The apparatus

Robert Millikan’s design is just a uniform electric field, which is a pair of parallel plates that lie horizontal with large potential difference. Then the oil drops are dropped in to the plates and the drops are suspended between the plates. By changing the voltage the oil drops can be made to rise and fall. A ring of insulating material is used to hold the plates together. The plates have four holes cut into them and three have a bright light shining through them, and the other has a microscope placed through it.

The oil is a type that is usually used in vacuum apparatus. This is because this type of oil has an extremely low vapour pressure. Ordinary oil would evaporate away under the heat of the light source, so the mass of the oil drop would not remain constant over the course of the experiment. Some oil drops will pick up a charge through friction with the nozzle as they are sprayed, but more can be charged by including an ionising radiation source (such as an X-ray tube).

[edit] Method

Initially the oil drops are allowed to fall between the plates with the electric field turned off. They very quickly reach a terminal velocity because of friction with the air in the chamber. The field is then turned on and, if it is large enough, some of the drops (the charged ones) will start to rise. (This is because the upwards electric force F_E is greater for them than the downwards gravitational force W, in the same way bits of paper can be picked up by a charged rubber rod.) A likely looking drop is selected and kept in the middle of the field of view by alternately switching off the voltage until all the other drops have fallen. The experiment is then continued with this one drop.

The drop is allowed to fall and its terminal velocity v₁ in the absence of an electric field is calculated. The drag force acting on the drop can then be worked out using Stokes' law:

where v₁ is the terminal velocity (i.e. velocity in the absence of an electric field) of the falling drop, η is the viscosity of the air, and r is the radius of the drop.

The weight W is the volume V multiplied by the density ρ and the acceleration due to gravity g. However, what is needed is the apparent weight. The apparent weight in air is the true weight minus the upthrust (which equals the weight of air displaced by the oil drop). For a perfectly spherical droplet the apparent weight can be written as:

Now at terminal velocity the oil drop is not accelerating. So the total force acting on it must be zero. So the two forces F and W must cancel one another out.

F = W implies:

Once r is calculated, W can easily be worked out.

Now the field is turned back on, and the electric force on the drop is

where q is the charge on the oil drop and E is the electric field between the plates. For parallel plates

where V is the potential difference and d is the distance between the plates.

One conceivable way to work out q would be to adjust V until the oil drop remained steady. Then we could equate F_E with W. But in practice this is extremely difficult to do precisely. Also, determining F_E proves difficult because the mass of the oil drop is difficult to determine without reverting back to the use of Stoke's Law. A more practical approach is to turn V up slightly so that the oil drop rises with a new terminal velocity v₂. Then

[edit] Millikan's experiment and cargo cult science

Richard Feynman said in a commencement lecture he gave at Caltech in 1974^[5]

We have learned a lot from experience about how to handle some of the ways we fool ourselves. One example: Millikan measured the charge on an electron by an experiment with falling oil drops, and got an answer which we now know not to be quite right. It's a little bit off because he had the incorrect value for the viscosity of air. It's interesting to look at the history of measurements of the charge of an electron, after Millikan. If you plot them as a function of time, you find that one is a little bit bigger than Millikan's, and the next one's a little bit bigger than that, and the next one's a little bit bigger than that, until finally they settle down to a number which is higher.

Why didn't they discover the new number was higher right away? It's a thing that scientists are ashamed of - this history - because it's apparent that people did things like this: When they got a number that was too high above Millikan's, they thought something must be wrong - and they would look for and find a reason why something might be wrong. When they got a number close to Millikan's value they didn't look so hard. And so they eliminated the numbers that were too far off, and did other things like that. We've learned those tricks nowadays, and now we don't have that kind of a disease.

As of 2008, the accepted value for the elementary charge is 1.602176487(40) x 10⁻¹⁹ coulombs,^[6] where the 40 indicates the uncertainty of the last two decimal places. In his Nobel lecture, Millikan gave his measurement as 4.774(5) x 10⁻¹⁰ statcoulombs,^[7] which equals 1.5924(17) x 10⁻¹⁹ coulombs. The difference is less than one percent, but it is more than five times greater than Millikan's standard error, so the disagreement is significant.