A-Level Physics • Light • Measurement

Fizeau’s Terrestrial Measurement of $c$

Paris, 1849: a cogwheel becomes a stopwatch — fast enough to “time” light.

I. The Hinterland: A Very French Race

Fizeau wasn’t just “the cogwheel guy”. Before the speed-of-light work, he and Léon Foucault helped push early photography forward — including the first photograph of the Sun, with sunspots visible.

In 1848 he suggested something conceptually modern: if a star is moving towards or away from us, its light should shift in colour (spectral lines shift), turning light into a velocity probe — an early form of what we now call the Doppler effect for light.

Then in 1849 he went after the headline: a terrestrial value of $c$ using a rotating toothed wheel and a distant mirror about 8.6 km away, so the beam travelled about 17 km on the round trip.

What he actually observed: in an eyepiece, the returning light was bright at low wheel speed. As he increased the rotation rate, the returning light dimmed to (near) darkness at a particular speed (first eclipse), then reappeared at a higher speed, then dimmed again (higher orders). The observable is brightness (time-averaged intensity) — but near the transition you’d also notice flicker.

Legend

● Source

◴ Wheel (representative)

▮ Distant mirror

— Pulse (visual guide)

The sim shows intensity (what the observer judges). The pulse is just a visual guide.

Eyepiece: brightness (with flicker) • below: return-gate trace —%

100% = sunlight-bright glare. 0% = pitch black. The trace shows “gap open” vs “tooth blocks”.

Observed intensity —%

—

Intensity = fraction of emission times that land in a gap on return.

Number of teeth $N$

N = 720

Mirror distance $D$ (m)

D = 8633 m

Wheel frequency $f$ (rev/s)

f = 0.00 rev/s

$\omega = 0$ rad s$^{-1}$

Show pulse Slow mode

Round-trip time —

First-eclipse prediction —

$c$ from $4DNf$ —

II. The Derivation: Timing by Geometry

Step 1: Round-trip light time

The light travels to the mirror and back, total distance $2D$.

t_{\text{light}}=\frac{2D}{c}.

Step 2: Tooth/gap step angle

A wheel with $N$ teeth has $N$ gaps, so there are $2N$ equal rim segments. One segment corresponds to

\Delta\theta_{\text{seg}}=\frac{2\pi}{2N}=\frac{\pi}{N}.

Step 3: First eclipse condition

The first eclipse happens when the wheel turns by exactly one segment while the light is away (gap $\rightarrow$ adjacent tooth):

\omega\,t_{\text{light}}=\frac{\pi}{N}.

Using $t_{\text{light}}=2D/c$ and $\omega=2\pi f$:

2\pi f\left(\frac{2D}{c}\right)=\frac{\pi}{N}\quad\Rightarrow\quad c=4DNf.

The eyepiece shows brightness (time-averaged intensity). Near the transition, the chopping can produce noticeable flicker.