Millikan's Oil-Drop Experiment

Introduction

Robert Millikan performed an experiment, crucial to modern physics since in the experiment he established that charge is quantized or comes in "bundles" of irreducible units.  He was awarded the Hobel Prize for physics in 1923 for this work.  The smallest unit of charge is e=1.6 × (10^(-19)) Coulombs.  (By the way, the Coulomb is defined through the Ampere current unit such that 1 Amp=Coul/Sec.)  Here you repeat Millikan's famous experiment to gain an appreciation for some of the difficulty of the work of experimental physics.  Since the smallest unit of charge (the charge of electron) is so small, the force on small droplets must be observed otherwise the gravitational force would dominate the electric force.

The Apparatus

Millikan's apparatus is basically a capacitor and d≃7.6 mm is the distance between the plates.  (This distance can be determined accurately by measuring the thickness of the plastic spacer.)

Very small oil drops are introduced through a hole in the upper capacitor plate with a sprayer and a single drop is picked by the experimenter for further study.  The radius of the drop is about a micron = (10^(-6)) m and the precise radius is determined next.

Experiment #1:
Falling Drop with Zero Field:

The field is turned off in this experiment and the steady velocity of fall of the drops is observed.  In the circumstance of constant fall velocity (Subscript[v, s]), the downward force of gravity, mg equals the upward force of air resistance k(Subscript[v, s]):

Thus we have the equation

mg = (Subscript[kv, s])

where g=9.8 m/sec is the acceleration of gravity, m is the mass of a droplet of oil, (Subscript[v, s]) is the constant velocity and k is a constant (see below).

Aside:  The measuring scale viewed through the telescope has major divisions 0.5mm apart and minor divisions 0.1 mm apart.  For example, it might take a drop t=19.3 sec to fall a distance x=0.5 mm so that a typical steady state velocities of fall (the fall rate) (Subscript[v, s])would be 2.9 ×(10^(-5)) m/sec. (Make sure to convert to S.I. units.)

x = 0.5*(10^(-3));
t = 19.3;
(Subscript[v, s]) = (x/t)

The force of air resistance k(Subscript[v, s]) is sometimes called Stoke's Law and the force increase linearly with the velocity (Subscript[v, s]).  The constant k is given by

k = 6  πηa

where a is the radius of the drop of oil and η is the shear viscosity of air.  The shear viscosity η  depends upon the temperature of the air so it is important to measure the temperature T during your experiment (see the attached Table of η versus T).  A typical value is η=1.8×(10^(-5))N-s/(m^2) in S.I. units.

Finally the mass of the droplet can be written

m = ρ ((4/3) π(a^3))

since Ω=(4/3) π(a^3) is the volume Ω of a spherical droplet of radius a and ρ=886 Kg/(m^3) is the density of the oil used for the droplets.

Homework #1

Using equations (1), (2) and (3) we can determine the numerical value for the radius a of a typical droplet.  Check the calculation below and show that the formula is correct.

ρ = 886. ;
η=1.8*(10^(-5));
a = (Sqrt[((9 η (Subscript[v, s]))/(2 ρ g))])

What is the unit of the radius a?  Calculate the mass m of the drop.  What are the units of this mass?

m = ((4 π (a^3) ρ)/3)

The weight mg of a typical droplet in Newtons is

m g

Experiment #2:
Droplet at Rest-Determine the Charge

The experimenter next adjusts the voltage until the same single drop is stationary (does not move).  The downward force of gravity, mg is just balanced by the electric field force qE thus

mg  =  qE

There is no air resistance force k(Subscript[v, s]) because (Subscript[v, s])=0  for a stationary drop.  Make sure you read the voltage Φ on the digital meter, then the electric field E is given by

E = (Φ/d)

where the distance between the plates is about d≈7.6 mm and a typical voltage is Φ=205 volts.

Homework #2

Combine equations (4) and (5) to obtain the charge q is Coulombs

d = 7.6*(10^(-3));
Φ = 205. ;
g = 9.8;
q = ((m g d)/Φ)

Difficulty of Static Measurement

The procedure above assumes the voltage can be adjusted so that the drop will be at rest.  However, it is difficult to visually determine if a drop is at rest or just moving very slowly.  Also, if the net force is zero, the drop can still move at a constant speed (if it initially has an initial speed).  For these reason, it is better to do the measurement of the charge a different way.

Droplet Moving Upward with Constant Velocity

Experiment #2 is replaced by one where an electric field has a large enough size to cause the droplet to move upward with a constant velocity.  In this case, there are three forces:  the gravitation force mg acts downward, the electric field force qE acts upward, and the air resistance force kV acts opposite the direction of motion (which is downward in this case).

The equation relating the above forces is

qE = mg + kV

As before, the electric field E is given by the voltage Φ

(Subscript[E, 0]) = (Φ/d)

but now (Subscript[Φ, 0])=385 Volts so that

(Subscript[Φ, 0]) = 385;
d = 7.6*(10^(-3));
(Subscript[E, 0]) = ((Subscript[Φ, 0])/d)

Also, the upward speed V is calculated using

X = 0.5*(10^(-3));
T = 22.2;
V = (X/T)

Finally, from equation (6)

k = 6 π η a;
q = ((m g + k V)/(Subscript[E, 0]))

Homework

Suppose in Experiment #3 the electric field is downward, then you can still use the data.  Write the equation that balances the forces in this case and remembere that the air resistance kV is always opposite the direction of motion.

Laboratory Exercise

Select a drop and make five measurements according to the protocol of Experiment #1 and average your result for the radius a and mass m.  Then take five measurements according to the protocol of Experiment #2 and determine the charge q on the drop.  Average your results.  Calculate the average value of q and the sample standard deviation.  Is the charge on the drop an integral multiple of e the charge on an electron?

Repeat the above for a second drop.

Graph useful for finding the viscosity of air inside the Millikan Oil Drop chamber once you know the temperature.

Table for finding the temperature inside the millikan oil drop chamber: