Evidence for the Quantization of Electric Charge and Measurement of the Fundamental Unit of Electric Charge

Transcription

1 PHYSICS 286: Modern Physics Laboratory SPRING 2009 (K. Kumar and A. Dinsmore, March 2009) Experiment 7: The Millikan Oil Drop Experiment: INTRODUCTION Evidence for the Quantization of Electric Charge and Measurement of the Fundamental Unit of Electric Charge In this laboratory experiment you will measure the electric charge on a series of oil drops and demonstrate that this charge is quantized. That is, the charges are integer multiples of a fundamental charge, presumably the charge of the electron. If this result is combined with the value of e/m measured for the electron in the Physics 285 laboratory, you will have also measured the mass of the electron. This experiment is a modern version of the classic experiment done by Robert Millikan during the period 1909 to 191 at the University of Chicago. He demonstrated that the electric charges on the observed oil drops were integer multiples of a fundamental unit of charge, and determined the magnitude of this fundamental charge. For this work, he was awarded the 192 Nobel Prize in Physics only the second American to receive the prize. It is of interest to note that Millikan discarded one observation from his experiment, and was so complete in his reporting that he mentioned this in a publication: "I have discarded one uncertain and unduplicated observation apparently on a singly charged drop, which gave a value of the charge on the drop some 0 percent lower than the final value of e." Robert A. Millikan, Phil. Mag. 110, 209(1910) We now know that the neutron and proton are not fundamental particles, but are each made up of three quarks. Each quark has an electric charge of magnitude 1/ or 2/ of the charge of an electron. Did Millikan actually observe a free quark charge on an oil drop? Although many searches have been performed, the observation of isolated quarks has never been demonstrated. There is an ongoing research effort at the Stanford Linear Accelerator Center (SLAC) to search for isolated fractional charged particles using a version of the Millikan experiment. The experiment uses a charge coupled device (CCD) video camera, and is completely automated in order to process macroscopic amounts of material. To learn more about this research see: PHYS 286 Spring 2009 Millikan Oil Drop Page 1 of 7

2 OVERVIEW University of Massachusetts, Amherst When oil drops are sprayed from an atomizer, some obtain an electric charge as a result of friction. The basic idea is to measure the terminal velocity of an oil drop when: (a) falling at speed v 1 in the absence of any electric field, subject only to the downward gravitational force and an upward resistive drag force (from Stoke s law), and (b) rising at speed v 2 in the presence of an electric field, subject to the downward gravitational force, a resistive drag force, and an upward electrical force. The two main parameters that describe the behavior of each drop are the size and amount of charge. Measurement of the two speeds, v 1 and v 2 allows the determination of both the radius and charge of the oil drop. The formulas are derived in Appendix I. The derivation of Stoke s Law for the resistive drag force assumes the moving object is much larger than the average distance between collisions with molecules in the air (the mean free path). For the drop sizes used in this experiment, this assumption is not exactly correct and a radius-dependent correction must be applied to the viscosity. For a drop size of 1 µm the correction is about 20%. The correction is smaller for larger drops. (This is described in Appendix I.) APPARATUS Millikan observed oil drops by eye through a microscope. We employ a modern adaptation of the Millikan apparatus due to Hoag, in which the oil drops are illuminated by a laser, and observed on a TV monitor. Oil drops are sprayed from an atomizer into a vapor tower above the observing chamber, and enter the chamber through a small hole. The top and bottom of the chamber are flat metallic plates to which a DC voltage is applied to provide the electric field. Sides of machined bakelite provide about a 5 mm separation, and keep the top and bottom plates parallel to one another. The front and back of the chamber are glass. A high-quality telescope focuses the image of drops falling through the chamber onto a CCD video camera directly connected to the TV monitor. The distance scale is provided by an array of 2µm tungsten wires. The wires are positioned so that the virtual image formed by its reflection from the front glass plate of the chamber is in the plane of the falling drops, and is thus also focused on the CCD camera. The drops and the wire array are both displayed on the TV monitor, and there is no parallax or magnification error. The center-to-center spacing of the wires is 1/80 inch (0.175 mm), set by the pitch of the #0-80 screws around which they are wound. PHYS 286 Spring 2009 Millikan Oil Drop Page 2 of 7

3 PROCEDURE University of Massachusetts, Amherst CAUTION - HIGH VOLTAGE Do not touch the metal plates forming the observing chamber unless: (a) the high voltage power supply switch is set to OFF, and (b) the reversing switch is in the middle position. In this position the plates are disconnected from the power supply and short circuited. 1) Record the oil density and the plate separation for your apparatus. Also record atmospheric pressure in cm of mercury, and the room temperature. Record the temperature again at the end of the lab period, and use the averages. 2) With all power off, examine the apparatus. Then turn on the power strip. Verify that the reversing switch is in the middle position, and that the power supply switch is off. The digital voltmeter (DVM) connected to the plates should read zero. Remove the vapor tower, and observe how the adjustable aperture (see fig.) works. You will observe that the upper plate has a symmetric array of holes. Pass a pin through the central hole, and verify that it is in focus on the TV monitor. The horizontal wires should also be in focus. Remove the pin. Replace the vapor tower with the aperture open. ) We now want to understand the polarity of the electric field. Look at the wires connecting the plates of the chamber to the DVM. What is the direction of the electric field when the DVM reading is positive? In order to have an upward electric force on a negative charge, should the DVM reading be positive or negative? 4) Next we want to find and trap a slowly falling oil drop. Squeeze the atomizer bulb a few times under the desk until a good spray is obtained. Squeeze the atomizer ONCE into the vapor tower, and replace the tower cap. Turn the power supply STANDBY switch ON, and set the voltage to about 500 V. Watch the falling drops, which appear as bright spots on the TV, until you find a very slowly falling drop, one that takes 10 to 20 seconds to fall from the top to bottom of the screen. Longer is better, because it means a smaller drop, and most likely a smaller charge. This will make the quantization of charge more obvious. Be patient, this may take several minutes. While you wait, estimate the waiting time for a good drop to appear on the screen, given the height of the tower and the wire spacing. PHYS 286 Spring 2009 Millikan Oil Drop Page of 7

4 5) When you see S L O W L Y falling drops: Experiment with the reversing switch until you find a drop that falls slowly in zero field, AND that rises when the voltage is on. Practice trapping a drop, having it fall from one wire to the next with V = 0, and then rising above the first wire with V 0. Once you have trapped a drop, close the aperture in the tower to prevent other drops from falling through the field of view. You may want to adjust the voltage so that the rise time is also conveniently long. It should be possible to obtain 10 or more measurements of the rise and fall time for each oil drop. (As a minimum, you should obtain at least 5 measurements per drop). Set the reversing switch to the middle position, and set the power supply switch to OFF. 6) Comments Terminal velocity is achieved in a very short time. This should not be a problem if you let the droplet start falling when it is above the wire. The best results are obtained with drops that fall slowly. Drops that fall slowly are small drops (r v 1 ) that have small mass, and are likely to have small charge. You want drops with small charge because it is easier to distinguish between q = 2e and q = e than, for example, between q = 12e and q = 1e. Charge Changes: On occasion, the rise time, t 2, will change abruptly owing to a change in the charge on the drop. In this case the total charge on the drop is probably small, and the change is very likely to be a single charge. Brownian Motion: You will observe that large, rapidly falling, drops fall smoothly through the field of view but that small, slowly falling, drops appear to bounce around. Indeed, they do bounce around! The smallest drops are small enough that they are visibly affected by collisions with the molecules in the air. The resulting random motions, known as Brownian motion, are superposed on the otherwise uniform motion. (Along with the theory of the photoelectric effect and special relativity, the theory of Brownian motion was one of Einstein s three great contributions in 1905.). The Brownian motion causes a larger variation in the measured values of the rise and fall times than would otherwise be expected. 7) You are now ready to take data: Open the tower aperture, and wait for another slowly falling oil drop. Turn the power supply switch ON. When a drop is trapped, close the tower aperture. Adjust V if necessary, and record the value. Obtain 10 (or more) sets of measurements if possible. Estimate how accurately you can measure a single time value with the stop clock. 8) Before collecting a large amount of data for many drops, you may wish to go through the calculations for the first drop(s). You will need the calculations in Appendix I. The use of a spreadsheet or similar computerized table is recommended. If you are reasonably comfortable with this, it should save time and help avoid arithmetic errors. Calculate the average fall time for the drop t 1, and the average rise time t 2. Calculate the standard deviations of the measured times. Then find the error of the means. (Recall this from the Radioactivity lab.) PHYS 286 Spring 2009 Millikan Oil Drop Page 4 of 7

5 Calculate the values of v 1 and v 2 as v = d/ t for the fall and rise of the drop (d is the center-to-center distance between wires: d = 1/80 inch = mm). Also calculate the uncertainty in v 1 and v 2. Calculate the overall constant C 1 from Eqn.7a of Appendix I. You need do this calculation only once. Check the units. Calculate the radius of the drop from Eqn. of Appendix I. Calculate C 2, the viscosity correction factor to the charge, from Eqn.7b. You can now calculate the charge on the drop from Eqn.7. Is the value of the charge at all reasonable? If not, go back and figure out where you lost all those powers of 10, and where errors were made in units. Start a summary table where you will tabulate these results for all drops studied. 9) Take more data after you are certain that you can make sense of the data obtained for the first charge. Open the tower aperture and watch for slowly falling drops. At some point, you may have to open the tower, and spray in more oil. Turn the power on, and take measurements for additional oil drops as in (7) above. Overall, this is not an easy experiment. Large statistical fluctuations are introduced into the data by the Brownian motion. Good results require patience, care, and observation of many oil drops (at least 10; 15 is better if you have time). 10) During the first week, take a few sets of good data which you can analyze later. When you have finished measurements for the day Set the reversing switch to the middle position, and set the power supply switch to OFF. Remove the vapor tower and wipe any excess oil off the top of the chamber. Replace the tower and cap. Record the temperature and air pressure again; use an average in calculations. Turn the power strip off. PHYS 286 Spring 2009 Millikan Oil Drop Page 5 of 7

6 DATA and ERROR ANALYSIS The first important point is that the known electron charge (e = C) plays no role in the data analysis. The goal is to arrive at your own measurement of the fundamental charge. Only after you have done that will you compare your determination with the accepted value of e. A second important point is that the quality of the data will affect how straightforward it is to analyze it. Here is where having slowly-moving droplets can help by giving you data with 1, 2, or times the fundamental charge. If the analysis of your first day s data does not go smoothly, you can use the second day to find slower droplets. 1) For each drop: Calculate the values of v 1 and v 2, and follow the steps in procedure (8) above to calculate the charge on the drop. Calculate the uncertainties in the determination of v 1 and v 2. Calculate the uncertainty in the determination of the charge of each drop, due to the uncertainties in v 1 and v 2. The constant C 2 depends on the drop radius, which in turn depends on v 1. Evaluate the uncertainty in C 2 to determine if it is large enough to include in the uncertainty in the charge. Similarly, evaluate the uncertainty in C 1. Either calculate the uncertainty in charge to include the uncertainties in C 1 and C 2, or present a quantitative argument for ignoring them. v 1 F drag W B 2) Now we want to ask if the measured charges are quantized. There is no well-defined procedure for this, but the following steps seem reasonable. First, draw a line which will be a charge axis. Pick a scale that will include all measured charges, and label the charge axis in Coulombs. Mark each measured charge on or near the axis. If there were any charge changes, include the values of Δq also. The charge quantization should be visually obvious. Now we will be more quantitative: Make a histogram of the values of q and Δq (if any). You might see peaks corresponding to integer values of e. If so, you could measure the average charge for each histogram peak. Then plot this value vs. the number of the peak (the first being #1, etc.). The slope of this line should be e. ) Your lab report should at least contain the following information and answers to these questions: The value of the fundamental unit of charge determined from the data. The experimental uncertainties in the result. How does your value of e compare with the accepted value of x C? Is this an acceptable result, considering experimental uncertainties? Think back over the entire experiment. Are there potential sources of error that you did not include, perhaps because there was no way to check or measure them? Discuss additional measurements that would allow these other sources of error to be determined. Discuss to what extent your data and analysis have (or have not) provided convincing evidence for the quantization of electric charge. APPENDIX I: CALCULATION of DROP RADIUS and CHARGE PHYS 286 Spring 2009 Millikan Oil Drop Page 6 of 7

7 In the absence of an electric field the net (upward) force on a drop falling at speed v is: 4 F $ ( W $ B) = 6"# rv $ " r (! $ )g (Eqn.1) drag oil! air Where W-B is the buoyant force on the drop (its weight minus that of the air that it displaces). In steady state, the net force is zero and the drop falls at a constant terminal velocity, v 1. Setting the net force to zero and, we obtain: 4 ( )! r # g 6 rv oil $ # air =!" 1 (Eqn.2) The radius is found to be: 9" v r = 1, where ρ = (ρoil ρ air ) ρ oil. (Eqn.) 2! g Thus, measurement of the terminal velocity v 1 allows calculation of the drop radius. We now consider the effect of an electric field, E = V/D, where V is the voltage difference between plates separated by D. The net force on the drop when rising at velocity v is: V qe! Fdrag! ( W! B) = q! 6"# rv! ( W! B). (Eqn.4) D Note that this time F drag < 0 because the drop moves upward. In steady state, the net force is zero and the drop rises at the terminal velocity v 2 : V 4 q =! r # g + 6!" rv2 = 6!" rv1 + 6!" rv2. (Eqn.5) D (For the last step, we used Eqn 2.) Now, using Eqn. for r, the charge is: D 18# D " v1 q = 6#" r( v1 + v2) = ( v1 + v2) (Eqn.6) V V 2! g Thus, by measuring v 2 and v 1, we can calculate the charge on the drop. v 2 F drag qe B W The viscosity of air is temperature dependent: $. " T kg = (18.27 # 10 6 ) 291K m! s A correction is required for the fact that the drop size is comparable to the mean free path. The viscosity must be divided by the factor (1 + b/pr), where b = 6.17 x 10 6 m cm, and p is the atmospheric pressure in cm of Hg. The net result is that the viscosity is reduced and the terminal velocity is increased. Finally, we can write the charge as: C ( v v ) V 2 q = C v (Eqn.7) " C1 = 18# D, (Eqn.7a) 2! g! / 2 C 2 = (1 + b / pr) (Eqn.7b) We have defined two constants: C 1, which is the same for all drops, and C 2, which is the viscosity correction to the charge, and is different for each drop. PHYS 286 Spring 2009 Millikan Oil Drop Page 7 of 7