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Millikan Oil-Drop Experiment Physics 2150 Experiment No. 2

University of Colorado

Introduction

The fundamental unit of charge is the charge of an electron, which has the magnitude of ! = 1.60219 x 10-19 C. With the exception of quarks, all elementary particles observed in nature are found to have a charge equal to an integral multiple of the charge of an electron. (Quarks have charge!/3, but they are never observed as free particles). By convention, the symbol ! represents a positive charge, so the charge ! of an electron is ! = !. The first person to measure accurately the electrons charge was the American physicist R. A. Millikan. Working at the University of Chicago in 1912, Millikan found that droplets of oil from a simple spray bottle usually carry a net charge of a few electrons. He built an apparatus in which tiny oil droplets fall between two horizontal capacitor plates while their motion is observed with a microscope. A voltage applied between the plates creates an electric field, which can exert an upward force on a charge droplet, counteracting the force of gravity. By studying the motion of a droplet in response to gravity, the electric field between the plates, and viscous drag due to the air, Millikan was able to calculate the charge on the droplet. He published data on 58 droplets and reported that the charge of an electron is ! = (1.5920.003) x 10-19 C. Millikans value was a bit low (three ! below the accepted value) because he used a slightly inaccurate expression for the drag force due to air. For this work and later research on the photoelectric effect, Millikan received the Nobel Prize in 1923.

In recent years, some historians have suggested that Millikan improperly threw out data which yielded charges of a fraction of an electrons charge; i.e. that he selected his data in order to get the answer he wanted. Fortunately for Millikans reputation, he kept excellent lab notebooks and it has been possible to re-construct all his measurements and calculations. Millikans notebooks do contain much data, which he never published, but there is no evidence that he fudged his results. All of his data, both published and unpublished, has been re-analyzed by CU physicist, Prof. Allan Franklin, who finds that all of Millikans data, properly analyzed, yields a result which agrees perfectly with the modern value of !. Experimental Apparatus Our experimental arrangement differs significantly from Millikans in two ways. Instead of oil drops, we use micron-sized latex spheres. And instead of peering through a microscope in a darkened room, we use a CCD video camera and watch the motion of the spheres on a television screen.

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A solution of the latex spheres in water is contained in a small squeeze bottle, called an atomizer. One squeeze produces a mist of latex spheres, most of which have a few extra charges, picked up in collisions with the plastic hose of the atomizer. The diameter of the latex spheres is 1.0700.005!m and their density is ! = 1.0500.005 g/cm3, according to the manufacturer. We will need the density for

our calculations. The diameter will not be used in our calculations but will be used only as a check on our results. The mist of latex spheres falls through a small opening in the top plate of a pair of horizontal metal plates. A variable voltage is applied across the plates, producing an electric field. A switch box allows the field to be turned on and off or reversed.

The tiny spheres are illuminated by a powerful lamp that shines through three infrared-absorbing glass plates. Without these heat-absorbing plates, the concentrated light from the lamp would heat the air in the sample chamber, causing convection currents which would disturb the motion of the spheres. Millikan himself was greatly concerned about heating due to the light source. Lacking IR-absorbing glass, he passed the light through several feet of water, which absorbs IR. The plates are in an air-tight housing with glass windows on the sides for viewing the sample volume between two plates. As the latex spheres fall through the still air between plates, they are viewed on a television monitor hooked to a video camera with a long working-distance microscope lens. A scale, taped to the face of the television monitor, allows one to measure the distance a sphere has moved. The TV scale is calibrated by pointing the camera at a scale with a known spacing.

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Theory At sufficiently low speeds, the drag force !! on a sphere of radius !, traveling with speed ! through a viscous medium of viscosity !, is given by Stokes Law, !! = 6 ! ! ! ! 1) where the minus sign indicates that the direction of the drag forces is opposite to the direction of the velocity. The derivation of this expression from a hydrodynamic theory assumes that the speed is sufficiently low so that the flow of the viscous medium around the sphere is laminar (smooth). At high speeds, the flow becomes turbulent and Stokes Law is invalid. In this experiment, the viscous medium is air and the sphere is a micron-sized latex sphere. Since air is not an ideally uniform fluid, Stokes Law must be corrected slightly. The sphere may fall freely through the spaces in between the air molecules: hence, the retarding force is decreased slightly. Millikan found empirically that Stokes Law must be modified: !! = ! ! (2) where the constant ! is given by ! = ! ! ! !

!! !! ! (3)

where ! is the atmospheric pressure and ! is an experimentally determined constant. If ! is measured in torr (1 torr = 2 mm Hg) and ! is in meters, then the value of ! is ! = 6.17 x 10-5 torr m. (Millikan used the slightly incorrect value of 6.25 x 10-5 torr m, his only seriously systematic error.) At 23, the viscosity of air is ! = 1.828 x 10-5 N sec/m2 per degree Centigrade.

When the external electric field is off, the sphere falls with a terminal speed vf, under action of two forces: the force of gravity downward (magnitude mg), and the drag force upward (magnitude !! = K vf). (In addition, according to Archimedes Principle, there is an upward buoyant force due to the weight of the displaced air; however, in this experiment, the buoyant force is negligibly small and can be ignored.)

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At the terminal speed, the drag force on the sphere equals the force of gravity, !" = ! !! . (4) The mass ! can be written in terms of the known density ! and the unknown radius ! of the sphere,

m = !!

! !! !. (5)

Combining Eqs. (3), (4), and (5) yields a quadratic equation in !, which has the solution

! = !!

!" ! !!! !

+ !!

! !

!. (6)

Thus, the radius of the sphere can be computed from a measurement of the fall speed. Once the radius is known, the mass can be computed from Eq. (5). If the charge on the sphere is ! and the electric field between the plates is ! = !

! ,

where ! is the plate separation and ! is the applied voltage, then there is an additional force on the sphere of magnitude !" = ! !

!. If the sign and magnitude of the field are such to

make the sphere rise, it will quickly acquire a terminal speed !! , determined by the equation

! !!= !" + !!! . (7)

Adding Eqs. (4) and (7) and solving for the charge ! yields ! = !

!! (!! + !!), (8)

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where the constant ! is found for a particular sphere from Eqs. (3) and (6). If the voltage is adjusted so that the force from the electric field equals the weight of the sphere, then the sphere will be suspended, stationary, neither rising nor falling. The suspension voltage !! is given by ! !!

!= !" = ! !

! ! !! !. (9)

Solving for ! yields ! = ! ! !

! ! ! !! !!

. (10) Eqs. (8) and (10) provide independent determinations of the spheres charge; however, both depend on the spheres radius !, determined from Eq. (6). Procedure Before you begin taking data, you must write down two pre-lab calculations:

1. Using Eq. (9), compute the suspension voltage as a function of the number ! of electron charges on the sphere. Assume that the radius of the sphere is as given by the manufacturer. The separation of the plates ! is approximately 6 mm. The exact value is marked on the sample chamber. Once you have !! = !!(!), a measurement of the suspension voltage immediately gives you the number of charges on the sphere.

2. Compute the fall velocity !! , assuming that the radius and density are as given by

the manufacturer. Use Eqs. (3), (4), and (5). Knowing !! , you can compute how long a droplet should take to fall a given distance on the TV monitor. Occasionally, two latex spheres stick together, producing a non-spherical mass which is useless for measurement. This double-sphere is heavier and falls more quickly than a single sphere. You should verify that your sphere is falling at the correct speed, lest you waste time taking useless data on a double-sphere.

Now, on to the experiment Begin by leveling the sample chamber to ensure that the applied electric field is exactly vertical. DANGER! Before making any adjustments on the sample chamber, be certain that the high voltage is turned down and the voltage switch is OFF. 500 volts will produce a shock that you will remember for a long time. Place the small metal tripod table (stored in a small gray wooden case) on the top plate of the sample chamber and then place a spirit level on the tripod table. Level the chamber by adjusting the three leveling screws on the base.

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Calibrate the scale on the TV screen by pointing the camera toward the 1mm calibration scale. The camera is focused by adjusting the distance between the camera and the target. Do NOT attempt to focus by turning the lens on the camera. The next task is to focus the camera on the sample volume and adjust the light source. Open the sliding lid on the top plate and insert a small wire into the sample chamber. Position the lamp for maximum illumination of the wire and focus the camera by adjusting its position with the focus screw on the camera mount. Also check that the height of the camera is such that the sample chamber is centered in the field of view and neither the top nor bottom plate are visible on the monitor. Place the nozzle of the atomizer directly over the hole on the top plate and give the atomizer one or two gentle squeezes. Close the sliding lid so the sample chamber is sealed. You should see several spheres, appearing as points of light on the TV monitor, slowly falling. Out-of-focus spheres will appear as dim blobs. You can adjust the focus with the focus screw (NOT the lens, remember). If you do not see any spheres, it is probably because the light source needs adjusting. Try tilting or rotating the light slightly for maximum illumination of the spheres. In viewing the spheres on the TV monitor scale, you will encounter considerable parallax. The glass TV screen is quite thick and has a curved surface. To avoid parallax error, you must position your head so that your line of sight is perpendicular to the glass surface. You can do this by observing the reflection of your head in the monitor screen and positioning yourself so that the reflection of your eye coincides with the sphere you are observing. The spheres take about 60 seconds to fall through the field of view of the monitor. Set the voltage around 50V and flip the on/off switch and the reversing switch to see the effect on the spheres. Spheres with large charge (q 10 e) will move very quickly when the field is applied. Look for a sphere that responds relatively slowly, i.e. one with a small charge. Spheres with one or two charges will require a very high voltage to reverse. You can get a quick estimate of the number of charges on a sphere by measuring the suspension voltage (pre-lab question 1 above). If the number of charges is greater than 10, do not take data on that charge number. (If the charge number is too high, it can be lowered by using the UV light source and the procedure described below.) Data on spheres with very high charge numbers (! > 15!) are generally useless, because your data will not be precise enough to determine the exact charge number. Even sloppy data with a 15% uncertainty will allow you to determine whether the charge number is 3 or 4 (3 to 4 is a 30% change). But even very precise data with an uncertainty of 3% cannot allow you to determine whether the charge number is 49 or 50 (49 to 50 is a 2% change). With the field off, all the spheres should fall at the same rate. Any sphere that falls faster than the others is certainly a double-sphere and should be rejected. Adjust the voltage so that the rise time is 30 - 60 seconds, i.e. slow enough to time precisely with the stopwatch. Before taking data, time the fall of your chosen sphere through a fixed distance and compare with your computed fall time (pre-lab question 2 above). If the fall time is

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okay and the number of charges is less than 10, proceed to measure at least 5 rise times and 5 fall times. You will use the averages of the trials in your calculations. You will repeat these measurements for each new charge and/or new sphere. Also, for each charge, carefully measure the suspension voltage, !!. The spheres charge can be of either sign, so always record the sign of the applied voltage. The switch box is wired so that a sphere with a positive charge will rise when the polarity switch is in the POS position. A negative sphere will rise when the switch is in the NEG position. With the sphere suspended, you will notice that it does not remain perfectly stationary, but jitters about slightly. This is Brownian motion, due to the incessant and random pounding of air molecules on the sphere. This Brownian motion is the primary cause of the spread in your rise and fall times. You can change the charge on your sphere using the ultra violet light located behind the sample chamber. The Ultra Violet light can be damaging to your eyesight as well as your skin so precautions to minimize exposure to this light should be taken. The source is enclosed inside a cardboard box and this should not be removed, nor should you look directly at the source through the chamber. The UV light produced by the UV source will ionize the surrounding air, producing free electrons and positive ions. To change the charge on a sphere, start with the sphere near the top of the TV screen, turn on the UV lamp and allow the sphere to fall down inside the chamber. As the sphere falls, it sweeps through a volume of air and is likely to pick up an ion or electron, changing the magnitude and possibly the sign of its charge. After allowing the sphere to fall for 5 or 10 seconds, switch the field back on to see if the charge changed. Remember to start this procedure with the sphere near the top of the monitor so that there is plenty of time before the sphere falls below the monitor view. If the spheres charge happens to change to zero, you will need time to use the wand again and recharge the sphere. If your sphere has a high charge, use of the wand almost always reduces the charge number to between +10 and -10. Using the ultra violet light source to change the spheres charge, get data (rise times, fall times, and suspension voltage) on up to 5 different charges with the same sphere (and remember: charge numbers must be 10 or less). Then charge spheres and repeat. Your goal is to get full data for at least 10 charge/sphere combinations using at least 2 different spheres. Full data means a minimum of 5 rise times, 5 fall times, and the suspension voltage every time you change the charge and/or the sphere. The fall speed should be a constant for a given sphere, since the fall speed depends only on the sphere radius. In your analysis, use all the fall times for a given sphere to get the best value of the fall speed for that sphere and the best value of the radius ! [Eq. (6)] and the drag coefficient ! [Eq. (3)]. Check that the compound radius is close to the manufacturers value. (Do not use the manufacturers value of ! in your calculations! Use your values of !, one for each new sphere. The manufacturers ! is used only as a check on your results.) For each new charge, compute the charge on the sphere using Eqs. (8) and (10), which provide two independent determinations of the charge.

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Make a histogram of the charges you measured, as in the figure below. If the measured charges appear to be multiples of a minimum charge (!) then you can safely identify the charge number !.

With your data, make a plot of sphere charge ! vs. number of charges !. (Remember, ! is an integer!). Recall that ! and ! can be of either sign and that, by convention, the symbol ! represents a positive charge. Hence, an electron has charge ! = !, corresponding to ! = 1. Your plot of ! vs. ! should have at least 20 points, resulting from the 2 independent determinations of ! from each of the 10 different charge/sphere combinations. The slope of the plot (! vs. !) is the charge !. Use the Mathcad program linfit.mcd to determine the slope and the uncertainty of the slope. The uncertainty of the slope of (! vs. !) gives the uncertainty in the charge e due to random measurement errors only. This random error must be combined with an estimate of the systematic error to get the total error. In this lab, we will determine the uncertainty due to systematic errors numerically, rather than analytically. Ordinarily, if one has some computed quantity ! which is a function of several experimentally determined quantities !,!, each with a systematic uncertainty !x, !y, , then the systematic uncertainty in ! is computed from

!!!"! =!"!"

!"!+ !"

!" !"

!+ .

In this experiment, the computed charge of ! (from which we get ! = !/!) is a function of several variables (!, !, ) with systematic uncertainties. However, a few moments contemplation of eqns. (3), (6), and (8) quickly leads to the conclusion that computing the necessary derivatives !"

!", !"!"

, is an extremely messy task. Fortunately, we can use Mathcad to compute numerically the systematic error in ! due to each of the variables. For instance, consider the error in ! due to the error in !, !!! =

!"!"

!". To compute this numerically, simply change the value of ! in your Mathcad document to its extreme (maximum or minimum) value, update the document, and observe how much the compute ! changes. The change is !!!. Repeat this procedure for each variable and add the errors in quadrature to get the total systematic error,

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!!!"! = !!!! + !!! ! + = ! !!!"! .

In the last equation, we are assuming that there is no uncertainty in !. Summary of Important Numbers Density of latex spheres: ! = 1.050 0.005 g/cm3 Diameter of latex spheres: 1.070 0.005 !m [used to check your computed !, Eq. (6)] Separation of plates: ! = (marked on the apparatus) Viscosity of air: ! = [1.828 x 10-5 + (4.8 x 10-8) ! 23 ]N sec/m2 Acceleration of gravity: g = 9.796 m/sec2 (latitude 40, altitude 1 mile)

One last helpful hint: Mathcad has a default zero tolerance of 15, which means any number less than 10-15 is rounded to zero. To change this in your Mathcad document, make sure that no regions are selected, then click on Format, Result, Tolerance Questions

1.-2 (See two questions, middle of page 5)

3. Archimedes Principle states that the buoyant force of an object immersed in a fluid is equal to the weight of the displaced fluid. The density of air at Boulders altitude is !!"# 0.0011 g/cm3. Why is the buoyant force on the latex spheres negligible in this experiment? (Simply saying that the buoyant force is small is not an acceptable answer. The question is: small compared to what?)

4. Prove Eq. (6).

5. The drag force on the spheres is given by Stokes Law, Eq. (1), multiplied by a

correction factor !

!! !! ! . What is the size of the correction factor for our latex

spheres?

6. The transition from laminar (smooth) flow to turbulent flow in a fluid is determined by the dimensionless Reynolds number, defined as ! = ! ! !

!, where ! is the

density of the fluid, ! is its viscosity, ! is the speed of flow, and ! is a length which characterizes the geometry under study. For the problem of a sphere moving

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through a fluid, ! is the spheres diameter. Fluid flow becomes turbulent when the Reynolds number exceeds a critical value, which lies in the range 1000 - 4000.

Consider a plastic sphere of density 1.0 g/cm3 in a free fall through air. Roughly, what is the largest diameter sphere which falls with purely laminar flow? Since this is a rough calculation, you can ignore the correction factor in Eq. (3) and assume that Eq. (3) is simple ! = 6 ! ! !. Give the answer for the diameter both in !m and mm.

7. Show that, for this experiment, a plot of charge ! vs. charge number ! has the slope

!.