II. Millikan Oil Drop ExperimentWayne Saslow

Jorge D Morales

Fig. 1. Milikan Cross Section

Abstract—The experiment was devised by American physicistRobert A. Millikan in 1909. Millikan wanted to determinewhether electric charges occurred in discrete units which wereintegral multiples of some smallest charge e (which turns outto be carried by an electron or proton), and to measure thatcharge. The present apparatus uses nylon beads rather than theoil droplets used by Millikan, but in most other respects is thesame.

I. INTRODUCTION

The apparatus is shown in cross-section in Figure 1. Anatomizer is used to spray an aerosol suspension of tinyspherical nylon beads into an upper chamber. Most of thebeads receive a small net charge from friction as they emergefrom the nozzle of the atomizer. (Note that this is analogousto the charging of the belt on a Van de Graaf generator).Some of the beads fall through a hole in the center of a metalplate (upper electrode), so that they begin to fall through theair in the lower chamber.

Since the spheres are far too small to be seen by the nakedeye, an illuminator and a microscope are used to observethem (Figure 2). Each sphere is seen as a small point oflight. When the potential difference between the upper andlower electrodes is zero, the sphere falls slowly and steadilywith a terminal velocity, corresponding to the equilibriumbetween the downward force of gravity and the upwardviscous force of the air. (There also is an upward buoyantforce equal to the weight of the air displaced by the droplet,but it will be neglected because it is less than one-thousandththe weight of the droplet). The situation is similar to that of aparachutist falling through the air. When a potential differenceis applied to the electrodes, an additional electric force actsand the terminal velocity is changed (if the droplet is charged).

Fig. 2. Microscope View

As will be seen below, if the density of nylon and theradius of the droplet are known, and the terminal velocity ismeasured, with and without an applied electric field E, theforce of the electric field on the bead can be determined, andfrom this the charge q on the bead can be determined. Aseries of terminal velocity measurements will indicate groupsof q values that are seen to multiples of a smallest valuee, which is hypothesized to be the fundamental electroniccharge. The accepted value of e is 1.6× 10−19C.

A. Levitation: equilibrium between electric and gravitationalforces

First consider a droplet or radius R that is suspended atconstant vertical position by an exact balance between upwardelectric force and a downward gravitational force. Figure 3.

Now consider what happens to a solid sphere of radius R,if only a modest number of surface atoms have net charge.Assume that the surface charge density is . Then the potentialat the surface is:

V =(σ4πR2)

4πε0R=σR

ε0(1)

The electric field at the surface of the sphere is:

E =(σ4πR2)

4πε0R2=σ

ε0(2)

The significance of this electric field is that the limit tothe amount of charge that can be developed on an objectarises from its ability to ionize atoms of gas near the surface.The ionization process is a function of electric field. In dry

Fig. 3. Levitation

air, for example, the maximum surface breakdown field isEb ' 106V .

The maximum electric force that can be exerted on thesphere by an external electric field Eext is then:

Fe = (σ4πR2) · Eext = ε0EbEext · 4πR2 (3)

Now the gravitational force on the same sphere accumulatesfrom the force on all the matter within the sphere, so

Fg = ρ · 43πR3︸ ︷︷ ︸ ·g (4)

m

Since the gravitational force increases as R3 while themaximum electric force increases like R2, we need to workwith very small spheres in order to attain an equilibriumbetween these two forces.

If we knew the mass of the droplet, we could solve directlyfor the charge on the droplet. In practice, we will be using tinyplastic beads (instead of the oil droplets used by Millikan),but the size (and mass) of the beads varies significantly withinthe sample, so that for a precise measurement we would needto measure the mass of each bead. We can actually do thiswithin the Millikan experiment by observing the beads, notin equilibrium levitation, but in free fall in which the electricand gravitational forces are not quite in balance.

B. Free fall

Next consider a sphere of mass m, charge q, and radius rfalling under the influence of gravity only (acceleration ofgravity = g). As noted above, the speed of the sphere rapidly

increases until a constant terminal velocity v0 is reached.This happens when the weight mg of the sphere (minus thebuoyant force) is exactly equal (and opposite) to the frictionalforce exerted by the air. This problem was first considered bySir George Stokes, who derived an expression for the viscousforce Ff that acts upon a particle of radius r as it passesthrough a fluid with velocity v [1]:

Ff = 6πηrν (5)

The quantity η = 1.85E−5kgs/m is the coefficient ofviscosity of air at room temperature. Thus in the absence ofelectric field:

E = 0 : mg = 6πηrν0 (6)

Where ν0 is a positive number. . Note that the radius rof the sphere can be determined from this measurement ofv0 in the absence of electric field. The mass of the bead ism = 4

3πr3ρ. Note: you will need to find a value for the

density of nylon, either in a library source or the Internet. Wecan therefore solve Eq. 2 for the radius:

r = 3

√ην02ρg

(7)

Eq. 3 is Stokes Law, relating the radius of a sphericalbody to its terminal velocity as it falls in a viscous medium.(Where Ff = mg)

Stokes Law, however, becomes incorrect when the velocityof fall of the droplets is less than 0̃.1 cm/s. Droplets havingthis and smaller velocities have radii < 2µm, comparable tothe mean free path of air molecules. The derivation of StokesLaw assumes that the sphere radius is larger than this meanfree path. Since the velocities of the droplets used in thisexperiment will be in the range of 0.01 to 0.001 cm/s, thecoefficient of viscosity must be multiplied by a correctionfactor [2]:

ηeff =η

1 + bpr

(8)

where b = 6.17E−4mmHgcm is a constant,p = 760mmHg is the atmospheric pressure, and r isthe radius of the drop as calculated by the uncorrected formof Stokes Law. Substituting ηeff from equation (8) intoequation (7) and solving for the radius r gives:

r =

√(b

2p

)2

+9ην02gρ

− b

2p(9)

Fig. 4. Levitation

Fig. 5. Levitation

C. Electric and gravitational forces in equilibrium

Now suppose that you apply an electric field E, in adirection such to cause an upward force on the bead sufficientto cause it to move upward with a terminal velocity v+.Since the bead is now moving upward, the viscous force actsdownward (Figure 4). The equation of equilibrium is:

qE = mg + 6πηrν+ (10)

Now suppose that the sign of the E-field is reversed. Theviscous force now acts upward, Figure 5.

qE +mg = 6πηrν− (11)

where v− is the new (downward) terminal velocity. If Eq.6 is added to Eq. 7, we eliminate the gravity term and obtain:

qE = 3πηr(ν+ + ν−) (12)

But, since E = V/d, where V is the potential differencebetween the plates and d is the plate separation, this reduces to:

q =3πηdr(ν+ + ν−)

V(13)

In summary, a measurement of v0 and use of Eq. 5 leadsto a determination of r, and further measurements of v+ andv-, enable us to determine the charge q on the bead.

Fig. 6. Equipment

II. SETTING UP THE EQUIPMENT

NOTE! The power should be turned off when anyadjustments of the apparatus are being made.

1) Install the atomizer ring. This is done by unscrewingthe electrode housing set screws, removing the upperelectrode housing plate, placing the atomizer ring withopenings for the illuminator and microscope orientedproperly, replacing the upper electrode housing plate,and retightening the set screws. Insert the loose end ofthe narrow tube into the storage bottle.

2) Make sure that the color-coded D.C. voltage leads areplugged into the corresponding color-coded terminals onthe power supply.

3) Set the 3-way polarity switch to its mid-position, whichputs no voltage on the electrodes. Turn the on/off switchto on. The illuminating lamp should light.

4) Adjust the microscope by turning the focus-adjustingknob until an approximate mid-position is established.The scale divisions should be distinguishable from thebackground.

5) Set the electrode potential to 300 V using the voltageadjustment knob, but keep the polarity switch in its mid-position.

6) In order to use the atomizer, first loosen the set screws ofthe electrode housing so as to permit air to escape. Theninject nylon spheres into the chamber with the atomizerby covering the air hole of the spray bulb with a fingerand squeezing the bulb. The hole must be covered forsuccessful injection.

Spraying is usually difficult the first few times. It isrecommended that you make several test sprays beforeattempting to make measurements. While spraying nylonbeads into the apparatus, carefully look for the dots of lightin the microscope. Adjust the focus of the microscope untilyou can see them clearly. Note that beads continue enteringthe region between the plates for several seconds after thespraying has stopped. Practice until you are comfortable withthe microscope.

Obtaining a suitable bead will require some patience. Afterspraying a sample of beads into the chamber, wait for theturbulence to subside and for the fast-moving beads to moveout of the field of view. Select a small bead near the center ofthe field of view that is falling slowly, taking about 30 sec tomove 1 mm (the distance between two large divisions on thescale). Toggle the reversing switch between the up and downpositions and note whether the speed of the droplet changes.If it does, the droplet is charged and would be a suitablecandidate for measurement. Note that occasionally the speedof the droplet can change abruptly without your touching thereversing switch, indicating that the charge on the droplet haschanged by a discrete amount. Note that droplets are fallingwith various speeds. Selection of fall rate is the means bywhich you will determine the droplet size (using Eq. 5).

Choose a droplet that falls with about this rate with E = 0,and changes fall rate noticeably (but not too drastically) whenthe E-field is toggled on. The smallest changes in speed leadto the best measurements, since we are trying to measure thefundamental charge of a single electron.

With the equipment at hand your team will need to workin close coordination. One person is needed to observethe droplet and operate a stopwatch. A second person willrecord the stopwatch readings, the polarity of the E-field, andthe voltage reading. A third one will operate the atomizerand control the voltage. For good results it is essentialthat readings for E = 0, +, and − be made on the samebead, which means that the observer must keep the beadcontinuously in view.

III. PROBLEM 1LEVITATION

You are going to use the electric force to defy gravity. Inorder to accomplish this feat, you take into consideration afundamental difference between these two forces of nature.The gravitational charge (i.e. mass) of all matter has a singlesign, and gravitation is always attractive between objects. Butthe electric charges in matter have both + and sign, meaningthat the positive charges in nuclear matter attract the negativecharges of electrons, but like charges repel one another.

This property of opposite charges has the consequence thatatoms form electrically neutral building blocks of matter, andso almost all the matter in an object has net gravitationalcharge but no net electric charge. In general only some fractionof the atoms on the surface layer of an object can be givena net electric charge by either stripping off one electron oradding one electron.

A. ProcedureYou will be using nylon beads of 4µm diameter. You

should find the best value for the mass density of nylon, so

that you can estimate the gravitational force.

The spheres are given a small (random) amount of electriccharge as they are shot into the chamber through the atomizernozzle. The rubbing of each sphere against the side wallsof the nozzle and against one another is quite sufficient togive the spheres the amount of charged needed to be able tolevitate them.

In this problem, you will levitate a sphere, and make anapproximate measurement of its total charge by measuringthe electric field needed for levitation.

Rotate the roles of operator, observer, and data loggerduring the experiment so that each partner observes thelevitation. You should make a histogram of the chargemeasurements on successive spheres (at least 8), and see ifyour measurements can all be characterized as an integertimes a fundamental unit of charge, hence verifying thatelectric charge is quantized in matter.

IV. PROBLEM 2

MEASUREMENT OF RISING AND FALLING DROPS

Millikan did this classic experiment using droplets of oilsprayed from an atomizer. He was able to make quite smalldroplets (down to 1̃ µm) but the droplet sizes were notuniform. In order to make consistent measurements, he hadto simultaneously measure the droplet size and the chargestate. He devised the procedure described above, in which foreach droplet one measures the velocity for free fall and theterminal velocity when an external field is applied to makethe sphere rise or fall. You will follow his procedure to makesuch simultaneous measurements.

A. Procedure

1) Record the time to fall 1 mm for a single dropletunder the three conditions = 0, +, and −. Calculatethe corresponding velocities v0, v+, and v-, and thencalculate the charge q.

2) Repeat Step 1 for as many different droplets as you can.3) Plot a histogram of the observed charge values. They

should fall into equally spaced clusters, correspondingto multiples of a single fundamental charge e.

4) Select a well-behaved droplet and measure v+ for severaldifferent accelerating voltages V. Make a plot of v+ vs.V. Interpret the slope and intercept of this plot. Wouldyou have expected the slope and intercept to be the sameif you had selected a different bead?

5) Prepare a histogram of the distribution of charge stateson the beads that you observe. The friction chargingmechanism is statistical in nature. The distribution ofnumber of electrons on each bead should be a Poissondistribution. Find a reference on Poisson distributions,

calculate the mean charge of your distribution, and thencalculate the Poisson distribution that corresponds to thatmean. Compare to your histogram.

B. Cleaning up after the experiment

It is important that the atomizer ring and atomizer becleaned after the measurements have been completed.Disassemble the ring and electrode structure by reversingStep 1 of the setup sequence. Clean the electrodes and ringby washing with water and drying with tissue. Clean thepump by immersing the end of the small tube in a beakerof clean water and pumping several times. Withdraw it fromthe beaker and pump until it stops spaying water. Restore theapparatus to the state in which you found it.

REFERENCES

[1] Saslow, Wayne. Laboratory 4. Millikan Oil Drop Experiment. Jan 2009.http://visual.physics.tamu.edu/vp208/LabManual/Lab42005ATK.pdf

[2] L. Page. Introduction to Theoretical Physics. New York, Van Nostrand, ch 6.

[3] R.A. Millikan, The Electron. Chicago, University of Chicago Press, ch 5.