Millikan's Math

Once the drop was suspended, Millikan knew that the force up (caused by the electric field) must be equal to the pull down (caused by gravity). Thus:

Millikan knew (thanks to Isaac Newton) that the force of gravity on an object depended on the mass of the object and the gravitation force of the planet (Earth, obviously). So:

where m = mass and g is the gravitation acceleration constant. He further knew that he could determine the mass of the drop using the volume of the drop* and the density of the oil.

Millikan also knew that the force of the plates on the drop depended on the electric field strength (which is voltage over distance) and the charge on the drop. Specifically, he knew that:

or, simplified,

Setting these two equations equal gave Millikan this formula:

Since all values in the equation were known (or measurable) other than charge, Millikan could solve for the charge of the drop.

Unfortunately, Millikan did NOT know how many extra electrons any particular drop held. As a result, he had to perform the experiment over and over and over again. When he had gathered an immense number of data points, and knowing that each drop must hold an integer number of electrons, he was able to find the least common multiple of his charges and, thus, the charge on a single electron.

Then, knowing the charge on the electron, Millikan was able to use Thomson’s charge to mass ratio to also find the mass of an electron.

*Millikan allowed the drop to fall without the electric field. The drop quickly reached terminal velocity. By measuring the terminal velocity of the drop and using an equation involving the coefficient of friction of air, Millikan could find the radius of the drop. Then, assuming the drop was essentially spherical, he could determine the volume of the oil in the drop.