Physics · Optics · Grade 9

Measuring the speed of light

How was light, travelling 300 thousand km every second, ever measured?

The situation

Mission control sent a “stop!” command to the rover on Mars. Radio waves also travel at the speed of light, but depending on how far away Mars is, the command spends from about 3 minutes to 22 minutes on the way — so the rover detects obstacles itself. On Earth, light goes round the equator more than 7 times in a second. How did people measure such a speed in the 17th century with a telescope, and in the 19th with a toothed wheel and a rotating mirror?

Topic

Measuring the speed of light. Rømer and the moons of Jupiter, Fizeau's wheel and the Michelson experiment

Key ideas

Rømer's method
Fizeau's wheel
v = c/n

Formulas

c = \frac{2 L}{τ}

The basis of each method: measure the time $τ$ that light takes to travel a distance $L$ and back.

c = \frac{Δ d}{Δ t}

Rømer's method: the eclipses of Io are delayed by $Δ t$ when the Earth–Jupiter distance increases by $Δ d$ .

c = 4 N L ν_{1}

Fizeau's wheel: $N$ teeth, distance $L$ ; the light is blocked for the first time at frequency $ν_{1}$ (by the time it returns, the wheel has turned by half a tooth period).

c = 16 L ν_{0}

Michelson's 8-sided mirror: by the time the light returns, the prism has turned by 1/8 of a revolution and the image stays in place.

v = \frac{c}{n}, u = \frac{c}{n} \pm (1 - \frac{1}{n^{2}}) v

In a medium light is $n$ times slower. In flowing water (Fizeau, 1851) not the whole water speed $v$ is added, only a fraction $(1 - 1 / n^{2})$ — this follows from the velocity addition law of the theory of relativity for $v ≪ c$ .

Theory

Rømer: a moon of Jupiter — a clock in the sky

Io orbits Jupiter in 1.77 days and enters its shadow on every orbit. In 1676 Ole Rømer noticed that the eclipses are delayed when the Earth moves away from Jupiter and come early when it approaches: the light has to cover an extra distance. For the diameter of the Earth's orbit (2 AU) the delay is $2 AU / c \approx 16.6$ min. On the stage you will see on the graph by how much the observed time of each eclipse differs from the calculated table ( $O - C$ ). In the months when Jupiter appears close to the Sun (angle less than 20°) it cannot be observed. Model: orbits are circular, Jupiter and Io are drawn enlarged.

Fizeau's toothed wheel (1849)

Light passes through a gap in a 720-tooth wheel, is reflected from a mirror 8 633 m away, and must pass through the wheel again. If the wheel turns slowly, the beam returns through the same gap. As the speed increases and the wheel turns by half a tooth period while the beam returns ( $τ = 2 L / c \approx 57.6$ µs), a tooth blocks the returning beam — the eyepiece goes dark: $ν_{1} = c / (4 N L) \approx 12.06$ rev/s. At twice the speed the beam passes through the next gap, at three times it is blocked again. With this method Fizeau obtained $c \approx 3.13 \cdot 10^{8}$ m/s.

Rotating mirror: Foucault and Michelson

In Foucault's method (1862), by the time the beam reflected from a fast-rotating mirror returns from a distant concave mirror, the mirror has turned by an angle $ω τ$ and the returning beam is deflected by $2 ω τ$ — the image on the scale shifts: $Δ = 2 ω τ R$ . Michelson (1926) used an 8-sided mirror between Mount Wilson and Mount San Antonio (≈ 35 km): if by the time the beam returns the prism has turned by exactly 1/8 of a revolution, the next face is in exactly the place of the previous one and the image does not shift. Finding this frequency, he obtained $c = 299 796$ km/s.

Light is an electromagnetic wave; slower in a medium

According to Maxwell's theory, light is an electromagnetic wave, and its speed in vacuum is $c = 1 / \sqrt{ε_{0} μ_{0}}$ . In a medium it is ** $n$ times slower**: $v = c / n$ (in water ≈ 225 thousand km/s, in diamond ≈ 124 thousand km/s); in 1850 Foucault showed by experiment that light is slower in water than in air. A wave carries energy: energy density $w = ε_{0} E^{2}$ , intensity $I = c w$ . For example, if the intensity of sunlight at the top of the Earth's atmosphere (the solar constant) is ≈ 1361 W/m², the root-mean-square value of the field strength is ≈ 716 V/m.

Fizeau's experiment (1851) and the addition of velocities

If velocities added according to Galileo, light along the stream would have to travel at $c / n + v$ . Fizeau sent two beams through flowing water in opposite directions and measured the shift of the interference fringes: the shift turned out much smaller than Galileo's prediction and matched the Fresnel coefficient $f = 1 - 1 / n^{2} \approx 0.44$ . This result was later derived exactly from the velocity addition law of the theory of relativity: $u = (c / n + v) / (1 + v / (n c))$ . Other phenomena of this kind — the aberration of stars (Bradley, 1729) and the relativistic Doppler effect — are in the “Special relativity” experiment.

The constancy of the speed of light

The Michelson–Morley experiment (1887) showed that the motion of the Earth does not affect the speed of light: $c$ is the same in all inertial frames of reference. This is the basis of Einstein's theory of relativity and the limit of applicability of classical mechanics: as speeds approach $c$ , Galilean addition becomes wrong, and no body can move faster than $c$ . Since 1983, $c = 299 792 458$ m/s has been an exact number, and the metre is defined through it.

Did you know?

In 1676 Ole Rømer found that the eclipses of Io are delayed when the Earth moves away from Jupiter, and estimated that light crosses the diameter of the Earth's orbit in about 22 minutes; the modern value is about 16.6 minutes.
Since 1983 the metre has been defined through light: the metre is the distance travelled by light in vacuum in 1/299 792 458 of a second. So c = 299 792 458 m/s is no longer a measured quantity but an exactly defined number.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

More experiments in this section

Car mirrors and the periscope — Grade 7
Laser, water and optical fibre — Grade 7
The eye and spectacles — Grade 7
Shadow and eclipses — Grade 7
Optical instruments — Grade 9
Interference and diffraction — Grade 11
Polarisation of light — Grade 11
Laser physics and holography — Grade 11
Luminous flux and illuminance — Grade 11
Infrared, UV and X-rays — Grade 11