PHYS-102 LAB 2

Millikan Oil Drop Experiment

1. Objective

The objectives of this lab are to demonstrate the quantized nature of the electrical charge ofan electron and to see how this charge directly interacts with an external Electric Field.

2. Theory

In 1909, Robert Millikan performed an experiment for which he received the Nobel Prize.At the time, the notion of subatomic particles such as electrons and protons was not yetwidely accepted. Although scientists like J.J. Thomson had already advanced the idea of“quantized” particles, it was believed that all of classical electromagnetic theory could befully explained by the concept of continuous charge. As such, no principle of quantizationwas necessary.

Figure1a: This is the original apparatus that Millikan used to perform the experiment.

Figure 1b: Simplified schematic of Millikan’s oil-drop experiment.

2.1 Overview

Millikan’s oil-drop experiment is illustrated schematically in Figure 1b. Tiny droplets of charged oil are suspended against the downward force of gravity using the interaction of electrical charges on the droplets and the electric field E between two charged parallel plates. By measuring the electric field E, the charge on the droplet could be deduced. By repeating this experiment multiple times, the total charge on the resulting droplets are measured to be integral packets of the electronic charge e = 1.6 x 10-19 C. The strategy of this experiment is straightforward: measuring the force experienced by a charged particle (oil droplet) in a known electric field E allows one to calculate the charge on the particle. However, the force exerted by typical electric fields in such a physical arrangement on a particle carrying only one or several excess electrons is very small. For example, a field of 100 volts per cm would exert a force of only 1.6 x l0-8 dyne (1.6x 10-13 N) on a particle bearing one excess electron. Thus, the success of the Millikan Oil Drop experiment depends on the ability to measure minute forces accurately. To do this, we carefully monitor the behavior of tiny droplets immersed in a gravitational and electric field. The observation of the velocity of the drop rising and falling in an electric field then permits a calculation of the net force on the sphere, and therefore, the net charge it carries. In this experiment, we determine the total charge on a droplet. By analyzing the data obtained, the charge of a single electron can be determined. It is important to select only spheres which rise and fall slowly under the action of the applied electric field. Such spheres have a very small number of excess electrons. The number of such spheres should be tracked and their respective charges calculated. If the charges on these spheres are demonstrated to be integral multiples of a certain smallest charge, then this is an indication of the quantized nature of charges.

2.2 Theory The free body diagram of all the forces on an isolated droplet is shown in Figure 2. As the sphere is falling in mid-air and has reached its terminal velocity vf, it experiences no acceleration. Since the net acceleration is zero, the sum of the forces on the particle must also be zero.

Figure 2: Forces acting on a falling droplet 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 weight.

Because the sphere is in a state of equilibrium (zero acceleration), it is obvious that: Ffr + Fbuoy = Fg [1] where Fg = mg [2]

The frictional force on the sphere can be modeled by a force that is proportional to the velocity of the sphere and pointing in a direction opposite to the motion: Ffr = kvg [3] Stokes’ law determines the relationship of the proportionality factor k to the particle radius r and the viscosity of the surrounding air: Ffr = 6πrηeff vg [4] The actual air viscosity η can be found by using the following experimentally determined formula: η = 1.8479 ×10−4 + 0.00275 ×10−4 (T − 72°) [6] where T is the temperature of the lab room in degrees Fahrenheit. According to Archimedes’s Principle, the buoyant force Fboy is equal to the weight of the fluid (air, in this case) displaced by the sphere:

FfrFbuoy

Fg

m

Fbuoy = volume_air ! density_air ! g

Fbuoy =43"r3#

$%&'( )airg

[7]

where ρair is the density of air.

Using Equations [4], [5] and [7] in Equation [1], one is able to determine the velocity of a falling droplets. Using this, Millikan figured out how big each droplet was, as there is no control over the size of the droplets created by spraying oil in the air. Using Equation [1], we find:

6!r"eff vg +

43!r3#

$%&'( )airg = mg

6!r"eff vg +43!r3#

$%&'( )airg =

43!r3)bead

#$%

&'( g

[8]

where ρbead is the density of the oil drop. One solves this equation for vg:

vg =29r2 (!bead " !air )

#eff

g [9]

Thus, we can obtain the radius r:

r =9vg!eff

2("bead # "air )g [10]

This was how Millikan originally determined the radius r of each individual oil drop.

2.3 Imposing an Electric field:

To suspend the spheres, the electric force that is to be imposed on the sphere must be pointed upwards. This, in turn, implies that the applied electric field E must be pointed downwards because the excess electrons on the spheres are negatively charged. The oil droplets will achieve a constant upwards velocity vup. Hence, the frictional force points downwards. This is summarized in Figure 3.

Figure 3, left: All the forces acting on our latex bead when we apply an electric field

pointing down. Figure 3, right: Forces on our bead when we reverse the polarity across the plates so

that the electric field is now pointing up. Since the particle is moving at constant velocity vup the sum of the forces must add to zero. From Figure 3: Fbuoy + Fel = Fg + Ffr [11]

where: Ffr = 6πrηeff vup [12] In terms of the electric field E, Fel must be: Fel = qE [13] where E is related to the potential difference V between the plates: V = Ed [14] The term d is the spacing between the plates (in our case 5mm). One can then substitute Equations [2], [7], [12] and [13] into Equation [11] to determine the charge on the droplet:

43πr3⎛

⎝⎜⎞⎠⎟ρairg + q

Vd

⎛⎝⎜

⎞⎠⎟=43πr3⎛

⎝⎜⎞⎠⎟ρbeadg + 6πrηeff vup [15]

We can simplify our derivation by switching the electric field direction. This is done by switching the polarity on the apparatus. This time the droplet will drift down at velocity vdown and the force due to the electric field will point in the opposite direction. Hence, Fel will now point in the opposite direction, and because the droplet is drifting down at a constant velocity, the frictional force must also be pointing in the opposite direction:

FfrFbuoy

Fg

m

Fel

F

fr

Fbuoy

Fg

m

F

el

For this reverse configuration, the analog of Equation [15] is:

43πr3⎛

⎝⎜⎞⎠⎟ρairg + 6πrηeff vdown = q

Vd

⎛⎝⎜

⎞⎠⎟+43πr3⎛

⎝⎜⎞⎠⎟ρbeadg [16]

By subtracting Equation [16] from Equation [15], we obtain:

43πr3⎛

⎝⎜⎞⎠⎟ρairg + q

Vd

⎛⎝⎜

⎞⎠⎟−

43πr3⎛

⎝⎜⎞⎠⎟ρairg + 6πrηeff vdown

⎛⎝⎜

⎞⎠⎟=

43πr3⎛

⎝⎜⎞⎠⎟ρbeadg + 6πrηeff vup − q V

d⎛⎝⎜

⎞⎠⎟+43πr3⎛

⎝⎜⎞⎠⎟ρbeadg

⎛⎝⎜

⎞⎠⎟

[17]

Simplifying the expression yields:

q Vd

⎛⎝⎜

⎞⎠⎟− 6πrηeff vdown = 6πrηeff vup − q

Vd

⎛⎝⎜

⎞⎠⎟

[18]

We can isolate the excess charge q:

q = 3πr dV

⎛⎝⎜

⎞⎠⎟ηeff (vup + vdown ) [19]

The radius of the droplet r has to be calculated using Equation [10] and the free fall velocity vg.

3. Experimental Procedure

3.1 Setup and Procedure:

On the successfully recorded AVI video, three parts of the trajectory of the SAME droplet should be traceable: free falling (1), moving up (2), moving down (3). Therefore, the droplet’s trajectory should not disappear from the screen, and the droplet should move slowly for the given voltage (the smaller charge the better). The number of useful droplets can vary but it should be a minimum of five (5).

The charge of the droplets can be positive or negative so some droplets will move up while the others move down for the same voltage polarity. Both can be used for charge calculation.

1. Connect the microscope to the computer via the USB port.

Note that you need to use the Video Capture software instead of the software described below.

2. The camera program, BW Microscope AP, should open automatically. If not, double click its icon on the Desktop to see a signal from the camera.

3. Go to File – Set Capture File. Create a name for the video and save it to either the Desktop or Student Documents. Input 100 Mbytes for the Capture File Size. Caution: Once the initial filename has been set, repeated saved recordings will overwrite the original file unless you rename the file prior to recording each time.

4. On the Millikan Apparatus, ensure that the electrode polarity switch is in the “off” (MIDDLE) position. Adjust the electrode voltage knob to 400 V initially.

5. Produce a small puff of the oil mist into the drift volume and watch for the droplets to appear on the monitor. Some droplets should slowly fall down. Be sure the nozzle is horizontal; you should not be able to see it on the video screen.

6. After a few seconds, toggle the switch to the positive (UP) position to apply voltage and to make the droplets move up. Make sure you can see a few relatively slow-moving droplets. They will be traced later using different software.

7. Before the droplets of interest leave the field of view, switch the polarity of the voltage and make the droplets move down.

8. After you finish practicing, start recording by going to Capture – Start Capture or the Play (>) tool button.

9. Be sure to record at least one free fall, upwards-, and downwards-moving segment for at least five of the droplets.

10. To stop recording, select Capture – Stop Capture, or the Stop (x) button. Your video will be saved as an .avi file with the name created in Step 3.

11. Turn off the high voltage and power off the apparatus.

After the AVI file is created, it should be opened with ImageJ, a particle analysis software. 1. To open the video file, go to File – Import – AVI. Select your file. Select the box

next to Convert to Grayscale to make the file size more manageable. 2. You will need to scale the video. Go to Analyze – Set Scale. Enter the following:

Distance in Pixels: 480 Known Distance: 3 Pixel Aspect Ratio: 1.0 Unit of Length: mm

Be sure to check the box next to Global so the scale is applied to all frames.

Manual Tracking – repeat process for each droplet

1. Go to Analyze – Set Measurements. Select only the “Invert Y coordinates” option. 2. Select the Multi-Point tool in the toolbar (right-click on the Single Point tool to

switch). 3. Advance the video to shortly after the start of the droplet rising or falling – click on

the particle. Record the time of the click (in the upper left corner). 4. Follow the droplet to the end of its rising or falling – click on the particle again.

Record the time. 5. Repeat this process for all trajectories of this particle.

6. Once all positions have been selected, go to Analyze – Measure. A window will appear with the x- and y-coordinates for each of the points. Copy the data into an Excel spreadsheet, and enter the corresponding times (to determine velocities).

7. Select Analyze – Clear Results to erase the data to track the next particle. Select the Rectangle tool in the toolbar and click on the screen to clear the points.

Automatic Tracking – w/ MTrack2 plugin 1. The video needs to be binary for the plugin to follow the particles. Go to Image –

Adjust – Threshold. Move the lower bar until all that remains are well-defined particles. Be sure the “B&W” option is selected, and click “Apply”. In the “Convert to Mask” window, select both “Calculate Threshold for Each Image” and “Black Background”.

2. To start the plugin, go to Plugins – MTrack2. Enter the following values: Minimum Object Size (pixels): 5 Maximum Object Size (pixels): 20 Maximum Velocity: 100 Minimum track length (frames): 100

Select “Save Results File”. If you would like to see a video with labels of the analyzed particles, select “Show Labels” and “Show Positions” as well.

3. Create a name for the results file – save it in the same location as your video. 4. Import the text file into Excel.

a. Open Excel. b. Select the “Data” tab. Choose “From Text” under the “Get External Data”

subcategory. c. Click “Finish”. d. Select a starting cell for the data and click “OK”.

5. Look through the various particle trajectories to find those that are moving up and down. Graph the chosen trajectories to locate the points at which the particles change direction.

6. Record the y-positions and frames (frames are taken 0.03s apart) at the turning points to determine the velocities of the particles.

LAB-02 Millikan’s Oil Drop Experiment

Name:_______________________ Sec./Group__________ Date:_____________

4. Prelab

1. Performing this experiment in air is tricky because air itself can be ionized and because ofthis, the beads may discharge over time, leading to erroneous results. If you could somehow put the entire apparatus in unionizable gas, what term(s) in your equations for the charges or for the individual forces would need to be modified?

2. Along the same lines, one might imagine we may be able to have even better results if wecould do this in a vacuum! Will this strategy work?

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

NOT REQUIRED FOR 154.

LAB-02 Millikan’s Oil Drop Experiment Name: _______________________ Sec./Group__________ Date:_____________ 5. Data Determine η. Record this here and place it into the appropriate row/column of your Excel spread sheet.

! = 1.8479!10!! +  0.00275  !10!! ! − 70!!!"#$ !!"!        

For each droplet of interest calculate r using Equation [10]. Use ρbead = 0.8 g/cm3. Then calculate the charge of that droplet.

! = 3!" !!! !!" + !!"#$ , d = 5 mm, V = 400 V

T = ______________ oF d = __________________ mm = _______________ m ! = ___________________!"#$(!)

!"! = _______________________ !  (!)!!

Droplet # vg (mm/s) vup (mm/s) vdown (mm/s) r (m) q (C)

1 2

LAB-02 Millikan’s Oil Drop Experiment Name:_______________________ Sec./Group__________ Date:_____________ 6. Analysis

1. Use Excel and your data to estimate the smallest divisible unit for the electron. Is there some number ‘e’ for which your charges are integer multiples thereof? Explain carefully. 2. Explain the experimental errors that you may have encountered in this experiment.

7. Just for the Fun of It

Plates on a van de Graaf Generator In the laboratory, you will find a van de Graaf generator and some aluminum pie plates. The van de Graaf generator is a device that can deposit large amounts of charges on its metal sphere. It does this by turning a belt and around an axle, close to metal brushes inside both the sphere and on the chasis. As a result, the sphere is held at very large voltages (several kilovolts) when fully charged so it is important not to touch it while being grounded or simply standing on the floor, or else one can get electrically shocked. Do not touch it unless the sphere is fully discharged by the TA using a grounded screwdriver. Your TA will now stack several aluminum pie plates on the sphere. Before the generator is turned on, predict what will happen to the stack of plates. Will they begin sparking ? Will they be attracted to the sphere ? Will they begin to float ? After making your prediction, ask your TA to charge the generator. Observe what happens and explain. What does this tell you about the strength of the electrostatic forces versus gravity. After the TA turns off the generator, make sure he grounds it first before you might attempt to touch the sphere.