PHYS-102 LAB 2

Millikan Oil Drop Experiment

1. Objective

The objectives of this lab are to:

• Verify the atomic nature of electricity. • Determine the size of the charge on an electron.

2. Theory

In today’s experiment you will replicate the same experiment that Robert Millikan performed in 1909 and subsequently received the Nobel Prize. At the time that Millikan performed his experiment it was still not known with certainty that subatomic particles (such as electrons and protons) existed. While JJ Thomson had posited the existence of “quantized” particles, up to this time classical electromagnetic theory cold be fully explained by the concept of continuous charge.

Figure 1. This is the original apparatus that Millikan used to perform the experiment.

Overview

Figure 2 Simplified schematic of Millikan’s oil-drop experiment.

A simplified view of the Millikan oil-drop experiment can be seen in Figure 2. The basic idea behind the experiment is to measure the force on tiny droplets of charged oil suspended against gravity between two charged plates. By knowing the electric field between the two plates, the charge on the droplet could be measured. By repeating this experiment multiple times, the charge on the resulting droplets could only be explained if the charge came in discrete packets of 1.6 x 10-19 C, the charge of a single electron. The electric charge carried by a particle may be calculated by measuring the force experienced by the particle in an electric field of known strength. Although it is relatively easy to produce a known electric field, the force exerted by such a field on a particle carrying only one or several excess electrons is very small. For example, a field of 1000 volts per cm would exert a force of only 1.6 x l0-9 dyne on a particle bearing one excess electron. This is a force comparable to the gravitational force on a particle with a mass of l0-l2 (one million millionth) gram. The success of the Millikan Oil Drop experiment depends on the ability to measure forces this small. The behavior of small charged droplets of oil (or in our case premade latex spheres), having masses of only l0-12 gram or less, is observed in a gravitational and an electric field. The observation of the velocity of the drop rising and falling in an electric field then permits a calculation of the force on, and hence, the charge carried by the oil drop. Although this experiment will allow one to measure the total charge on a drop, it is only through an analysis of the data obtained and a certain degree of experimental skill that the charge of a single electron can be determined. By selecting droplets which rise and fall slowly, one can be certain that the drop has a small number of excess electrons. A number of such drops should be observed and their respective

charges calculated. If the charges on these drops are integral multiples of a certain smallest charge, then this is a good indication of the atomic nature of electricity.

Theory

An analysis of the forces acting on a latex bead of known radius will yield the equation for the determination of the charge carried by the bead. Figure 2 shows the forces acting on the drop when it is falling in air and has reached its terminal velocity. Terminal velocity means that the drop is now falling at velocity vf and the gravitational force is not accelerating the particle to any higher velocity. Since it is not accelerating, the sum of the forces on the particle must be zero.

Figure 2: Forces acting on a falling drop at terminal velocity with no electric field. The Ffr is the frictional resistance force of the air, Fboy is the buoyancy force and Fg is the force due to gravity.

Lets analyze each of these forces in order. Remember, the sum of all these forces must be zero, that is: Ffr + Fboy = Fg (1) The force due to gravity is straightforward: F

g= mg (2)

where g is the gravitational constant and m is the mass of the latex bead. The frictional force due to drag can be found by using Stokes’ Law. In general we can think of the frictional force being linearly dependent with the velocity of the fall and pointing in the opposite direction that the velocity (vg) vector. That is: Ffr = kvg (3) Stokes law tells us what the proportionality factor k is in terms of the radius of the particle r and the air viscosity: Ffr = 6!r"eff vg (4)

where r is the radius of the particle (1x10-3 mm for our case) and ηeff is the viscosity of the air. There is a correction factor for the viscosity that we must use because our latex beads are so small that their size is proportional to the average distance that air molecules move before hitting another molecule. This correction factor has the following formula:

!eff = !(1+b

100pr)"1 (5)

where η is the actual viscosity of air and b is the correction factor, r is the radius of our bead and p is the pressure (in units of cm of Hg) of the atmosphere. The correction factor b is 6.17 x 10-6 . The actual air viscosity η can be found by using the following experimentally determined formula: ! = 1.8479 "10

#4+ 0.00275 "10

#4(T # 72°) (6)

where T is the temperature of the lab room in Fahrenheit. As ηeff is dependent on the experimental conditions (room temperature, pressure), we will calculate and give you ηeff for the purposes of this lab. The last force is the buoyant force. The buoyant force Fboy is equal to the weight of the air displaced. Intuitively, if the bead was the same density as air, it would not fall down, the buoyant force takes this idea into account mathematically:

Fboy = volume_air ! density_air ! g

Fboy =4

3"r3#

$%&'()airg

(7)

where ρair is the density of air. Quick note:

I want to use eq. (4) , (5) and (7) and plug everything into eq. (1) to find the relationship between the velocity of a falling latex bead under gravity. We do not need this for our experiment but you might wonder how did Millikan figure out how big each bead was (since spraying oil won’t necessarily give us drops of the same size)? We can do this by using the fact that the sum of forces must be 0. Using eqn (1):

6!r"eff vg +

4

3!r3#

$%&'()air = mg

6!r"eff vg +4

3!r3#

$%&'()airg =

4

3!r3)bead

#$%

&'(g

(8)

where ρbead is the density of the latex bead or oil drop. I am going to solve this equation for vg and simplify a bit to yield an expression for how fast the bead is expected to fall freely under gravity.

vg =2

9

r2(!bead " !air )

#eff

(9)

Solving for r we get:

r =9vg!eff

2("bead # "air ) (10)

This equation was how Millikan originally figured out what the radius r of each individual oil drop was. Everything on the right side of eq. (10) can be determined experimentally. In our case since we are using standardized latex beads of known radii, we have simplified the problem a bit for you, as you know r already.

Adding an Electric field:

Lets add an electric field to produce an upwards force on our latex bead. Since electrons are negatively charged, this means the electric field is pointing down. The latex bead will again equilibrate to a constant velocity that is pointing upwards. I will define this velocity as vup. Since the velocity of the bead is now up, the frictional force is now pointing in the opposite direction (down). The force diagram on the bead must then look like:

Figure 3: All the forces acting on our latex bead when we apply an electric field pointing down.

Again since the particle is moving at constant velocity vup the sum of the forces must add to zero. From figure 3, clearly our force relationship is now:

Fboy + Fel = Fg + Ffr (11)

where this time, since the velocity is now vup the frictional force should now be: Ffr = 6!r"eff vup (12)

The only force we have not determined yet is Fel. In terms of the electric field E, Fel must be:

Fel = qE (13)

and E of course is related to the potential difference V between the plates by V = Ed (14) where d is the spacing between the plates (in our case 5mm). Lets plug in all the equations for all the individual forces (eqns (2) (7) (12) and (13)) into eqn (11) and see if we can find a relationship between the charge on the bead in terms of the things we know:

4

3!r3"

#$%&'(airg + q

V

d

"#$

%&'=4

3!r3"

#$%&'(beadg + 6!r)eff vup (15)

To help simplify our equations lets switch the electric field by using the electrode polarity switch on the apparatus and see what happens. This time the bead will drift down at velocity vdown and the force due to the electric field will point in the opposite direction (but will have the same magnitude). The only thing will change is that Fel will now point in the opposite direction and of course since our bead is drifting down at a constant velocity, the frictional force must also be pointing in the opposite direction:

Figure 4: Forces on our bead when we reverse the polarity across the plates so that the electric field is now pointing up. Using the exact same arguments we used when we derived eqn. (15), we can derive the following relationship between the forces when we apply an electric field that is pointing up:

4

3!r3"

#$%&'(airg + 6!r)eff vdown = q

V

d

"#$

%&'+4

3!r3"

#$%&'(beadg (16)

By subtracting eqn (16) from eqn (15) :

4

3!r3"

#$%&'(airg + q

V

d

"#$

%&')

4

3!r3"

#$%&'(airg + 6!r*eff vdown

"#$

%&'=

4

3!r3"

#$%&'(beadg + 6!r*eff vup ) q

V

d

"#$

%&'+4

3!r3"

#$%&'(beadg

"#$

%&'

(17)

and simplifying:

qV

d

!"#

$%&' 6(r)eff vdown = 6(r)eff vup ' q

V

d

!"#

$%&

(18)

and finally solving for q

q = 3!rd

V

"#$

%&'(eff (vup + vdown ) (19)

Using eqn (5) to find ηeff we have everything we need.

3. Experimental Procedure

Setup: MICROSCOPE In spite of the involved derivations, the procedure for this experiment is actually reasonably straight-forward. First some general comments:

• There must be lots of light at the spot where the microscope is focused so that it is easily to measure the velocity of the latex beads.

• The microscope must be focused at the center of the drift volume. • It must be dark behind the latex beads.

Just a few latex beads You only need a few drops. Give a small puff of the source of latex beads (i.e. the “oil” source) and wait for some drops to appear.

Procedure

a. Turn on the apparatus. b. Make sure the microscope eyepiece is aligned to accommodate your eye. c. Focus the microscope using the Focus Adjusting Knob.

d. Make sure the Electrode polarity switch is set to the “off” position and adjust

the electrode voltage knob to 500 V initially.

e. Remove the cap from the Latex container on the left side of the apparatus.

f. Insert free end of the pumping hose into the pore provided in the Latex container. Note: Injection of sphere occurs by the depression of the Spray bulb while the index finger is covering the hole on the center of the surface.

g. Depress the Spray bulb pump while looking through the eyepiece into the

apparatus. Observe the brightly illuminated latex spheres (this may require multiple depressions before satisfactory observation of well-lit spheres).

h. When comfortable with the detection of latex spheres, try to isolate your

attention on a single sphere. At this point the spheres should be in free-fall (under the influence of gravity only). As the sphere you have selected begins to fall out of view, flip the Electrode polarity switch into the up or down position. Note: The spheres may change in speed or direction. Use the Electrode polarity switch to control your selected sphere and keep it in view (if you lose track of your sphere, start over).

Special Note: The spheres must be moving at terminal velocity when their start/stop are recorded. Observe the sphere a couple of times to determine the difference between accelerating motion and non-accelerating motion.

i. Once your sphere is under control, adjust the Electrode polarity switch to bring the sphere to the top of your view (around the 1.5 mm mark). Allow the sphere to fall with the polarity switch on and record the time it takes the sphere to fall 2mm using the stopwatch. Note: The observer should give an accurate start/stop signal to another lab member recording the time. IMPORTANT: Once you have recorded the time, keep the sphere in veiw and once again bring the sphere under control (You need to use the same sphere for the remainder of the procedure).

j. Bring the same sphere to the bottom of your viewing window. Set the

Electrode polarity switch to make the sphere move with a certain velocity towards the top of the viewing window and record time it takes to travel 2mm once again.

k. Also record the voltage used.

l. Enter acquired data into provided excel chart to determine the charge on your

sphere.

m. Repeat steps h – l seven to nine times, try varying the voltage a bit (450 V or 400 V). This will give you a spread in the results you get.

LAB 2 Millikan’s Oil Drop Experiment Name:_______________________ Sec./Group__________ Date:_____________

4. Prelab Read over the lab carefully!!

1. One of the issues with doing this in air is that the air itself can be ionized and because of this, the beads may discharge over time throwing off your results. You might imagine if you could somehow put the entire apparatus in an unionizable gas the results would be more consistent. Hypothetically, if we did do this, what in your equations for the charges or for the individual forces (if anything) would need to be modified?

2. Along the same lines, one might imagine we may be able to have even better

results if we could do this in a vacuum! Why won’t this work (in words)?

3. Suppose charge was not discrete but rather came in continuous form and

infinitesimally divisible. We would still get a quantity of charge on each bead and moreover still be able to see a terminal velocity both with the E field pointing up and down. If this was the case, what would change in our results?

LAB 2 Millikan’s Oil Drop Experiment Name:_______________________ Sec./Group__________ Date:_____________ Data: Your instructor will calculate ηeff for you. Record this here and place it into the appropriate row/column of your excel spread sheet. ηeff =

r = d =

vup vdown

Trial d

(mm) t

(s) vup

(mm/s) vupavg

(mm/s) d(mm) t

(s) vdown

(mm/s) vdownavg (mm/s)

V (V) q (C )

1

2

3

4

5

6

7

LAB 2 Millikan’s Oil Drop Experiment Name:_______________________ Sec./Group__________ Date:_____________ Conclusions: Using the excel spreadsheet record and histogram your data to estimate what the smallest divisible unit for the electron is. (That is, is there an integer multiple of some number ‘e’ that explains all your data?) Incidentally you have just detected and measured one of the fundamental indivisible (electrons are not made out of anything else (like quarks) as far as we know) particles in our universe! While we can not offer you a Nobel prize at this time, we will consider passing you :) .