Millikan's oil drop experiment: measuring the charge of the electron

A human hair is about 70 micrometres thick. Now picture an oil drop thirty-five times smaller than that: about 2 micrometres across. You cannot see it with the naked eye, but shine a strong light on it from the side and look through a microscope, and it shows up like a star glowing against a dark background. Between 1909 and 1913, at the University of Chicago, Robert A. Millikan held these “stars” still between two metal plates and used them to measure one of nature’s most basic numbers: the charge of the electron. In this article we explain the idea behind the experiment, the apparatus, the calculation and the parts that are still debated.

What was he trying to find?

In 1897 J. J. Thomson had shown that cathode rays are made of tiny negatively charged particles, and he had measured the ratio of their charge to their mass. But the charge and the mass were not known separately. There was also a bigger question: can electric charge be split into ever smaller pieces, like a liquid, or is there a smallest packet that cannot be divided?

Millikan’s answer was the second one. Every charge he measured came out as a whole multiple of a single number. Today we call that number the elementary charge and write it as $e$. The electron’s charge is $-e$ and the proton’s is $+e$.

The apparatus: two plates, an atomizer, a microscope

At the heart of the apparatus are two flat metal plates, lying horizontally with a small gap between them. In Millikan’s 1913 paper the plates were about 22 centimetres across, with a gap of roughly one and a half centimetres. When the plates are connected to a voltage supply, a uniform electric field forms between them:

Here $V$ is the voltage between the plates (in volts) and $d$ is the distance between them (in metres).

The other parts of the apparatus:

• Atomizer: It works like the pump on a perfume bottle; squeezing the rubber bulb sprays a very fine mist of clock oil. Some of the drops pick up a charge by friction as they are sprayed. A few drops drift through a small hole in the top plate into the space between the plates.
• Light source: It lights the drops from the side. Millikan passed the light through a water cell so that its heat would not stir the air.
• Microscope with a scale: The eyepiece has horizontal lines. A stopwatch measures how many seconds the drop takes to cross the space between two lines.
• X-ray source: It ionizes the air between the plates. When the drop picks up one of these ions its charge changes, so different charges can be measured with the same drop. Millikan used X-rays and radium for this.

The drop’s radius: Stokes’ law

The first problem with the calculation is this: the drop is so small that we cannot measure its radius directly. Millikan found it from the speed at which it falls.

With the voltage off, the drop falls under gravity alone. The air pushes back on the falling drop with a drag force proportional to its speed. For a small sphere this force is given by Stokes’ law:

Here $\eta$ is the viscosity of air (about $1.8 \times 10^{-5}$ Pa·s), $r$ is the drop’s radius and $v$ is its speed. Because the drop is so light, this force balances its weight in less than a thousandth of a second, and the drop then falls at a steady speed, its terminal speed. At that point the weight equals the drag:

$\rho$ is the density of the oil. So timing how long the drop takes to fall between two lines is enough to find its radius, and therefore its mass. We explain air drag and terminal speed in more detail in our article on free fall.

Millikan also noticed that Stokes’ law drifts slightly for very small drops: the gaps between air molecules become comparable to the size of the drop. So he applied a correction to his results (known today as the Cunningham correction).

Holding the drop still: qE = mg

Now let’s switch the voltage on. If the top plate is positive and the bottom plate negative, a negatively charged drop feels an upward electric force: $F = qE$. If we raise the voltage slowly, we reach a value at which the drop stops in the microscope. At that moment the two forces balance:

The mass comes from Stokes’ law, $d$ from a ruler and $V$ from the voltmeter. That gives the drop’s charge $q$.

In Millikan’s actual measurements the drops were usually not held exactly still; he measured how fast a drop rose with the field on and how fast it fell with the field off, and worked out the charge from those two speeds. The idea is the same, but measuring speeds is more precise than catching the exact balance. The simulation below uses the balance method because it is easier to follow.

Try it in the simulation

The apparatus below is a simplified model of Millikan’s. The front of the top plate is drawn cut away so you can see into the gap. Set the voltage with the slider or by typing it in the box: if the drop rises in the microscope, lower the voltage; if it falls, raise it. When the drop hovers, press “Measure and record”. Then ionize the drop or spray a new one and measure again. The measured charges pile up in the histogram at multiples of e.

Worked example

Let’s work through the simulation’s starting values. The plate gap is $d = 6$ mm and the oil density is $\rho = 870$ kg/m³.

1. Fall speed: With the voltage off, the drop falls 0.5 mm (five scale divisions in the microscope) in about 4.74 seconds. $v_0 = 0.5 \text{ mm} / 4.74 \text{ s} \approx 1.054 \times 10^{-4}$ m/s.
2. Radius: $r = \sqrt{9 \cdot 1.8\times10^{-5} \cdot 1.054\times10^{-4} / (2 \cdot 870 \cdot 9.81)} \approx 1.00 \times 10^{-6}$ m, or 1 micrometre.
3. Mass and weight: $m = \frac{4}{3}\pi r^3 \rho \approx 3.64 \times 10^{-15}$ kg, $mg \approx 3.58 \times 10^{-14}$ N.
4. Balancing voltage: The drop stops at 446 V. $E = 446 / 0.006 \approx 74{,}300$ V/m.
5. Charge: $q = mg / E \approx 4.81 \times 10^{-19}$ C.
6. How many e? $q / e = 4.81 / 1.602 \approx 3.00$. This drop carries three extra electrons.

In a real measurement the small error of the stopwatch makes the result come out as something like 2.98 or 3.02. The common divisor of many measurements, however, gets close to $e$.

Why is every charge a multiple of e?

Millikan made hundreds of measurements and never found a charge such as $1.5e$ or $2.3e$. When the same drop caught an ion from the X-rays, its charge also changed each time by exactly $e$ or $2e$. This was direct evidence that electric charge is quantized: it comes in packets that cannot be divided.

In 1913 Millikan found about $1.592 \times 10^{-19}$ C for $e$. The value accepted today is:

Since 2019 this number has had exactly this value by definition in the SI system of units. Millikan’s result was about 0.6% smaller than today’s. Most of the difference came from the value he used for the viscosity of air, which was slightly too low.

For this work and his measurements on the photoelectric effect, Millikan received the 1923 Nobel Prize in Physics.

Harvey Fletcher’s contribution

One name stayed in the background of this story for a long time: Millikan’s doctoral student Harvey Fletcher. The first attempts used water drops, and because water drops evaporate and shrink quickly, a measurement could last only a few seconds. According to Fletcher’s published memoir, the idea of using oil for the drops, and the first oil drop apparatus, were his. Because oil does not evaporate, a single drop could be watched for hours.

As Fletcher tells it, Millikan proposed that of the papers coming out of their joint work, the one on the elementary charge would carry only Millikan’s name and the one on Brownian motion only Fletcher’s; Fletcher accepted this for his doctoral thesis. Fletcher asked for the memoir to be published after his death; it appeared in Physics Today in 1982.

The debate about data selection

In his 1913 paper Millikan wrote that the 58 drops he presented were not a selected group but all the drops he had measured over 60 days. Yet the laboratory notebooks, which the historian Gerald Holton studied in the 1970s, show that more than a hundred drops were measured in the same period and that some were left out with notes such as “very low” or “something wrong”.

This is sometimes summed up as “Millikan made up his data”; that is not true, but the story holds an important lesson. The physicist and historian Allan Franklin took the drops left out of the notebooks into account too and showed that the value of $e$ hardly changes; only the uncertainty grows a little. For many of the excluded drops there were also experimental reasons (a drop too small, a drifting voltage, a problem during the measurement). Even so, the phrase “all the drops” in the paper did not reflect what actually happened.

At the time, Felix Ehrenhaft in Vienna claimed to have measured “subelectron” charges smaller than $e$. This dispute, and the result Millikan expected, are thought to have influenced how he selected his data. Independent measurements in the following years confirmed Millikan’s result and did not support Ehrenhaft’s subelectrons.

The lesson for science is this: stating clearly why you left a measurement out matters as much as the result itself.

Summary

• Millikan measured the charge of oil drops floating between two horizontal metal plates (1909–1913).
• The drop’s radius comes from its fall speed with the field off, using Stokes’ law.
• When the drop hovers, $qE = mg$, which gives $q = mgd/V$.
• Every measured charge came out as a whole multiple of $e \approx 1.602 \times 10^{-19}$ C: electric charge is quantized.
• The idea of using oil was Harvey Fletcher’s; Millikan’s data selection is still debated, but his result was confirmed by later measurements.

For more, take a look at our physics articles.