Physics · Turning Points in Physics · A2 Level

Millikan's Oil-Drop Experiment

Thomson's cathode ray tube pinned down the ratio of the electron's charge to its mass, but not either quantity on its own. Robert Millikan's answer, arrived at over years of painstaking work watching microscopic oil droplets hover in an electric field, was to trap a single tiny drop of oil against gravity and read the electron's charge directly off the voltage needed to hold it still.

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Apparatus and method

Millikan's apparatus centred on two horizontal, circular metal plates mounted parallel to each other a few millimetres apart, connected to a variable, reversible high-voltage supply so a controlled uniform electric field could be set up between them, pointing either up or down. A fine spray (an atomiser) above the top plate sprayed a mist of tiny oil droplets, some of which fell through a small hole in the top plate into the space between the plates. As the oil left the atomiser's fine nozzle, friction between the oil and the nozzle left many of the droplets carrying a small electric charge — some positive, some negative, purely by chance, with no control over the exact amount.

A travelling microscope mounted to view the space between the plates side-on let a single chosen droplet be watched and timed as it moved. With the electric field switched off, a droplet released from rest quickly reaches terminal velocity as it falls under gravity, since the upward drag force from the air increases with speed until it balances the droplet's weight. Timing the droplet falling at constant terminal velocity over a measured distance gives that terminal velocity directly.

Side-view schematic of Millikan's apparatus. An atomiser sprays a fine mist of oil droplets above a horizontal top plate with a small central hole; one droplet falls through the hole into the region between the top plate and a horizontal bottom plate below it, both plates connected to a variable voltage supply; a microscope is drawn to the side, its line of sight aimed at the single droplet between the plates. — Millikan's apparatus: charged oil droplets sprayed above the upper plate fall into the field region and are observed through a microscope.

With the field switched on and adjusted (in the direction that pushes a negatively charged droplet upward), Millikan increased the voltage until the electric force on that particular droplet exactly balanced its weight, leaving the droplet stationary, held motionless in mid-air against gravity. The voltage needed to achieve this balance — together with the droplet's terminal velocity measured earlier with the field off — contained enough information to calculate the droplet's charge.

Finding the droplet's radius and mass

Before the charge can be found, the droplet's weight has to be known, which means finding its mass — and since the droplets are far too small to weigh directly, Millikan instead found each droplet's radius from its terminal velocity (measured with the field off) using Stokes' law for the drag force on a small sphere moving slowly through a fluid.

DerivationDroplet radius from terminal velocity

At terminal velocity $v$ (field off, falling under gravity), the droplet is in equilibrium: the upward drag force equals the downward resultant of weight and the small upthrust from the surrounding air. To a good approximation for a dense oil droplet in air, upthrust is neglected and drag is set equal to weight.

Stokes' law gives the drag force on a small sphere of radius $r$ moving at speed $v$ through a fluid of viscosity $\eta$ as $F = 6\pi\eta r v$.

The droplet's weight is $mg = \rho V g = \rho\left(\dfrac{4}{3}\pi r^3\right)g$, where $\rho$ is the density of the oil.

Setting drag equal to weight at terminal velocity: $6\pi\eta r v = \dfrac{4}{3}\pi r^3 \rho g$.

Cancelling $\pi r$ from both sides and rearranging for $r^2$:

$r^2 = \dfrac{9\eta v}{2\rho g} \quad\Rightarrow\quad r = \sqrt{\dfrac{9\eta v}{2\rho g}}$

Every quantity here — viscosity of air $\eta$, terminal velocity $v$ (measured directly), oil density $\rho$, and $g$ — is known or measurable, so the tiny droplet's radius can be calculated without ever weighing it.

Once $r$ is known, the droplet's mass follows from $m = \rho V = \rho\left(\dfrac{4}{3}\pi r^3\right)$, and hence its weight $mg$.

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Finding the droplet's charge

With the droplet's weight now known, balancing the droplet in the field gives its charge directly.

DerivationCharge from the balance condition

When the droplet is held stationary by the field, the upward electric force exactly balances its downward weight: $QE = mg$, where $Q$ is the droplet's charge.

The field strength between two parallel plates separated by distance $d$ with potential difference $V$ across them is $E = \dfrac{V}{d}$.

Substituting and rearranging for $Q$:

$Q = \dfrac{mgd}{V}$

$m$ comes from the Stokes' law analysis above, $d$ is the fixed, known plate separation, and $V$ is the balancing voltage read directly off the supply — so $Q$ can be calculated for this one droplet.

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Worked exampleCalculating a droplet's charge

A charged oil droplet of mass $m = 9.8\times10^{-15}\,\text{kg}$ is held stationary between two horizontal plates separated by $d = 5.0\,\text{mm}$, when the potential difference across the plates is $V = 612\,\text{V}$. Find the charge on the droplet.

Use $Q = \dfrac{mgd}{V}$, with $g = 9.81\,\text{m}\,\text{s}^{-2}$ and $d = 5.0\times10^{-3}\,\text{m}$.

$Q = \dfrac{9.8\times10^{-15}\times9.81\times5.0\times10^{-3}}{612}$

$Q = \dfrac{4.807\times10^{-16}}{612} \approx 7.85\times10^{-19}\,\text{C}$

Dividing this by the known elementary charge, $1.6\times10^{-19}\,\text{C}$, gives $7.85\times10^{-19}/1.6\times10^{-19} \approx 4.9$ — close enough to the integer 5 (allowing for rounding in the given data) to say this droplet carried a charge of exactly $5e$.

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The key result: charge is quantised

Millikan repeated this measurement for many hundreds of different oil droplets, each carrying a different, essentially random amount of charge picked up during spraying. When he tabulated all the resulting charge values, a striking pattern emerged: every single measured charge, across hundreds of independent droplets, came out as a whole-number multiple of one particular, smallest value — never a fraction of it, and never anything in between.

That smallest value is the elementary charge, $e = 1.6\times10^{-19}\,\text{C}$, the magnitude of the charge carried by a single electron. Millikan's result showed that electric charge is not a continuously variable quantity that could in principle take any value at all — it is quantised, built up only in whole-number multiples of this one fundamental unit, because any charged object's charge ultimately comes from a whole number of extra or missing electrons.

Charge quantisation

Combined with Thomson's specific charge $e/m$, Millikan's direct measurement of $e$ let physicists calculate the electron's mass, $m = e / (e/m) \approx 9.1\times10^{-31}\,\text{kg}$, for the first time — completing the picture of the electron as a specific, fundamental particle with a definite charge and a definite (extremely small) mass.

Why the field is switched off to find v, then on to balance

The two stages of the experiment measure two different things using the same droplet. Field off: the droplet falls under gravity and drag alone, reaching terminal velocity — this stage finds the droplet's size and mass, and involves no charge information at all. Field on: the droplet is balanced against gravity by the electric force — this stage finds the droplet's charge, using the mass already found in the first stage. Both stages are needed, in that order, for a single droplet before its charge can be calculated.

Key takeaways

Millikan sprayed charged oil droplets between two horizontal parallel plates and observed a single droplet through a microscope.
With the field off, the droplet falls to terminal velocity $v$; Stokes' law ($F=6\pi\eta r v$) balanced against weight gives the droplet's radius, $r = \sqrt{9\eta v/(2\rho g)}$, and hence its mass.
With the field on, the voltage is adjusted until the droplet is held stationary, where $QE=mg$; using $E=V/d$ gives the droplet's charge, $Q = mgd/V$.
Across hundreds of droplets, every measured charge was found to be a whole-number multiple of one smallest value — showing that charge is quantised.
That smallest value is the elementary charge, $e = 1.6\times10^{-19}\,\text{C}$, the charge of a single electron.
Combining Millikan's $e$ with Thomson's $e/m$ gives the electron's mass directly, $m\approx9.1\times10^{-31}\,\text{kg}$.
The experiment requires two separate stages on the same droplet: field-off timing for mass, then field-on balancing for charge.

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