Investigate, assess and model Millikan's oil drop experiment to determine the elementary charge and the quantisation of electric charge

What this dot point is asking

NESA wants you to describe Millikan's apparatus, explain the force balance on a charged oil drop between parallel plates ($qE = mg$ with $E = V/d$), use it to extract the charge on individual drops, and account for the observation that all measured charges are integer multiples of the elementary charge $e = 1.60 \times 10^{-19}$ C, with the conclusion that electric charge is quantised.

The answer

Why the experiment was needed

Thomson's 1897 measurement of $e/m$ for the electron was the charge-to-mass ratio, not the charge itself. To separate the two and find both the mass and charge of the electron, an independent measurement of $e$ alone was required.

The apparatus

Robert Millikan's 1909 experiment (refined through about 1913) used:

• A small chamber containing two horizontal parallel metal plates separated by distance $d$, with a small hole in the upper plate.
• A potential difference $V$ applied between the plates, creating a uniform vertical electric field $E = V/d$.
• An atomiser to spray tiny oil droplets above the upper plate. A few droplets fall through the hole into the space between the plates.
• A short-wavelength source (X-rays, or ionising radiation) to ionise some air molecules and so charge some droplets by attachment.
• A microscope to track individual droplets and a stopwatch to measure terminal velocities.

Two methods

Stationary method (the simplest to describe). Adjust the voltage until a chosen droplet hangs motionless. The electric force on the charge balances gravity:

So:

The mass $m$ of the droplet is found by switching off the field and measuring the terminal velocity of free fall through the air, then using Stokes' law (or, in modern presentations, treating the droplet density and radius separately).

Falling-and-rising method (Millikan's actual method). With the field off, the droplet falls at terminal velocity $v_g$ set by gravity vs viscous drag. With the field switched on (in the direction that drives the negative droplet upward), it rises at terminal velocity $v_E$ set by net electric force vs drag. Combining $v_g$ and $v_E$ eliminates the radius-dependent constants and gives the charge $q$ directly.

Results

Millikan measured thousands of drops over many years. Every measured charge was a positive integer multiple of a single value:

with $e \approx 1.60 \times 10^{-19}$ C. Drops with $n = 1$ (singly charged) were the most common, but $n = 2, 3, 4$ appeared often, and occasionally larger values. Sometimes a drop's charge would jump (after a momentary exposure to ionising radiation), but always to a different integer multiple of the same base unit.

The interpretation is direct: charge is quantised. The smallest unit of free charge in nature is $e$, and macroscopic charges are integer multiples of it.

Millikan's best value was $e = 1.592 \times 10^{-19}$ C, very close to the modern value $1.602 \times 10^{-19}$ C. Combined with Thomson's $e/m$, this fixed the electron mass at $m_e = 9.11 \times 10^{-31}$ kg.

Worked example: a heavier drop

A drop of mass $5.0 \times 10^{-15}$ kg is held stationary between plates 5.0 mm apart with potential difference 460 V. Find the charge on the drop.

Electric field: $E = V/d = 460 / 5.0 \times 10^{-3} = 9.2 \times 10^4$ V/m.

Force balance: $qE = mg$, so $q = mg/E = (5.0 \times 10^{-15})(9.80)/(9.2 \times 10^4) = 5.3 \times 10^{-19}$ C.

In elementary charges: $n = q/e = 5.3 \times 10^{-19} / 1.60 \times 10^{-19} \approx 3.3$.

The closest integer is 3, so the drop carries $3e = 4.8 \times 10^{-19}$ C. The 10% discrepancy in this textbook problem usually reflects measurement uncertainty rather than fractional charge.

Modern view

Charge quantisation in units of $e$ is observed in every macroscopic system. Quarks have charges of $\pm e/3$ and $\pm 2e/3$, but they are confined inside hadrons and cannot be isolated as free particles. The smallest free charge is the electron's $-e$ (or its antiparticle's $+e$), exactly the unit Millikan measured.

Try it: Electric field calculator for $E = V/d$ between parallel plates, and explore the force on a charged droplet between them.

Common traps

Confusing $E$ and $V$. Between parallel plates, $E = V/d$. The field is in V/m and is uniform between the plates; the voltage is the work per unit charge to move from one plate to the other.

Setting up the force balance with the wrong sign. The electric force must be opposite to gravity (upward, for a negative droplet) to balance it. Always check by drawing the free-body diagram.

Reporting a non-integer multiple of $e$ without comment. If your calculation gives $q/e = 2.4$, you should say either that experimental error puts the actual value at 2 or 3, or that the problem is testing your ability to round. Real charges are integer multiples of $e$.

Saying Millikan measured the mass of the electron directly. He measured the charge $e$. The mass of the electron then follows from Thomson's $e/m$.

Claiming the experiment proves the electron is the smallest charge in nature. It proves the smallest free charge is $e$. Quarks have smaller charges but are not free.

In one sentence

Millikan balanced the electric force $qE$ on a charged oil drop against gravity $mg$ between parallel plates, found that the charges on different drops are always integer multiples of $e = 1.60 \times 10^{-19}$ C, and so established both the value of the elementary charge and the quantisation of electric charge.