Millikan and the Charge of the Electron

Introduction

Thomson's experiments supplied a ratio of charge to mass. To separate those quantities, scientists needed an independent measurement. Millikan's oil-drop work measured the tiny charges carried by drops and showed that they occurred in whole-number multiples of a common unit. The reasoning combines mechanics, electricity and repeated observation.

Core explanation

A small charged drop experiences gravity and an electrical force when placed in an electric field. In a simplified model, a stationary drop can have its downward effective weight balanced by an upward electrical force. If the field magnitude and effective weight are known, the charge magnitude follows from q E = effective weight .

The full analysis is more careful than simply weighing a visible drop on a balance. Drop motion through air, drag, buoyancy and the relation between size and mass must be considered. These corrections are part of why the experiment is a precision measurement rather than a simple classroom observation.

Different drops do not necessarily carry the same total charge. Some have acquired more excess electrons than others. The important pattern is that measured charges can be described as q = ±ne , with n a whole number. Changes in a drop's charge also provide evidence for a recurring charge unit.

For calculations in this unit use e ≈ 1.60 × 10⁻¹⁹ C. The electron carries −e; the positive elementary-charge symbol e is its magnitude. A drop with three excess electrons therefore has charge −4.80 × 10⁻¹⁹ C, not a charge of minus three coulombs.

Once e is known, dividing it by the electron's charge-to-mass magnitude gives the electron mass. The two experiments answer complementary questions. The conceptual history and the discrete changes in drop charge are discussed in Millikan's Nobel lecture. No practical high-voltage or radiation procedure is needed to understand this inference.

Formulae

Charge on an ordinary drop: q = ±ne.

Number of excess electrons: n = q /e.

For an idealised stationary drop: q E = mg, when buoyancy is neglected.

Step-by-step reasoning

1. Express every measured charge in the same power-of-ten unit. 2. Identify a common step size consistent with all measurements. 3. Divide each charge magnitude by that step to check for whole numbers. 4. Use the sign of the charge to distinguish excess electrons from an electron deficit.

Visual explanation

Draw a drop with one downward arrow labelled effective weight and one equal upward arrow labelled electrical force. Beside it, draw a number line with allowed charge magnitudes e, 2e, 3e and 4e. Keep the force diagram separate from the charge-counting diagram.

Real-world analogy

A vending machine that accepts only identical tokens can hold one, two or three tokens, but not a fractional token. Measuring several token totals can reveal the unit size. Ordinary drop charges similarly reveal discrete electron counts, although electric charge is a physical property rather than a currency.

Real-world example

Electrostatic printing and coating technologies control charged droplets or particles using electric fields. Their motion depends on the relationship between charge, mass and field strength. Individual engineering droplets usually carry many electron charges, while the oil-drop idea focuses on resolving the small unit underlying that total.

Why?

Why measure many drops or several charge states of one drop? A single total does not reveal its smallest unit. A charge of 6 units could represent one large contribution or several smaller ones. Repeated discrete differences provide stronger evidence for the common increment.

Common misconception

“Each oil drop was one electron.” A drop contains a huge collection of atoms. Its net charge arises from a small imbalance between positive and negative charges, often caused by a few excess electrons.

Worked example

An idealised data set gives charge magnitudes 3.20, 4.80 and 8.00, all multiplied by 10⁻¹⁹ C. Dividing by 1.60 × 10⁻¹⁹ C gives 2, 3 and 5. These measurements are consistent with two, three and five elementary charges. They do not, by themselves, prove that no smaller common unit could fit; broader evidence establishes e.

Quick check

1. What is the charge of a drop with four excess electrons, using e = 1.60 × 10⁻¹⁹ C? Answer: −6.40 × 10⁻¹⁹ C, because each excess electron contributes −e.

Exam focus

State the assumptions of a simplified force balance. If buoyancy is explicitly neglected, mg is acceptable; do not silently treat an idealised equation as the entire historical analysis. Distinguish total drop charge from the charge on one electron.

Advanced insight

Fractionally charged quarks exist within composite particles, but they are not isolated charge carriers on ordinary oil drops. The whole-number electron-charge rule used here describes ordinary charged objects. Its context prevents a useful school-level rule from becoming an incorrect universal claim.

Summary

The oil-drop method connected electrical force with a mechanically inferred drop weight. Repeated charges and charge changes revealed a common elementary unit. Combining this independent charge measurement with the charge-to-mass ratio made an electron-mass estimate possible.

Practice questions

1. A drop has charge −8.00 × 10⁻¹⁹ C. How many excess electrons does it carry using the rounded value of e? Answer: Five, since 8.00 divided by 1.60 is 5. 2. Why does a stationary drop need a force balance? Answer: Its acceleration is zero, so its total force must be zero in the model. 3. Does an uncharged drop contain no electrons? Answer: No. It contains many electrons whose negative charge balances the positive nuclear charge. 4. Which additional measurement allows e to yield electron mass? Answer: The electron's charge-to-mass magnitude, because mass equals e divided by that ratio.