welch Cat. #0620B)
- power supply, with 400 to 500 VDC and a 6V filament supply; Pasco Model SF-9585 or Heath Model IP-32 or Teltron 801 are good.
- digital volt meter (METEX 3800 is good)
- atomizer
- mineral oil (Locke watch oil 1407, r = 0.8577 at 25°C); silicone oil is
**not**good. - ruled grating for calibration of microscope
- timer (the Cronus electronic timer is good)
- caliper, for measuring the plate separation

Details:

- There is a 5M resistor protecting the plates from the HV source.
- in the shorted position, there is no connection.
- replacement light bulbs: best is #44, 6.3V, .25 A, bayonet base; #47, .15A, would * probably do too.
- Common sources of problems include:

Leveling (the drop disappears from view after a few trips

Vibrations - cause the drop to be displaced

Parallax errors, due to not being focused in the plane of the calibration (Don’t refocus after calibration.)

Note: the following is the traditional way the experiment has been done. As a result of the real experimental difficulties, this year we will try an alternative approach as well. See below.

With a downwards electric field, the sum of the forces yields

6p h av_{down} = mg+qE_{down} **(7)**

or

q = ne = (6p h av_{down}–mg)/E_{down} **(8)**

With the electric field upwards, we get

q = ne = (6p h av_{up} + mg)/E_{up} **(9)**

We can combine this two, noting that **a** is determined by the free–fall time, and **v _{up}** and

q = ne = 6p h a/E_{down}(v_{down}+v_{up})**
**or

where

When the charge changes we have another expression that can help in the determination of n: From (7) we have

** **q’ = n’e = (6p h av’_{down}–mg)/E_{down}** (11)**

so

** **(n’-n)e = q’– q = 6p h a/E_{down }L(1/T’_{down}-1/T_{down})** (12)**

(Or one could do the same with rise times, or differences of up + down times before and after the charge change.) in any case, following the changes in charge often gives confirmation of the values of n, the number of charges on the droplet. Again the constant term out front only varies with drop size, so for a single drop, it will usually remain constant.