Speed of Light
How people timed the fastest thing there is.
A Constant Worth Measuring
From lanterns on hills to a constant fixed by definition.
Light is so fast that for most of history it seemed to arrive instantly. Measuring its speed took clever astronomy, then spinning wheels and mirrors, and finally led us to fix it by definition.
A very special speed
The speed of light in vacuum is a fundamental constant with a remarkable property: anything moving at this speed in one frame of reference moves at the same speed in every other frame. This forced a complete rethink of space and time, the basis of the special theory of relativity.
Fixed by definition
Since 1983 the speed of light has been defined to be exactly , and the metre is defined as the distance light travels in 1/299,792,458 of a second. So today, timing light over a path measures the length of the path, not c. Before 1983 the metre was defined separately, and c was a quantity to be measured.
Galileo’s lanterns
Galileo proposed an experiment with two people on distant hills, each with a covered lantern. A uncovers his lantern; B uncovers hers the moment she sees A’s light; A covers his the moment he sees B’s. The time between A uncovering and covering, together with the distance, would give the speed.
Bars on a logarithmic scale (1 µs to about 3 s). Ratio ≈ 3,997 to 1.
It couldn’t work: for hills 15 km apart, light takes only about 10⁻⁴ s for the round trip, thousands of times shorter than human reaction times.
Astronomy gets there first
- Rømer (1676) noticed that eclipses of Jupiter’s moon Io came progressively later as the Earth moved away from Jupiter: light needed extra time to cross the extra distance. His result was about 2.1 × 10⁸ m/s, too low but the right size.
- Bradley (1728) found that stars appear slightly shifted in the direction of the Earth’s motion (aberration of starlight), which gave a value quite close to the correct one.
The first measurements on Earth came in the 1800s with Fizeau, Foucault and Michelson: the next two lessons.
If each lantern operator takes 1/100 s to work the shutter, how far apart must they be for that delay to be only a tenth of light’s one-way travel time?
- 1c is the same in every frame: the basis of special relativity.
- 2Since 1983 c = 299,792,458 m/s exactly; the metre is defined from it.
- 3Galileo's lantern method failed: reaction times dwarf light's travel time.
- 4Rømer (Io's eclipses, ≈ 2.1 × 10⁸ m/s) and Bradley (stellar aberration) made the first estimates.
Check yourself
Fizeau's Toothed Wheel
Chop light into pulses, send them to a far mirror, and spin until the returning light hits a tooth.
In 1849 Hippolyte Fizeau timed light over a 17-kilometre round trip using nothing faster than a spinning gear wheel.
The apparatus
Light from a source is reflected by a half-silvered glass plate and focused at the rim of a toothed wheel. Light passing through a gap is made into a parallel beam by a lens, travels several kilometres (8.6 km in Fizeau’s experiment), and is focused by another lens onto a mirror at its focal plane. It returns along the same path to the wheel. If it finds a gap, it passes through the glass plate to an observer looking through a telescope.
The eclipse
Spinning the wheel chops the light into bursts. At low speed the image flickers. As the wheel speeds up, a speed is reached where, in the time light takes to go to the mirror and back, the wheel turns so that the next tooth sits where the gap was. Every burst then returns to a tooth, and the image disappears completely.
With n teeth, a tooth replaces the neighbouring gap after the wheel turns by π/n. At angular speed ω this takes (π/n)/ω, during which light covers 2D:
A wheel with 500 teeth is 8.0 km from the mirror. At what rate does the image first vanish?
- 1A toothed wheel chops light into pulses that travel D to a mirror and back.
- 2The image first vanishes when a tooth replaces a gap during the round trip.
- 3c = 4Dnν.
- 4Drawbacks: dim image, needs kilometres of space.
Check yourself
Videos, simulations & further reading (3)
Foucault's & Michelson's Rotating Mirrors
A spinning mirror turns a tiny time delay into a measurable angle.
A mirror spinning hundreds of times a second turns the tiny time light spends in flight into an angle you can measure with a microscope.
Foucault’s rotating mirror
Light from a source S passes through a glass plate and a converging lens, then strikes a plane mirror M₁ that can rotate fast. M₁ reflects it to a concave mirror M₂ whose radius of curvature R equals its distance from M₁. The central ray strikes M₂ normally and comes straight back to M₁, then back through the lens, and the glass plate reflects it to form an image I of the source.
If M₁ is still, the image doesn’t move. If it spins, it turns through a small angle Δθ in the time 2R/c the light spends going to M₂ and back. The returning beam is then reflected at 2Δθ to its old direction, and the image shifts to I′, a distance s away.
The formula
The light reflected by M₁ seems to come from the image of M₂’s centre in M₁, which moves by R·2Δθ. The lens images this onto the source’s plane with magnification a/(R + b) (a = source to lens, b = lens to M₁). With Δθ = 2Rω/c:
Fixed: lens to rotating mirror b = 5 m, source to lens a = 2 m.
R = 20 m, b = 5.0 m, a = 2.0 m and the mirror turns at 500 revolutions per second. Find the image shift.
Michelson’s rotating polygon
Michelson replaced the plane mirror with a polygon of N mirror faces. Light reflects off one face, travels a long path (to a distant concave mirror and back, many kilometres), and returns to strike the polygon at another face before entering a telescope.
If the polygon is still, the image is seen. Spinning it, the returning light generally finds a face at the wrong angle and the image flickers and vanishes. But at one speed the polygon turns exactly one face (2π/N) while the light is away, and the next face sits exactly where the last one was: the image is steady again. With D the path between the two reflections:
Light travels 70 km between reflections from an eight-sided rotating mirror. At what lowest speed is the image steady?
Michelson first used an octagonal mirror in 1879; the last of his series, finished by Pease and Pearson in 1935, used a 32-sided mirror.
- 1Foucault: a spinning mirror turns by Δθ = 2Rω/c while light makes a round trip; the image shifts by s.
- 2c = 4R²ωa/[s(R + b)]; fits in a lab, and showed light is slower in water.
- 3Michelson: an N-faced mirror; the image is steady when one face advances during the light's trip.
- 4c = DνN.