In the book of Experiment in Modern Physics, by Melissinos, at page 2, it is given that:

2. The Millikan Oil Drop Experiment

In 1909, R. Millikan reported a reliable method for measuring ionic charges. It consists of observing the motion of small oil droplets under the influence of an electric field. Usually the drops acquire a few electron charges and thus conventional fields impart to them velocities that permit isolation of a drop and continuous observation for a considerable length of time; further, the mass of the oil droplet remains almost constant (there is very slight evaporation) during these long observation times.

But how did they know this in that time?

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    The Faraday constant was known. Avogadro's number was also more or less known, so there was a reasonable estimate of how large the electron charge should be. But read Millikan's paper! – Pieter Feb 4 at 8:02
  • What do you mean by "this" in your question "But how did they know this in that time?"? Could you please be more specific? – Geremia Feb 5 at 19:28
  • "this" $:=$ "that drops would acquire a few electron charges". – onurcanbektas Feb 6 at 3:29

I think the way Millikan deduced the number of electrons was, aside from the current state of knowledge as Conifold relates, by taking lots and lots of measurements (certainly that's what I did when repeating the experiment in undergrad). When you plot the calculated charge on each droplet, you see the points on the graph are not a continuum but rather they show up at quantized separations. Since the uncertainty in the electron count for a droplet containing, say, 100 electrons is likely to be several electrons' worth, concentrate on the points with the fewest quantized charge units. [edit: to clarify, I mean the error bars on such drops are several electron charge units wide] From that you can make a pretty good guess there's no "fractional" charge that would occur if your estimate of the electron charge were twice the truth.
So, while the number of electrons in a given droplet may follow some statistical distribution, making use of the least ambiguous subset of all captured droplets leads to a more reliable answer.

One would think that in the second Melissinos and Napolitano would update on how "reliable" Millikan's method was. Even Wikipedia mentions that according to his own notes he "edited" the data he had to reduce the statistical error, see also Is Millikan's famous oil drop experiment a fraud?

In any case, the idea of electrons and ions was in circulation since 1890-s, when Lorentz's electron theory explained magneto-optics, the behavior of conductors and non-conductors, etc., using them. Thomson directly observed them in cathode (then Lenard) rays in 1897. Here is from Lorentz's Nobel lecture (1902) devoted to the electron theory:

"The theory of which I am going to give an account represents the physical world as consisting of three separate things, composed of three types of building material: first ordinary tangible or ponderable matter, second electrons, and third ether... The covibrating particles must, we concluded, be electrically charged; so we can conveniently call them "electrons", the name that was introduced later by Johnstone Stoney...

At the same time, a start was made on a general theory which ascribed all electromagnetic processes taking place in ponderable substances to electrons. In this theory an electrical charge is conceived as being a surplus of positive or negative electrons, but a current in a metallic wire is considered to be a genuine progression of these particles, to which is ascribed a certain mobility in conductors, whereas in non-conductors they are bound to certain equilibrium positions, about which, as has already been said, they can vibrate.

If we combine the results to which Zeeman's experiments lead with those which can be deduced from the colour dispersion of gases, on the hypothesis that it is the same type of electrons which is under consideration in both cases, we come to the conclusion that the charge of an electron is of the same order of size as the charge of an electrolytic ion. The mass, however, is much smaller... the principal thing is that, as we have remarked before, the electron is very small compared with the atom. The latter is a composite structure, which can contain many electrons, some mobile, some fixed."

Do not think that Einstein's 1905 theory immediately abolished ether, in 1900-s even Einstein himself occasionally used the ether talk. In any case, one could simply replace "ether" with "electromagnetic field" in most places. Millikan’s and Fletcher's apparatus had a parallel pair of horizontal metal plates separated by an insulator with a uniform electric field created in between. A fine mist of oil droplets was sprayed into a chamber above the plates. Oil drops were charged either through friction with the spraying nozzle or by ionising radiation from an X-ray tube.

According to Lorentz's theory, the charging was the result of a few extra electrons sticking to the droplet (too many would be repelled by the excess negative charge). By changing the voltage across the plates Millikan could make the droplets rise, fall or stand still. From the latter (he claimed) he determined the charges to be small integer multiples of $1.5924(17)×10^{−19}\,{\rm C}$, about 0.6% below the current value. Feynman describes the comical aftermath when subsequent experimenters gradually increased the Millikan's value to $1.602176487(40)×10^{−19}\,{\rm C}$:

"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..."

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    Interesting but I don't see this addressing the question. – Carl Witthoft Feb 6 at 12:57
  • @CarlWitthoft Isn't the question about the theory behind electrons attaching to droplets? – Conifold Feb 6 at 18:53
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    The milikan oil drop charge measurement is a standard physics lab experiment . I vaguely remember doing it back in 1959 using a microscope. The multiple charged droplets zoomed to the bottom very fast, and the few charges ones were drifting slowly enough to see differences in their velocity. Measuring accurateleyis a different problem. Here there is a measurement from current physics courses experiment web.pa.msu.edu/courses/2003spring/PHY192/… , page 10 – anna v Feb 7 at 5:05

Robert Millikan’s oil drop experiment presented the first measurement of the electron’s charge.

The first results came out in 1910, but the seminal work was a 1913 paper in the Physical Review.

Millikan reported a value for the fundamental electric charge that was within half a percent of today’s accepted value.

Backdrop:

J. J. Thomson discovered the electron in 1897 when he measured the charge-to-mass ratio for electrons in a beam. But the value of the charge and whether it was fundamental remained open questions.

Thomson and others tried to measure an irreducible electric charge by looking at clouds of water droplets. Using various techniques, they estimated the smallest charge that a droplet could hold, but the results were not entirely convincing because they relied on averages over many particles of various sizes.

“The evidence for a unitary charge was at the time very ambiguous,” says science historian Gerald Holton of Harvard University.

At the University of Chicago in the 1900s, Millikan and his graduate students realized that ramping up the electric field would disperse a water cloud, so that only a few droplets remained.

He decided to try isolating single droplets, but it soon became clear that single water droplets evaporated too quickly to make reliable measurements.

One of his students, Harvey Fletcher, found that long-lasting droplets could be made with a light oil that was used for lubricating clocks.

The oil drop experiment that Millikan and Fletcher designed had two chambers. In the upper chamber, an atomizer dispersed a fine mist of micron-sized oil droplets.

Individual droplets would fall through a pinhole into the lower chamber, which consisted of two horizontal plates, with one held 16 millimeters above the other.

The air in this chamber was ionized with x rays, so that ions or free electrons could be captured on the falling droplets.

A small window on the side allowed the experimenter to observe the droplets through a telescope. The droplets fell slowly enough—due to atmospheric drag—that the researchers could measure their downward speed by eye, using horizontal lines in the telescope. From this speed, they could estimate the size and mass of each droplet.

They then applied a high voltage across the plates and measured the upward speed of the droplet, to determine the electric force and ultimately the charge. Multiple measurements on a single droplet could be performed by repeatedly turning the electric field on and off.

The droplets had various amounts of charge on them (and they would often gain or lose charge during an observation), but the data showed that the charge was indeed quantized into integer multiples of a unit charge.

In 1910 Millikan published the first results of these experiments [1] (Fletcher was not included as an author, based on a deal the two struck [2]).

Millikan then made several improvements, including an empirical estimate of the drag forces. The culmination of this effort, reported in 1913, was a value of the fundamental charge with an error bar of just 0.2 percent.

The precision acquired was so great that “other experiments did not improve on his result until a decade later,” Holton says.

But Felix Ehrenhaft of the University of Vienna repeatedly challenged Millikan’s results, based on his own measurements of “sub-electron” charges on small metal particles. The dispute lasted for many years—known as the “Battle over the Electron”—but eventually most physicists sided with Millikan.

In more recent years, historians who have examined Millikan’s lab notes have said that he discarded some of the measurements to boost the evidence of a fundamental charge.

David Goodstein of the California Institute of Technology in Pasadena believes these accusations of fraud are unwarranted. He has analyzed the notes and says that Millikan excluded droplets because their observations were incomplete, not because their implied charge didn’t match his expectations [3]. “Millikan’s oil drop experiment is a classic example of outstanding physics done by one of the giants of his era,” Goodstein says.

Those scientists and other scholars who have carefully reviewed this case have failed to agree on whether Millikan was guilty of unethical behavior or "bad science" in the treatment and presentation of his data. One of the expressed opinions condemns Millikan on the simple basis of the fact that his published statement is at odds with what can be concluded from an uncritical examination of his laboratory notebooks.

Others exonerate Millikan on the basis of a careful analysis and interpretation of comments on the data that appear in the notebooks. In the opinion of these Millikan defenders, the assertion that all drops were presented in the paper refers to all of the data taken under those conditions when the apparatus was working properly. Some of the scientists who have commented on this case appear to permit Millikan much discretion in the use of his "scientific intuition" to decide which data to include or exclude. This latter view seems to be guided by the principle that any scientist who consistently gets what turns out to be the correct answer is doing "good" science.

For biographical information about Robert A Millikan and the history of the oil-drop experiment that will provide the context for this case we suggest that one can read:

"Robert A. Millikan," by Daniel J. Kevles, Scientific American, 240, pp 142-151 (January 1979).

"My Work With Millikan On the Oil-drop Experiment," by Harvey Fletcher, Physics Today, pp 43-47 (June 1982).

For a detailed analysis of the Millikan's work on the oil-drop experiment, including what he wrote in his laboratory notebooks see:

"Subelectrons, Presuppositions and the Millikan-Ehrenhaft Dispute" in The Scientific Imagination, by Gerald Holton (Cambridge, Mass: Cambridge Univ. Press, 1978).

http://www.onlineethics.org/cms/9726.aspx

https://en.wikipedia.org/wiki/Oil_drop_experiment

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