How did the values of charge-to-mass ratio of an electron and charge of an electron(Millikan's oil drop experiment) come?

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Solution

The experiment entailed observing tiny charged droplets of oil between two horizontal metal electrodes. First, with zero applied electric field, the terminal velocity of a droplet was measured. At terminal velocity, the drag force equals the gravitational force (due to the mass of the electrons), and these depend on the radius in different ways, so that the radius of the droplet, and therefore the mass and gravitational force, could be determined (using the known density of the oil).
Then an adjustable voltage was applied between the plates to induce an electric field, and the voltage was adjusted until the drops were suspended in mechanical equilibrium, indicating that the electrical force and the gravitational force were balanced. Now using the known electric field, Millikan and Fletcher could determine the charge on the oil droplet. By repeating the experiment for many droplets, they confirmed that the charges were all small integer multiples of a certain base value, which was found to be approximately $1.602 \times 10^{- 19} C$ . They proposed that this was the (negative of the) charge of a single electron.

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