✓ Revise Right course
Stokes' law gives the viscous drag force on a small sphere moving slowly through a fluid: F = 6πηrv. Notes, worked examples and 15 questions answered.
Stokes' Law and Significance of Millikan's Results is part of The discovery of the electron (A-level only) in A Level AQA Physics.
6 min
Reading time
21
Lessons in this topic
A Level AQA
Pathway
Reviewed by Revise Right
Content review
Stokes' law gives the viscous drag force on a small sphere moving slowly through a fluid: F = 6πηrv. Notes, worked examples and 15 questions answered.
Use this lesson to stay inside The discovery of the electron (A-level only) while connecting the detail back to the wider Turning points in physics (A-level only) topic.
Read the core explanations from Stokes' Law and Significance of Millikan's Results before testing yourself from memory.
In 1909, Robert Millikan suspended tiny oil droplets between charged plates and measured something nobody had pinned down before: the exact charge on a single electron. That number — $1.6 \times 10^{-19}$ C — is now one of the most important constants in all of physics.
To get there, Millikan needed to know the size of each droplet precisely. That is where Stokes' Law comes in. Before we reach the significance of the results, we need to understand the physics that made the measurement possible.
Picture a tiny sphere moving slowly through a thick fluid — honey, oil, or even air. The fluid resists the sphere's motion. George Gabriel Stokes showed in 1851 that, for a small sphere moving at low speed through a viscous fluid with laminar (smooth, non-turbulent) flow, the drag force has a beautifully simple form.
Notice that the drag force depends linearly on both $r$ and $v$. Double the radius and you double the drag; double the speed and you double the drag. This linear relationship is what makes Stokes' Law tractable for calculation — and it is only valid for small, slow spheres where the flow remains laminar.
The key assumption behind Stokes' Law is that the flow around the sphere is laminar — the fluid moves in smooth layers with no turbulence. For the microscopic oil droplets in Millikan's experiment (radii of order $10^{-6}$ m moving at fractions of a millimetre per second), this condition is comfortably satisfied.
Here is the clever part. When an oil droplet falls through air with no electric field applied, it accelerates briefly and then reaches terminal velocity. At that point, the net force is zero: the downward weight exactly balances the upward drag plus upthrust.
Writing out the force balance at terminal velocity (and noting that upthrust from air is small but included for precision):
Rearranging for the droplet radius $r$ gives:
This is the critical link between Stokes' Law and Millikan's experiment. By timing how fast a droplet falls, Millikan could calculate its radius — and from the radius, its mass. Knowing the mass was essential for the next stage: measuring the charge.
Worked example: Calculating Oil Droplet Radius Using Stokes' Law
An oil droplet falls through air at terminal velocity $v_0 = 1.2 \times 10^{-4}$ m s$^{-1}$. The dynamic viscosity of air is $\eta = 1.8 \times 10^{-5}$ Pa s, the density of the oil is $\rho = 900$ kg m$^{-3}$, and the density of air is $\rho_0 = 1.2$ kg m$^{-3}$. Calculate the radius of the droplet.
Answer: $r \approx 1.05 \times 10^{-6}$ m (1.05 μm)
Now that we can find the droplet radius from its terminal velocity, we have everything we need to understand Millikan's full method. The experiment has two distinct phases: measuring the droplet's mass (using Stokes' Law as above), and then measuring the charge on the droplet by applying an electric field.
The elegance of the stationary-droplet condition is worth pausing on. When the droplet hangs motionless, there is no drag force at all — the electric force and gravity are in perfect balance. This means the charge calculation $q = mgd/V$ does not require Stokes' Law at this stage; Stokes' Law was only needed earlier to find $r$ and hence $m$.
After repeating his measurements on hundreds of droplets, Millikan noticed a striking pattern. The charges he measured were never random. Every single value was a whole-number multiple of the same tiny quantity: $1.6 \times 10^{-19}$ C.
He never measured a charge of $2.4 \times 10^{-19}$ C, for instance — that would be 1.5 times the fundamental unit, and it did not appear. Charges came in steps: $1e$, $2e$, $3e$, $4e$, and so on, where $e = 1.6 \times 10^{-19}$ C.
Think of it like money. You can have 10p, 20p, or 30p — but not 15p, because coins come in fixed denominations. Charge works the same way: it comes in fixed units of $e$, and you cannot have a fraction of one.
This was a profound result. Before Millikan, the idea that charge might be quantised was theoretical — J.J. Thomson had discovered the electron in 1897 and proposed it carried a fixed charge, but nobody had measured that charge directly on individual particles. Millikan's experiment turned a hypothesis into a measured fact.
The significance of Millikan's experiment extends well beyond a single number. It represents a turning point in physics for several interconnected reasons, and AQA expects you to be able to evaluate these reasons with precision.
When evaluating significance in an exam, the strongest answers connect multiple consequences. Millikan did not merely measure a constant — he demonstrated a fundamental principle (quantisation) and enabled a cascade of further measurements (electron mass) that shaped twentieth-century physics.
Questions on this topic tend to fall into three categories: Stokes' Law calculations, description of Millikan's method, and evaluation of significance. Each has its own pitfalls.
Stokes' Law and Millikan's experiment together illustrate how a single, carefully designed measurement can unlock multiple fundamental constants and principles. The radius formula connects macroscopic observations (a falling droplet's speed) to microscopic quantities (the charge on a single electron) — a chain of reasoning that exemplifies the power of physics at its best.
Published by Revise Right. Use the official specification reference to check the requirements for your course.
Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Stokes' law gives the viscous drag force on a small sphere moving slowly through a fluid with laminar flow. The drag depends on the fluid's viscosity η and depends linearly on both the sphere's radius r and its speed v: double the radius or the speed and the drag doubles.
Quantisation of charge means electric charge does not vary continuously but only exists in discrete integer multiples of the elementary charge e = 1.6 × 10⁻¹⁹ C, so Q = ne. Millikan never measured a charge such as 2.4 × 10⁻¹⁹ C, which would be 1.5 times the fundamental unit.
Dynamic viscosity, η, is a measure of a fluid's resistance to flow, in units of Pa s (pascal-seconds). A higher value of η means a thicker, more resistive fluid; air has η ≈ 1.8 × 10⁻⁵ Pa s at room temperature.
The radius of an oil droplet comes from its terminal velocity v: r = √(9ηv ÷ (2(ρ − ρ₀)g)), where ρ is the oil density and ρ₀ the air density. For v = 1.2 × 10⁻⁴ m s⁻¹, η = 1.8 × 10⁻⁵ Pa s and ρ = 900 kg m⁻³, r ≈ 1.05 × 10⁻⁶ m.
Millikan's results gave the first direct measurement of the elementary charge e and proved that charge is quantised. Combined with Thomson's charge-to-mass ratio, his value of e allowed the electron mass, about 9.11 × 10⁻³¹ kg, to be calculated for the first time, and supported the foundations of quantum theory.
Stokes' law is only valid for small, slow spheres where the flow remains laminar, meaning the fluid moves in smooth layers with no turbulence. Millikan's oil droplets, with radii of order 10⁻⁶ m moving at fractions of a millimetre per second, this condition is comfortably satisfied.
Stokes' drag depends linearly on both the radius and the speed of the sphere: doubling the radius doubles the drag, and doubling the speed doubles the drag. This linear relationship makes Stokes' law simple to use, but it only holds for small, slow spheres where the flow remains laminar.
Laminar flow is smooth, non-turbulent flow in which the fluid moves in layers. Stokes' law assumes the flow around the sphere is laminar, a condition comfortably satisfied by Millikan's oil droplets, with radii of order 10⁻⁶ m moving at fractions of a millimetre per second.
Upthrust from the air is included when finding the radius of an oil drop for accuracy, because at terminal velocity the weight balances the drag plus upthrust. It gives r = √(9ηv ÷ 2(ρ − ρ₀)g), though air density, 1.2 kg m⁻³, is much smaller than oil density, so the difference is close to 900 kg m⁻³.
The mass of the electron was first calculated by combining Millikan's value of e with Thomson's earlier measurement of the charge-to-mass ratio e/m. Dividing e by e/m gives an electron mass of about 9.11 × 10⁻³¹ kg.
Millikan never measured a charge of 2.4 × 10⁻¹⁹ C because that would be 1.5 times the fundamental unit, and charge only comes in whole-number multiples of e = 1.6 × 10⁻¹⁹ C. Charges came in steps of 1e, 2e, 3e, 4e and so on, much as coins come in fixed units.
The stationary oil drop calculation, q = mgd ÷ V, does not use Stokes' law, because a motionless droplet has no drag force: the electric force and gravity are in perfect balance. Stokes' law was only needed earlier to find the radius, and hence the mass, from the terminal velocity.
Stokes' law was found by George Gabriel Stokes in 1851. He showed that for a small sphere moving at low speed through a viscous fluid with laminar flow, the drag force is F = 6πηrv, depending linearly on the radius and the speed.
Q = ne means any measured charge Q is a whole-number multiple of the elementary charge e = 1.6 × 10⁻¹⁹ C, where n is a positive or negative integer. Electric charge does not vary continuously; it only exists in discrete multiples of e, a fact shown by Millikan's results.
Millikan's experiment was important for quantum theory because the quantisation of charge was an early and direct demonstration that nature operates in discrete steps at the atomic scale, a principle that underpins all of quantum mechanics. It also supported the picture of matter as made of discrete charged particles.