Millikan's Oil Drop Experiment: Quantizing Charge

The Core Idea: A Microscopic Balancing Act

At its heart, Millikan's experiment is a beautifully simple idea. Imagine you have a tiny, charged particle. If you place it in a uniform vertical electric field ($E$), it will feel an electric force, $F_E=qE$. If the particle's charge ($q$) is negative and the upper plate is positive, this force will be directed upward. This upward electric force can be used to counteract the downward force of gravity, $F_g=mg$.

If you could adjust the electric field perfectly to suspend the particle in mid-air, you would have a simple equilibrium:

$F_E=F_g$
$qE=mg$

From this, you could calculate the charge: $q=\frac{mg}{E}$.

This simple picture contains the essence of the experiment. However, it conceals a formidable practical challenge: How do you measure the mass ($m$) of a single, microscopic oil droplet? You cannot place it on a scale. This is where the true genius of Millikan's method is revealed.

From Idea to Measurement: The Role of Air Resistance

Millikan realized he could determine the mass of the droplet by observing its motion through the air. Instead of trying to perfectly suspend the drop, he analyzed its motion under two distinct conditions.

To understand the setup and the physical principles, let's watch a video that provides a fantastic narrative of Millikan's work, from its historical context to the core physics.

As the video explained, Millikan's procedure was a two-act play:

Act 1: Field Off - Measuring the Mass
First, he would let a single oil drop fall under the influence of gravity through the air. The drop would quickly reach a constant terminal velocity ($v_f$) where the downward gravitational force ($mg$) was perfectly balanced by the upward viscous drag force from the air. Using a relationship known as Stokes' Law, which connects the drag force to the drop's radius and velocity, he could calculate the drop's radius ($a$) and therefore its mass ($m=\frac{4}{3}\pi a^3\rho$), where $\rho$ is the known density of the oil. The unknown mass was no longer an obstacle.

Act 2: Field On - Measuring the Charge
Next, Millikan would turn on the electric field, creating an upward force $F_E=qE$. This force would cause the negatively charged drop to move upward with a new terminal velocity ($v_r$). In this situation, the upward electric force is balanced by two downward forces: gravity ($mg$) and the drag force (which now also points down). By measuring this new velocity, $v_r$, he now had all the information needed to calculate the charge, $q$, on the drop.

For a deeper look into the iterative process of refining this method, the provided reading [PDF] The Oil Drop Experiment: A Rational Reconstruction is an excellent resource, particularly the section "Millikan’s Early Experiments" (resource LINK, section 1).

The Climax: Evidence for the Atom of Charge

Calculating the charge on a single drop was a remarkable achievement, but it wasn't the most profound discovery. The crucial insight came when Millikan and his student Harvey Fletcher repeated this procedure for hundreds of different oil drops.

When they analyzed the results, they found something astonishing. The calculated values for the charge $q$ were not random or continuous. They all clustered around integer multiples of a single, smallest value.

Let's watch a short video that clearly explains and visualizes this final, crucial step of data analysis.

The data looked something like this (from Millikan's own paper):

This was the smoking gun. The charge $q$ on any oil drop was always an integer multiple of a fundamental value, which he called $e$.

$q=ne\text{where }n=1,2,3,...$

This proved that electric charge is quantized—it exists in discrete packets, not as a continuous fluid. The smallest indivisible packet of charge is the elementary charge, $e$. Millikan's best value for this charge was remarkably close to the modern accepted value of $1.602\times 10^{-19}$ Coulombs.

With the value of $e$ finally known, physicists could take Thomson's $e/m$ ratio and calculate the mass of the electron for the first time, confirming it was incredibly small, about 1/1837th the mass of a hydrogen atom.

Guiding Assumptions and Scientific Discovery

Your interest in the philosophy behind science is particularly relevant here. Millikan's experiment is a prime example of what philosopher of science Imre Lakatos would call a "research program" guided by a "hard core" of assumptions. Millikan began with the strong hypothesis that an elementary charge existed.

As the text highlights, this guiding assumption was crucial. It allowed him to interpret his data—seeing integer multiples where others might have seen experimental noise—and to persevere through immense experimental difficulty. This was not "cheating"; it was theory-guided science at its best, where a powerful hypothesis helps to extract a clear signal from a noisy world. The self-correcting nature of science ensures that such assumptions are ultimately validated or falsified when other scientists repeat and refine the experiment.

Guided Discovery

Let's work through the logic of Millikan's data analysis. Imagine you have performed the oil drop experiment and collected the following values for the total charge ($q$) on four different oil drops:

• Drop A: $4.8\times 10^{-19}$ C
• Drop B: $8.0\times 10^{-19}$ C
• Drop C: $11.2\times 10^{-19}$ C
• Drop D: $6.4\times 10^{-19}$ C

Problem:

1. Finding the Fundamental Unit: If you assume that charge is quantized, what is the most likely value for the elementary charge, $e$, based on this data set? (Hint: Look for the greatest common divisor of these numbers).

2. Assigning Quantum Numbers: For each drop (A, B, C, and D), determine the integer $n$ (the number of excess electrons) that corresponds to its total charge, using the value of $e$ you found in part 1.

3. The "Captured Ion" Effect: Millikan often observed a drop suddenly change its velocity while the electric field was on. This happened when the drop captured a stray ion from the air. Suppose Drop D (with its initial charge of $6.4\times 10^{-19}$ C) is moving upward, and it suddenly captures a single electron.

   • What is the new total charge on Drop D?
   • Qualitatively, how would its upward velocity change (increase, decrease, or stay the same)? Explain your reasoning in terms of the forces acting on the drop.

4. A Thought Experiment: A rival scientist, Felix, performs a similar experiment but, due to a flaw in his apparatus, only ever manages to observe drops with an even number of excess electrons. His measured charges are $3.2\times 10^{-19}$ C, $6.4\times 10^{-19}$ C, $9.6\times 10^{-19}$ C, etc. What value would Felix likely report for the elementary charge? How does this illustrate the importance of a comprehensive and unbiased data set in science?

Conclusion

Today, we have unraveled the logic of Millikan's oil drop experiment, one of the cornerstones of modern physics. We saw how a simple idea of balancing forces, combined with a clever method for measuring mass, led to a profound discovery about the fundamental nature of our universe.

Key Takeaways:

• The experiment's core principle is the balance of gravitational, electric, and viscous drag forces acting on a microscopic oil droplet.
• Millikan's genius was in using a two-step measurement (with the field off, then on) to determine both the mass of the drop and its total charge.
• The crucial discovery was that the measured charges on hundreds of drops were not continuous, but were always integer multiples of a fundamental unit ($q=ne$).
• This proved that electric charge is quantized—it exists in discrete "atoms" of charge.
• The experiment provided the value of the elementary charge, $e$, which, when combined with Thomson's $e/m$ ratio, finally allowed for the calculation of the electron's mass.

We have now established the key components of the early 20th-century atom: a tiny, dense, positive nucleus (from Rutherford's experiment) and one or more very light, negative electrons (from Thomson's and Millikan's work). This leads us to our next major intellectual hurdle. According to classical physics, an electron orbiting a nucleus should continuously radiate energy, spiral inward, and cause the atom to collapse in a fraction of a second. But atoms are stable. In our next lesson, we will confront this crisis and examine Niels Bohr's "desperate fix"—a radical and brilliant, yet ultimately flawed, model that introduced the idea of quantization directly into the atom itself.