eduKate Learning Manual · Electricity × Atomic Physics × Measurement Science · Secondary → JC · Fall → Balance → Charge → Quantise
Wait, What? A Tiny Oil Drop Can Carry 1, 2, 3 or More Units of Charge — but Not 1.37 Units
Electric charge can look continuous in everyday life because macroscopic objects contain enormous numbers of charged particles. But at the microscopic level, ordinary observable charge transfer comes in discrete units.
The oil-drop experiment made that discreteness measurable. By watching tiny charged droplets move under gravity and an electric field, Robert Millikan and collaborators could infer each droplet’s charge. The values varied from drop to drop — yet they clustered near integer multiples of one common amount.
That common amount is the magnitude of the elementary charge, e.
drop falls → terminal speed reveals effective size → electric field pushes charged drop → force balance reveals q → many q values are compared → common spacing reveals e.
The Big Question
How can the motion of a microscopic droplet reveal that electric charge is quantised?
Quick Answer
A charged oil droplet experiences gravity, buoyancy, viscous drag and an electric force qE. First, its terminal falling speed can be used to estimate its radius and effective weight. Then an electric field is applied. If the field is adjusted so the droplet is stationary, the electric force balances the effective downward weight:
|q|E = Weffective
Repeating the measurement for many drops gives charges such as approximately e, 2e, 3e and so on. The common divisor is the elementary charge:
e ≈ 1.602176634 × 10⁻¹⁹ C
In today’s SI, that numerical value is exact because the coulomb is defined through the fixed elementary charge. Historically, however, the experiment helped determine the value from physical measurements.
What You Will Learn
- why terminal speed helps determine droplet size
- how gravity, buoyancy, drag and electric force interact
- why an electric field can suspend a charged drop
- how individual droplet charge is inferred
- why many drops are needed to establish quantisation
- why Stokes’ law needs a small-particle correction
- how Brownian motion and evaporation limit precision
- why charge can change in discrete jumps on one drop
- what the historical data-selection controversy does and does not imply
- why a robust result must survive changes in apparatus assumptions
Part 1 — A Falling Drop Quickly Reaches Terminal Speed
A tiny oil droplet falling through air accelerates only briefly. Viscous drag rises with speed until the net force becomes approximately zero. The drop then falls at terminal speed.
For a small sphere moving slowly enough through a viscous fluid, Stokes’ drag law gives:
Fd = 6πηrv
where η is air viscosity, r the drop radius and v its speed.
The downward gravitational force is partly offset by air buoyancy. At terminal fall:
effective weight = viscous drag
This allows the drop’s radius to be inferred from its measured terminal speed if the densities and viscosity are known.
Part 2 — Effective Weight Includes Buoyancy
The droplet displaces air. Therefore the downward force relevant to suspension is not simply mg.
For a spherical drop of volume Vd:
Weffective = (ρoil − ρair)Vdg
with:
Vd = 4πr³/3
Ignoring buoyancy would bias the inferred charge. At high precision, even a correction that seems small must be justified rather than casually discarded.
Part 3 — The Electric Field Adds a Controllable Force
Between approximately parallel plates separated by distance d with potential difference V, the ideal field magnitude is:
E ≈ V/d
A drop carrying charge q experiences electric force:
FE = qE
Choose the field direction so that the electric force points upward for the charged drop. Adjust the voltage until the droplet is approximately stationary. Then:
|q|E = Weffective
so:
|q| = Weffective/E
The experiment has converted an invisible microscopic charge into a balance among measurable mechanical quantities.
Part 4 — One Drop Does Not Establish Quantisation
Suppose one drop has inferred charge 4.80 × 10⁻¹⁹ C. That alone does not reveal the elementary unit. It could represent one large unit, two smaller units, three still smaller units or simply a continuous value.
Quantisation appears when many measured charges share a common spacing. For example:
- 1.60 × 10⁻¹⁹ C
- 3.20 × 10⁻¹⁹ C
- 4.81 × 10⁻¹⁹ C
- 6.41 × 10⁻¹⁹ C
Within experimental uncertainty, these values are approximately 1e, 2e, 3e and 4e.
The scientific claim is therefore not “this one drop had charge e.” It is “a collection of independently measured charges is consistent with integer multiples of a common elementary magnitude.”
A Quantitative Window — Infer the Integer
If a measured drop charge is 8.01 × 10⁻¹⁹ C, divide by e:
n ≈ (8.01 × 10⁻¹⁹)/(1.602 × 10⁻¹⁹) ≈ 5.00
The result is consistent with five elementary charges. Measurement uncertainty means we do not demand an infinite string of perfect zeroes after 5; we ask whether the uncertainty interval is consistent with an integer and inconsistent with nearby alternatives.
Part 5 — A Stronger Observation: One Drop Can Change Charge in Steps
Ionising radiation can cause a droplet to gain or lose electrons. If the same tracked droplet changes charge, the voltage required to balance it changes.
Those changes occur in discrete steps associated with adding or removing whole electrons.
same droplet + same mechanical mass + sudden new balance voltage → charge changed by a discrete amount.
This is especially persuasive because it reduces the chance that different droplet sizes are creating the apparent spacing.
Part 6 — Why Stokes’ Law Needed a Correction
Stokes’ law assumes a continuous fluid that sticks to the sphere’s surface in the usual way. But Millikan’s droplets were so small that their size could become comparable with the mean free path of air molecules.
At that scale, the continuum no-slip approximation becomes imperfect. The effective drag is lower than naive Stokes’ law predicts. A slip correction, often associated with Cunningham’s treatment, is required for accurate charge estimates.
This is an important model-boundary lesson: an equation can be correct within its regime and still generate a biased result when pushed outside it.
Part 7 — Brownian Motion Makes the Drop Jitter
Air molecules collide randomly with the droplet. Because the droplet is microscopic, those molecular impacts produce visible Brownian fluctuations in its position.
The observed trajectory is therefore not a perfectly smooth line. Timing one short fall can give a noisy velocity estimate. Repeated timing and averaging reduce random error, but Brownian motion sets a real measurement challenge.
Again, the experiment becomes stronger when the learner separates the mean drift caused by forces from the random microscopic fluctuations superimposed on it.
Part 8 — Why Use Oil Rather Than Water?
Water droplets evaporate relatively quickly, changing radius and therefore mass while the measurement is underway. A low-volatility oil reduces that problem.
Even oil drops can slowly change, so long measurements still require attention to stability. A parameter that drifts during an experiment can masquerade as a force change if the analysis assumes it stayed fixed.
The Historical Carrier — Millikan, Fletcher and the Experiment’s Development
Robert Millikan’s oil-drop work in the early twentieth century produced a precise value for the elementary charge and strong evidence for charge quantisation. Harvey Fletcher, then Millikan’s graduate student, made important contributions to developing the oil-drop method.
Millikan received the 1923 Nobel Prize in Physics for work on the elementary electric charge and the photoelectric effect.
The historical record has also generated debate about how Millikan selected and reported oil-drop observations. Later historians examined laboratory notebooks and found that not every observed drop appeared in the published analysis, prompting discussion about data-selection criteria and scientific reporting.
The careful conclusion is not that charge quantisation disappears. The value of e was independently reproduced and is foundational to modern physics. The lesson is about transparency: strong results should be accompanied by clear inclusion criteria so readers can understand how observations became the published dataset.
Part 9 — The Elementary Charge Is Now Exact in SI
Since the 2019 redefinition of the International System of Units, the elementary charge is assigned the exact numerical value:
e = 1.602176634 × 10⁻¹⁹ C exactly
This reverses the historical measurement relationship. Millikan-era experiments helped determine e in coulombs. Today the coulomb is linked to the fixed value of e, and experiments test electrical measurement chains against the SI framework.
RFE Stress Test — Quantisation or Analysis Artefact?
- Radius model: does the inferred charge remain consistent after the correct slip correction?
- Field calibration: are plate spacing and voltage known well enough?
- Brownian motion: are enough observations averaged to estimate drift rather than random jitter?
- Evaporation: is droplet size stable during the measurement?
- Charge-change test: do single drops change balance conditions in discrete steps?
- Dataset transparency: are inclusion/exclusion rules stated before interpreting common multiples?
- Independent replication: do different methods recover the same elementary charge?
A strong quantisation claim does not come from forcing noisy values onto integers. It comes from a physical model, transparent uncertainty and repeatable clustering around one common charge unit.
Observation vs Inference
Observation: individual droplets rise, fall or remain suspended at specific electric fields.
Mechanical inference: force balance and drag analysis yield a charge q for each drop.
Quantum inference: many q values are consistent with integer multiples of a common elementary charge e.
Common Misconceptions and How to Repair Them
- “Millikan saw electrons directly.” Repair: he observed droplets and inferred charge from their motion.
- “A suspended drop has no forces on it.” Repair: forces are present but their vector sum is zero.
- “One measured charge proves quantisation.” Repair: the evidence is the common spacing across many measurements and discrete changes.
- “Stokes’ law is exact for any tiny drop.” Repair: the continuum approximation requires a slip correction at sufficiently small sizes.
- “The published historical dataset contained every observed drop.” Repair: historical notebooks show selection occurred; the scientific issue is whether criteria and uncertainty were justified and transparent.
- “The controversy means e is doubtful.” Repair: charge quantisation and the value of e have extensive independent confirmation.
Checkpoint Questions
- Why does a droplet reach terminal speed?
- Why must buoyancy be included?
- What is the electric force on a charge q?
- What condition holds when a drop is suspended?
- Why are many droplets needed?
- Why does Stokes’ drag require a correction for very small droplets?
- What does a discrete charge change on one droplet add to the evidence?
Apply It — Same Drop, Twice the Charge
A droplet is suspended in field E. It then gains one or more electrons so its charge magnitude doubles while its size is essentially unchanged. To suspend it again, the required field magnitude is approximately halved because |q|E must still equal the same effective weight.
Unfamiliar Transfer — Discreteness Hidden Inside a Continuous-Looking Signal
The oil-drop logic generalises. Macroscopic current looks continuous because astronomical numbers of elementary charges move each second. Chemical concentration looks continuous even though matter is molecular. Light intensity can look smooth even though photons are detected discretely.
A recurring scientific question is therefore:
is the apparently continuous quantity fundamentally continuous, or is it the average of many discrete events?
Answer Key
1. Drag grows until it balances effective weight. 2. Displaced air exerts an upward force. 3. F = qE. 4. Electric force balances effective weight. 5. Quantisation is inferred from a common unit across many q values. 6. The air continuum/no-slip assumptions weaken when droplet size approaches molecular mean-free-path scales. 7. It shows charge changes on a fixed mechanical object occur in discrete increments.
Can You Explain WHY?
Explain why seeing several different droplet charges can reveal one universal elementary charge. A strong answer should connect terminal velocity → radius/effective weight → electric balance → q for each drop → uncertainty → integer multiples → common e.
Singapore Secondary and JC Science Bridge
Secondary Physics supplies electric charge, fields, forces and terminal motion. JC Physics adds quantitative fields, particle models, uncertainty and evidence evaluation. Millikan’s experiment forces those topics into one chain where the unseen electron is never directly observed but its charge becomes measurable.
Deep Science Windows
- Brownian motion: microscopic random collisions connect molecular kinetic theory to measurement noise.
- Stokes drag and Knudsen corrections: fluid models change when particle size approaches the gas mean free path.
- Single-electron devices: modern electronics can control transport one electron at a time.
- Quantum Hall metrology: electrical standards connect e and h to extraordinarily precise resistance measurements.
- Fractional quasiparticle charge: in specialised many-body systems, emergent excitations can carry fractional effective charge without changing the elementary charge of the electron itself.
Evidence Boundaries
The simple suspension equation hides the difficult part: determining droplet size accurately and controlling fluid corrections, Brownian noise and geometry. Historical interpretation should also separate the robustness of charge quantisation from debates over data selection and attribution. Today’s exact SI value of e is a definition, not a new oil-drop measurement.
Manual Summary — KNOW → CONNECT → EXPLAIN → APPLY → CHECK
- KNOW: drag, buoyancy, qE and effective weight control drop motion.
- CONNECT: terminal fall estimates size; electric balance estimates charge.
- EXPLAIN: multiple measured charges reveal a common elementary unit.
- APPLY: predict how charge or electric field changes suspension conditions.
- CHECK: slip correction, Brownian motion, evaporation, calibration, selection criteria and replication.
Teaching Guide for Parents, Tutors and Teachers
Why this opening works: the surprising part is not that droplets can be electrically moved; it is that many apparently arbitrary charges resolve into integer multiples of one hidden unit.
- Central reasoning model: falling drop → size → effective weight → electric balance → q → common divisor.
- Teaching sequence: terminal velocity → Stokes model → buoyancy → field force → suspension → repeated q values → quantisation → model corrections.
- Diagnostic question: “Why can one suspended drop not prove the existence of e?”
- If stuck: begin with a larger object balanced by two opposite forces before shrinking to the droplet.
- Ready for more: introduce Cunningham correction, Brownian uncertainty, SI redefinition and single-electron transport.
Quiet Teaching Standard: do not teach the experiment as qE = mg alone. Require buoyancy, size determination, uncertainty and the many-drop argument that actually establishes quantisation.