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Thomson's 1897 cathode-ray work established the specific charge e/m of the electron with high precision but it could not separate e from m. To complete the picture, somebody had to measure either the charge or the mass of the electron independently — and the charge turned out to be the easier target. Between 1909 and 1913, Robert Millikan at the University of Chicago designed and refined an experiment of breathtaking simplicity in principle: balance the gravitational weight of a tiny charged oil droplet against the electric force of a known field, and measure the charge directly. The pay-off was even larger than getting e numerically right. By repeating the experiment with thousands of droplets carrying different numbers of trapped charges, Millikan showed that the charge always came in integer multiples of a fundamental quantum e ≈ 1.60 × 10⁻¹⁹ C — direct experimental proof that charge is discrete, not continuous.
Spec mapping: This lesson covers AQA 7408 section 3.12.1 — Millikan's experiment, the balance of weight and electric force on a charged oil droplet, the use of terminal velocity in the absence of an electric field to determine droplet radius via Stokes' law, the quantisation of charge as integer multiples of e ≈ 1.60 × 10⁻¹⁹ C, and the determination of the electron mass when combined with Thomson's specific charge. (Refer to the official AQA specification document for exact wording.)
Synoptic links:
- Section 3.12.1 (discovery of the electron): Millikan's e combined with Thomson's e/m yields m = e ÷ (e/m) ≈ 9.11 × 10⁻³¹ kg — closing the loop on the electron's two basic parameters. The two experiments are partners.
- Section 3.5.1.5 (uniform electric fields): Millikan's parallel plates are the canonical example of a uniform field E = V/d acting on a point charge. The force-balance calculation re-uses the same formula students meet in Year 12.
- Section 3.4.1.5 (terminal velocity and Stokes' law): the falling-droplet phase of the experiment is an application of Stokes' drag in a viscous fluid (here, air) — the same physics that governs raindrops, sediment in water and the centrifuging of blood.
The oil-drop chamber in its mature 1913 form consisted of two horizontal brass plates of diameter about 22 cm, separated by an insulating ring giving a plate gap of d ≈ 16 mm. The upper plate was perforated with small holes through which fine droplets of low-vapour-pressure oil (Millikan used a clock oil — dense, slow-evaporating, low viscosity sensitivity) were sprayed by an atomiser. The atomisation process itself charged most droplets by friction, with a smaller number additionally ionised by an X-ray source aimed into the gap. A horizontal microscope, mounted at right-angles to the gap and back-illuminated by a small lamp, allowed individual droplets to be tracked as bright pinpoints against a dark background.
A DC voltage V (variable up to a few thousand volts) could be applied across the plates, producing a uniform vertical electric field E = V/d in the gap. A reversing switch let the operator flip the field direction or switch it off entirely. The experiment ran in two phases for each droplet: (1) field off, droplet falls under gravity reaching a terminal velocity in air; (2) field on, droplet either rises, falls more slowly, hovers, or is accelerated downward depending on the sign and magnitude of its charge.
graph TD
A["Atomiser<br/>(charges droplets)"] --> B["Upper plate (+)<br/>(perforated)"]
B --> C["Droplet observation region<br/>(microscope)"]
C --> D["Lower plate (-)"]
E["X-ray source<br/>(extra ionisation)"] --> C
F["Variable HV supply<br/>+ reversing switch"] --> B
F --> D
style B fill:#ef4444,color:#fff
style D fill:#1d4ed8,color:#fff
style E fill:#f59e0b,color:#fff
With the field off, the droplet falls under gravity. After a fraction of a second it reaches terminal velocity v₁ where its weight is balanced by viscous drag. For a small sphere moving slowly through a fluid, Stokes' law gives the drag force as
F_drag = 6πηrv
where η is the viscosity of air (≈ 1.8 × 10⁻⁵ Pa s at 20 °C) and r is the droplet radius. At terminal velocity:
mg = 6πηrv₁
The mass of the droplet is m = ρ × (4/3)πr³ where ρ is the density of the oil (≈ 900 kg m⁻³). Substituting and rearranging for r:
r = √[ 9ηv₁ / (2ρg) ]
So one careful measurement of terminal velocity — typically by timing the droplet across the field of view of a microscope reticle of known scale, say 1 mm in 5–20 seconds — gives the droplet radius (and hence mass) without ever needing to know the charge.
In Millikan's 1913 paper, he refined this with a small Cunningham slip correction to Stokes' law because the droplet radius is comparable to the mean free path of air molecules — but the spec treats Stokes' law in its unmodified form and that is what is examined at A-Level.
The operator then turns on the voltage with the polarity chosen so that the electric force on the droplet opposes gravity. By adjusting V, the droplet is brought to a halt (or — in Millikan's original version — given a slower terminal velocity v₂ in the opposite direction; both methods work and the stationary version is simpler to present). At the balance point:
QE = mg → Q = mg/E = mgd/V
Q is the total charge on the droplet, an integer multiple of e. Combining with the radius from Phase 1:
Q = (4/3)πr³ ρ g d / V
— every quantity on the right-hand side has been measured. Millikan repeated this for thousands of droplets across his 1909-1913 papers.
When Millikan plotted Q values for many droplets, they did not form a continuum. Instead, every measured Q fell within experimental error of an integer multiple of a single base value:
Q = n × e, with e ≈ 1.60 × 10⁻¹⁹ C and n = 1, 2, 3, 4, …
Droplets carrying n = 1 (a single elementary charge) were the most informative; droplets with larger n confirmed the integer pattern. When a droplet's charge changed mid-experiment (because of an ion captured from the X-ray-ionised air), the change in Q was always also an integer multiple of e. Charge is quantised. The agreement with the modern value of e (1.602 × 10⁻¹⁹ C) is within about 1% of Millikan's published mean — a remarkable result for an experiment whose limiting precision was set by the variability of air viscosity, droplet evaporation, and convection currents in the chamber.
graph LR
A["Q = 1.60 × 10⁻¹⁹"] --> B["n = 1"]
C["Q = 3.20 × 10⁻¹⁹"] --> D["n = 2"]
E["Q = 4.80 × 10⁻¹⁹"] --> F["n = 3"]
G["Q = 6.40 × 10⁻¹⁹"] --> H["n = 4"]
style B fill:#27ae60,color:#fff
style D fill:#27ae60,color:#fff
style F fill:#27ae60,color:#fff
style H fill:#27ae60,color:#fff
Combined with Thomson's e/m ≈ 1.76 × 10¹¹ C kg⁻¹, Millikan's e fixes the mass of the electron:
m_e = e / (e/m) = (1.60 × 10⁻¹⁹) / (1.76 × 10¹¹) ≈ 9.11 × 10⁻³¹ kg
That mass, smaller than a hydrogen atom by a factor of 1836, was the first sub-atomic mass scale established by experiment.
An oil droplet of radius r = 1.5 × 10⁻⁶ m and density ρ = 880 kg m⁻³ is held stationary between parallel plates separated by d = 12 mm and across which a potential difference V = 410 V is applied. Calculate (a) the mass of the droplet, (b) the charge on the droplet, (c) the number of elementary charges.
Solution.
(a) m = ρ × (4/3)πr³ = 880 × (4/3) × π × (1.5 × 10⁻⁶)³ = 880 × 4.19 × 3.375 × 10⁻¹⁸ = 1.24 × 10⁻¹⁴ kg (3 s.f.).
(b) At balance, QE = mg, with E = V/d.
Q = mgd/V = (1.24 × 10⁻¹⁴) × 9.81 × (0.012) / 410 = 3.56 × 10⁻¹⁹ C (3 s.f.).
(c) n = Q/e = (3.56 × 10⁻¹⁹) / (1.60 × 10⁻¹⁹) ≈ 2.22.
That is suspiciously not an integer. Two interpretations: (i) there is small experimental error in V or r and the true n is 2; (ii) the droplet has been mis-measured and n is genuinely between two values. In a real Millikan run, the operator would average over many droplets and the histogram would peak sharply at integers. For the purpose of an exam answer, the n = 2 reading is the correct conclusion — small measurement error nudged the apparent Q above 2e.
A droplet falls through 0.50 mm of the microscope field in 18.0 s with the field off. The viscosity of air is η = 1.81 × 10⁻⁵ Pa s and the oil density is 920 kg m⁻³. When a voltage V = 530 V is applied across plates separated by d = 15 mm, the droplet rises to a balanced halt. Find the droplet radius, mass, and charge, and identify n.
Solution.
Terminal velocity: v₁ = 0.50 × 10⁻³ / 18.0 = 2.78 × 10⁻⁵ m s⁻¹.
Radius from Stokes' law balance, mg = 6πηrv₁ with m = (4/3)πr³ρ. Combining:
r² = 9ηv₁ / (2ρg) = 9 × (1.81 × 10⁻⁵) × (2.78 × 10⁻⁵) / [2 × 920 × 9.81] = 4.53 × 10⁻⁹ / 18 050 = 2.51 × 10⁻¹³ r = 5.01 × 10⁻⁷ m (≈ 0.5 µm).
Mass: m = (4/3)π × (5.01 × 10⁻⁷)³ × 920 = (4/3) × π × 1.26 × 10⁻¹⁹ × 920 = 4.84 × 10⁻¹⁶ kg.
Charge: Q = mgd/V = (4.84 × 10⁻¹⁶) × 9.81 × (0.015) / 530 = 1.34 × 10⁻¹⁹ C ≈ 0.84 e.
The expected n = 1 droplet gives a slight under-reading because of the Cunningham slip correction at small r — at radii ≪ mean free path of air, Stokes' law over-estimates drag and so the inferred r is too small, the mass too small, the charge too small. This is the systematic effect Millikan eventually corrected for in his 1913 paper. At A-Level, n = 1 is the correct identification and the small discrepancy is an acceptable real-experiment feature.
In 1909, the natural assumption was that electricity, like water or air, was a continuous fluid that could be subdivided arbitrarily finely. Many physicists — including, briefly, Felix Ehrenhaft, who ran his own competing oil-drop experiments and claimed to find sub-electron "fractional" charges — believed the discreteness of charge was a statistical artefact, not a fundamental fact. Millikan's measurements, with hundreds of droplets giving integer multiples of the same e, were decisive: charge is a quantised property of matter, and the unit is the charge on the electron (or, with opposite sign, the proton).
This shaped 20th-century physics in three concrete ways. First, it gave the chemical concept of "valence" — the integer count of bonds — a physical mechanism: bonds are integer-charge transfers or shares. Second, it set the stage for the quantisation of energy in Bohr's atom (1913, the same year as Millikan's mature paper) — once charge is quantised, the orbital angular momentum that supports a stable atom must also be quantised. Third, it gave subsequent particle-physics measurements a benchmark: every charged elementary particle discovered since (muon, tau, quarks, W bosons) carries a charge that is either e, −e, or — in the case of quarks — ⅓ or ⅔ of e, with the integer-multiples constraint preserved at the level of hadrons.
A modern subtlety: quarks carry fractional charges (+2/3 e for up, −1/3 e for down) but they are never observed in isolation — they are always confined into hadrons whose total charge is an integer multiple of e. Millikan's result — at the level of free particles — therefore stands unmodified.
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