Download 1 Determining the Charge of an Electron: The Millikan Oil Drop

Determining the Charge of an Electron: The Millikan Oil Drop Experiment
Trent H. Stein, Michele L. Stover, and David A. Dixon
Department of Chemistry, The University of Alabama, Shelby Hall, Tuscaloosa, AL, 35487-0336
Millikan’s original apparatus
Introduction:
A key elementary quantity is the actual charge of the electron. In 1897, J. J. Thomson showed
that cathode rays are what we now call electrons. He measured the charge to mass ratio of the
electron by using crossed electric and magnetic fields. In addition, he showed that the mass of
the electron was small, more than 1000 times smaller than the hydrogen atom. This means that
atoms were composed of negative charges and positive charges. Thomson envisioned that the
electrons were embedded in a “pudding” of positive charge to make an atom. For this work,
Thomson received the Nobel Prize in Physics in 1906. Thomson also invented the mass
spectrometer. E. Rutherford shot alpha particles at a gold foil and showed that only a few back
scattered. This led to the nuclear model of the atom in 1911. Rutherford had already won the
Nobel Prize in Chemistry in 1908 for his studies of the disintegration of the elements and the
chemistry of radioactive processes. At this point, a critical question was the charge of the
electron. R. Millikan, then came up with a relatively simple approach to measure the charge of
the electron while working at the University of Chicago. The experiment was initially done with
water droplets but changed to oil droplets because the water evaporated too quickly. The
experiments were done in 1909 and 1910 and Miilikan published his best work on this in 1913.
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He received the Nobel Prize in Physics in 1923 for his work on determining the charge of the
electron and the photoelectric effect. At this time, he had recently moved to help found the
California Institute of Technology.
What is the experiment?
The goal of the experiment is to use a force one knows, gravity, to balance the electrical force
which is the unknown.
When particles (polystyrene spheres) are vaporized by an atomizer (think a perfume sprayer),
some acquire extra electrons and some lose electrons due to frictional processes similar to
drawing a comb through your hair on a dry day. Some particles will acquire more charge than
others and when they are exposed to an electric field in the region between a pair of electrodes,
they experience a force proportional to their acquired charge q and to the applied electric field E.
This electrostatic force FE is given in equation (1).
FE = qE
(1)
The bold type indicates a vector quantity as the force and the electric field have a direction. The
electric field is oriented so that for a positive charge, F is in the same direction as the electric
field. If the charge (q) is negative, then the direction of F is opposite to the electric field. We can
calculate the strength of the electric field, E, if we know the potential difference (V in volts)
across the plates and the separation of the plates (d in meters). The equation to determine the
magnitude of the electric field is given in equation (2).
E = V/d
(2)
Our goal is to balance FE against any opposing forces so that the particle does not move. What
other forces are present? The obvious one is FG, the force on the particle due to gravity, which
pulls the particle down while FE is pushing the particle up for our given choice of charge and
direction of E. There is another force, FVD, the viscous damping force which is always opposite
to the direction of motion. If we choose our coordinate system (remember that these are all
vectors so the forces have a direction) so that vertically upward is positive then our force
equation for the polystyrene sphere at equilibrium becomes:
FE - FG - FVD = 0
(3)
FE = FG + FVD
(4)
and
Now let’s look at each force in detail. By combining equations (1) and (2), we get equation (5)
FE = qV/d
(5)
where our unknown charge q is given in terms of known quantities V and d if we know FE.
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The next force to consider is the force due to gravity given in equation (6).
FG = mg
(6)
where g is the acceleration of gravity (9.8 m/sec2). What is the mass of our particle, a single
polystyrene sphere? Do you think it would be easy to weigh a single sphere? Probably not very
accurately as it is very light, so we must express the mass of a single sphere in terms of a bulk
property of the material that is easier to measure. The density (ρ) of the polystyrene used is given
on the bottle label, as is the diameter of the spheres (in microns, 1 micron = 1 x 10-6 m). Given
this data, the mass of a single sphere is the density (ρ) times the volume (Vol, equation (7)).
m = ρ(Vol)
(7)
with Vol the volume of the sphere given in equation (8) (remember that the equation for Vol has
the radius r in it).
Vol = (4/3)πr3
(8)
Substituting (8) into (7) and the resulting equation into (6) gives equation (9).
FG = ρ(4/3)πr3g
(9)
The last force we need to look at is the viscous damping force. Viscous drag is experienced by a
body of any shape but the force is only readily calculable for a sphere. For this, we will need the
equation worked out in 1845 by Sir George Stokes, which is known as Stokes’ Law. Stokes’ Law
gives us the viscous damping force on a sphere moving in an ideal fluid. What is our fluid? Air!
Assuming that air is an ideal fluid does not affect our experimental error significantly (Why
not?). The Stokes equation equivalent for viscous drag on a sphere is given by equation (10).
FVD = 6πηrvT
(10)
where η is the viscosity of the fluid (for dry air at 760 mm of mercury η = 1.81x10-5 N sec/m2), r
is the radius of the sphere and vT is the terminal velocity of the particle which we will measure.
Note that vT is our vector quantity. Substituting equations (5), (9), and (10) into our force
equation (4) gives equation (11), where we now only use the magnitudes of the forces.
qV/d = 6πηrvT + ρ(4/3)πr3g
(11)
Solving for the charge, q, gives equation (12).
q = [6πηrvT + ρ(4/3)πr3g]
(d/V)
(12)
In the course of the experiment we will measure the needed unknown, the terminal velocity, vT,
of the particles and the experimental variable, the voltage across the electrode plates V. The
other quantities are known. Using equation (12), it is possible to calculate the charge of a single
electron, which also illustrates the quantized nature of charge. Why is this latter idea important?
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Experiment:
Basic idea: The basic apparatus is shown in Figure 1. A small puff of opaque polystyrene
spheres of known size distribution is delivered to the electrode assembly by the atomizer. When
the spheres enter the space between the electrodes, the incident light is scattered by the spheres
and some of this light is redirected toward the microscope. Because the spheres are on the order
of one micron in diameter, they are too small to readily measure their size with the fixed
magnification of this microscope. However, each sphere or small cluster will be easily visible as
a bright point of light that will drift slowly downward under the influence of gravity. At this
point, after a few short quick pumps of the atomizer bulb, the view through the microscope
should look like a dark sky with many stars slowly moving downward. The microscope in this
optical system is fitted with a reticle or eyepiece micrometer which can be used with or without
calibration to observe the behavior of neutral and charged spheres in the presence or absence of a
constant electric field between the electrode plates. The polarity of the electrodes can be changed
by using the three position switch on the device. The particle response will be rapid motion
toward the plate with the charge opposite that of the particle. Quick reversal of the electrode
polarity (inversion) can be used to trap individual spheres for use in repeated timing of the path
velocity for a single sphere. This allows the user to make repeated measurements on the same
sphere, thus, allowing statistically valid data to be collected for measuring the charge on the
particle.
Atomizer
Voltage display
Microscope
Light block
Main chamber
Voltage
knob
Figure 1. Millikan Apparatus
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Polarity
switch
Safety Instructions:

This apparatus uses high voltage. Do not touch any exposed metal surfaces in the
experiment.

Do not put anything metal into the hole in the left side of the main chamber when the
voltage is on (the voltage knob is not turned to its counter-clockwise limit).

Do not remove your safety glasses when performing this lab. You may lift them up when
you are looking through the microscope, but they need to remain on your face. As soon as
you are done looking through microscope, the glasses need to be back over your eyes.
Equiment:




Millikan apparatus
Microscope
Atomizer


Bottle of polystyrene spheres in
liquid suspension
Plastic machine screw
Stopwatch
Procedure A: Apparatus Set Up and Initial Results
1) Carefully set up the Millikan Apparatus (figure 1) by following steps 2-6.
2) Begin by plugging in the Millikan Apparatus into an electrical outlet. Make sure that the
voltage knob is turned off (to the counter-clockwise limit) and the polarity switch is in its
central (neutral) position before plugging it in. Do not force the voltage knob as it will
stop turning at this limit. Press the power button.
Figure 2. Main chamber
Figure 3. Atomizer
3) To assemble the microscope, pull back on the black focusing knob, retracting the rubber
roller. Slide the microscope tube into the focusing block and release the knob to trap the
tube (Figure 2).
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4) Initial focus may be achieved by inserting the plastic machine screw into the hole on the
left side of the main chamber (Figure 2). Moving the eyepiece by hand while the knob is
retracted provides coarse adjustment. Turn the engaged knob for fine focus. Once a good
focus has been achieved, remove the machine screw.
5) Unscrew the cap and fill the atomizer with a sufficient volume of the latex particles to
cover the bottom of the large vertical atomizer barrel (Figure 3). Do not exceed the max
fill level as the air mix port must not be obstructed. Replace the cap.
6) Place the atomizer into the hole in the main chamber. Insert the delivery tube into the
hole on the left side of the chamber (figure 2). The delivery tube will only go in about 1
½ cm. Do not force the delivery tube past this point. Look through the microscope and
make sure that you cannot see the tube. If you can see it, pull back on the delivery tube
until you cannot see it.
7) At this point, your set up should look like Figure 1. If it does not, ask your TA for help
before proceeding with the experiment.
8) Squeeze the atomizer bulb quickly two times. Look through the microscope and note the
spheres which are visible as tiny points of light. If necessary, rotate the microscope tube
so that the reticle is right side up. Sharpen the focus for the individual viewing. Slide the
knobs on the top and back of the light block to adjust the amount of light entering the
electrode chamber (Figure 1).
9) To observe the polystyrene sphere behavior in the applied electric field, turn the voltage
adjust knob clockwise for a ½ turn. While looking through the microscope, move the
polarity switch from the center (neutral) position to one side (charged).
10) Squeeze the atomizer bulb 2 times quickly. Turn the voltage knob clockwise as far as it
can go. While looking through the microscope, move the polarity switch to charged.
Answer Procedure A Question 1.
11) Squeeze the atomizer bulb 2 times quickly. Turn the voltage knob counter clockwise to
the beginning of the white semi-circle. While looking through the microscope, move the
polarity switch to one side (charged). Answer Procedure A Question 2.
12) Wait 5 minutes. Look through the microscope and answer Procedure A Questions 3 - 4.
13) Turn the voltage off and the polarity switch to neutral. Squeeze the atomizer bulb 2 times.
Look through microscope. Answer Procedure Step A Question 5.
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Questions: Procedure A
1) What do you see? Describe the speed and direction of the particles. Are all of the
particles moving in the same direction with the same speed? If not, explain why.
2) What do you see? What happened to the magnitude of the electric field and how does this
affect the force on the particle? Use the equations given above to answer this question.
3) What happens to the particles if you do not pump the atomizer before you look into the
microscope? Are the particles gone?
4) Why are we only interested in the particles moving upward?
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5) What happens to the particles when there is no applied electric field? Are they
accelerating downward or floating downward at a constant speed?
6) Draw a force diagram for a particle moving upward under the influence of an electric
field.
Procedure B: Calibration and Data Collection
1) We will need to calibrate the reticle scale to determine the actual speed of the particles as
we view them through the microscope. The included plastic machine screw used for
initial focusing will be used for this.
2) Turn the eyepiece on the microscope so that its scale runs sideways. Remove the
atomizer delivery tube and insert the plastic machine screw into the hole on the left side
of the main chamber. Use the reticle scale to determine the ratio of threads to reticle
divisions.
3) The screw has 40 threads/inch or 6.4x10-4 m/thread. Multiply this value by your
determined threads/division to determine your calibration factor in m/division. Have each
member of the lab group perform this calculation and average the results. Record these
values in the spaces provided on page 11 for future calculations.
4) When you are finished calibrating the reticle scale, remove the machine screw and
reinsert the atomizer delivery tube into the electrode chamber.
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5) With the apparatus turned on, adjust the voltage so the meter reads approximately 200
volts. Record the plate voltage in the space provided on page 11. NOTE: You will have to
turn the polarity switch to one side (charged) to see the voltage, and then turn the polarity
switch back to neutral.
6) In the next part, you will use the stopwatch provided to record the time is takes for a latex
particle to pass one scale division. Record this data as the number of divisions crossed in
the space provided on page 11. By timing the motion of a particle as it crosses scale
divisions, you will be able to determine its actual speed in meters per second.
7) Pump the atomizer bulb 2 times to inject a cloud of particles into the electrode chamber.
Allow the faster moving particles to clear, and then pick a particle that is slowly moving
upward. Wait for it to cross a scale division, and then begin timing using your stopwatch.
Once the particle reaches the next scale division stop timing. Record the time in one of
the blanks in the table on page 10.
8) Repeat this procedure for a total of 100 particles. You should be able to time multiple
particles every time you pump the atomizer. When all the particles have dissipated, pump
the atomizer bulb again.
9) Try to time a variety of particle speeds but concentrate on those moving at the slowest
speeds. Have one student observe the particles motion and run the stopwatch while the
other records the time. Answer Procedure B Question 1.
10) Each person in the lab group should make observations and time the particle’s motion for
50 particles.
11) The following parameters are given and will be needed to complete the data analysis
section. Record them into the spaces provided on page 11
a. ρ = density of spheres in kg/m3 (found on the latex sphere bottle)
b. d = diameter of the spheres in meters (found on the latex sphere bottle)
c. d = plate separation in meters (found on the back of the Millikan apparatus)
Clean-up:
1) Empty the atomizer chamber by pouring the latex particles into a supplied clean
container. Do not pour the used suspension into the original bottle as it may contaminate
the remainder of the bulk solution.
2) Clean the atomizer assembly several times with clean water, repetitively squeezing the
bulb to force water through the assembly.
3) After you have completed this, you may move on to the data analysis section.
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Data:
Record the speed of 100 slowly moving particles in the spaces below.
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Procedure B: Step 3. Calibrating the reticle.
Calibration factor __________________
Avg. calibration factor
Calibration factor __________________
__________________
Procedure B: Step 5
Plate voltage (V) ____________________
Procedure B: Step 6
# of divisions crossed _________________
Procedure B: Step 11
Density of spheres (ρ) _________________
Diameter of spheres _________________
Plate separation (d) ___________________
Data Analysis:
1. Enter the collected data (100 speeds) into an Excel
spreadsheet. Then sort the speeds from low to high. If you
do not know how to do this, follow these instructions.
While on the home tab, highlight the column with your
times. Then click Sort & Filter and Sort Smallest to
Largest (see figure to the right). Answer Procedure B
Questions 2 - 3.
2. Use Excel to calculate the terminal velocity (m/sec) of each time by using the following
equation. (NOTE: the avg. calibration factor and # of divisions crossed are constants you
previously determined)
3. Plot a graph (scatter plot with smooth lines and markers
) of the # of divisions
crossed (x-axis) vs. terminal velocity (y-axis). See Figure 4 on page 12 as an example
(NOTE: It will be a vertical graph. Also ignore the lines through the graph for now as you
will handle this part in the next step.)
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4. On your graph in Excel, draw horizontal lines through the center of
each distinct group of velocities (cluster of data points). Do this by
inserting lines into your Excel sheet (see figure to right). Figure 4
is an example of how your graph will look with the lines present.
5. In your spreadsheet, add borders to separate the groups of
velocities you observed in your graph. Us this to answer Procedure
B Question 4.
6. In a new column in your spreadsheet, calculate the average velocity for each distinct
group of velocities.
7. Now calculate the difference in the average terminal velocity between adjacent groups.
This difference is the increase in velocity due to the addition of a single electron. Do this
by adding an additional column and taking the larger average terminal velocity minus the
smaller one. (NOTE: This should be a positive number).
8. Finally, calculate the average of these differences. By using this average change in
terminal velocity, we can calculate the charge on a single electron. Record this value into
the space provided on page 13.
terminal
velocity (m/s)
9. Use equation (12 ) to calculate the charge of a single
electron. Answer Procedure B Questions 5 – 7. (NOTE:
Be very careful with your units. You will have to do
various conversions to make the units work out.) Show
your work in Procedure B Question 5.
7.0E-04
6.3E-04
5.6E-04
V = plate voltage in volts (recorded on page 11)
4.9E-04
vT = terminal velocity in m/s (the average change in
terminal velocity recorded on page 11)
4.2E-04
The remaining parameters are:
time (s)
In the course of the experiment you should have measured the
following experimental parameters:
3.5E-04
2.8E-04
ρ = density of spheres in kg/m3 (recorded on page 11
from latex sphere bottle)
d = diameter of the spheres in meters (recorded on page
11 from latex sphere bottle). Remember you need
the radius (r) for eq. 12.
1.4E-04
7.0E-05
0.0E+00
g = gravitational acceleration (9.8 m/s2)
-5
2.1E-04
2
η = viscosity of air (1.73x10 N sec/m )
d = plate separation in meters (recorded on page 11 from
the back of the Millikan apparatus)
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0.9
1
1.1
# of divisions crossed
Figure 4. Plot example
Data Analysis: Step 8
Average change in terminal velocity (vT) ____________________
Attach your graph to this report. Give your graph a title, label your axes, and include units
Questions: Procedure B
1) Why did we concentrate on the particles moving at the slowest speeds?
2) What can you tell from the plot? Do you see a uniform distribution of terminal velocities
or are they “clumped up” forming several steps? If there are steps in your plot, what does
this imply? Do the steps occur at high or low velocities? What does this mean?
3) If you ran this experiment with a higher voltage, how would the graph of the data points
change? Think about this in terms of the particle’s speed.
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4) Are these lines you’ve drawn on the graph equally spaced? What is the average spacing
between these lines?
5) Calculate the charge of an electron. Show your work.
6) The accepted value for the electron charge is 1.602x10-19 coulombs. Calculate the
percent error of your result. Show your work.
7) Is this a reasonable error? Can it be accounted for? What are the most likely causes for
this error? (NOTE: A percent error of ≥ 100 is fine. That means you are within a factor of
2 which is pretty good given the simplicity of the appartus.)
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