*The Millikan Oil Drop Experiment *

I. Introduction

II. Theory

III. Procedure

**I. Introduction**

In 1909, Robert Millikan along with Harvey Fletcher and other physicists introduced a cloud of oil droplets via an atomizer into a cylindrical chamber. At the bottom of the cylinder a concentrated number of droplets entered a small hole in the center of a metal disk, and floated down under the influence of gravity towards another metal disk, where they were viewed through a microscope. Determining the terminal velocity of a drop allowed for the calculation of the drop's radius. A potential difference was then created across the two metal disks of varying voltage, and the drop - if it had a net charge - could be seen to change direction and begin to rise. Measuring the new terminal velocity allowed Millikan to calculate the value of the charge on the oil droplet. This was repeated for several different charge values, which, upon inspection, were discovered to be integer multiples of a fundamental number - the charge of the electron.

**II. Theory**

The Millikan oil drop experiment investigates the charge on a single droplet of oil. When a droplet falls through the air, it is acted on by three forces, the gravitational force, a buoyant force, and a drag force, as shown in Figure 1.

The net force is F_{net} = (F_{b}+F_{d} - F_{g}).

The force due to gravity F_{g} is

F_{g} = mg = (4/3)πa^{3}ρg,

where m, a, and ρ are the respective mass, radius, and density of the droplet and g is the acceleration due to gravity.

The buoyant force as given by Archmmede's principal is

F_{b} = m_{air}g = (4/3)πa^{3}ρ_{air}g.

Here, ρ_{air }is the density of the air displaced by the
droplet.

The drag force given by Stoke's law is

F_{d} = -kv = 6πrηv,

where k is the drag coefficient, η is the dynamic or absolute viscosity
of air, and v is the velocity of the droplet. The dynamic viscosity of
air is a function of temperature and is computed using Sutherland's
formula (Crane, 1988): η = η_{0}(a/b)(T/T_{0})^{3/2},
where η_{0} is the reference viscosity in centipoise at
reference temperature T_{0}, C is Sutherland's constant, a =
0.555T_{0}+C, and b = 0.555T+C. For standard air, Sutherland's
constant is 120. For a temperature of 20°C, this gives η a value of
1.842∙10^{-5} Pa∙s.

For objects with radii on the order of the mean free path of air
molecules, the viscosity of air, η, must be multiplied by the correction
factor η_{eff }= (1+b/pa)^{-1}. The new term η_{eff}
is the effective viscosity, where b is a constant, p is the atmospheric
pressure, and a is the radius of the drop. The value for b is 8.20∙10^{-3
}Pa∙m. The drag force becomes

F_{d} = 6πrηv/(1+b/pa).

When the droplet reaches terminal velocity, the net force is zero:

F_{net} = (4/3)πa^{3}(ρ')g - 6πaηv/(1+b/pa) = 0,
where ρ' = ρ-ρ_{air}.

This can be solved for a to find the radius of the droplet experimentally:

a = sqrt[(b/2p)^{2} + 9ηv_{g}/2g(ρ')] - b/2p.

Under the influence of an electric field, if there is a net charge on the droplet, there will be an electric force.

In the case that the electric force causes the droplet to rise, the forces will be as shown in Figure 2.

The resultant net force becomes

F_{net} = (F_{E} + F_{b} - F_{g} - F_{d})
.

The electric force is

F_{E} = qE = qV/d.

When the drop reaches terminal velocity v_{E} under the
influence of the electric field, the net force is again zero.

Solving for k from the freefall terminal velocity gives k = (4/3)πa^{3}ρ'g/v_{g}

where v_{g} is the terminal velocity during freefall. Solving
the net force under the influence of the electric field for the charge
q, and substituting in the expression for k allows one to calculate the
charge on the droplet:

q = (d/V) (4/3) πa^{3}ρ'g {sqrt[(b/2p)^{2} +
9ηv_{g}/2gρ'] - b/2p}^{3 }(v_{g}+v_{E})/v_{g}.

**III. Procedure**

The oil drop simulation is very similar to the actual oil drop experiment, and will be performed in the same manner. Unlike the actual experiment however, in which the various velocities must be determined by using a stopwatch, the simulation will simply record the velocities with the click of a button.

To begin, click the 'Spray Oil' button. You will notice a cloud of oil drops falling very slowly, at slightly different speeds. Choose the drop that you wish to measure and click it.

1. Measure the free-fall velocity of the drop. This can be done with a stopwatch by hand, or by simply clicking the 'Capture' button. The freefall velocity will then be displayed to the right.

2. Move the Voltage slider back and forth, observing the drops. An appropriate drop has a velocity that is significantly influenced by the electric field, indicating a large number of charges. A different drop can be selected at this time if desired (after changing drops, repeat Step 1). Measure the vE, the velocity when the electric field is present, again by using a stopwatch or by clicking the 'Capture' button.

3. To calculate the charge, plug the respective velocities into the formula for q to solve for the charge on the drop. This is also achieved by simply clicking on the 'Calculate Charge' button. The value is recorded in the table to the right.

4. To change the charge on the droplet, click the 'Ionize' button. This is representative of allowing the alpha particles of the decay of barium or thorium to enter the viewing area and strip electrons from the drop. When this happens, the plotting panel will briefly flash red.

5. Repeat steps 2 - 3 to determine subsequent charges.

6. Click the Save File button to save the list of charges to a text file. This file will be exported to MatLab to be analyzed. Instructions for this can be found in the MatLab documentation.