# Fizeau–Foucault apparatus

Figure 1: Schematic of the Foucault apparatus. Left panel: Light is reflected by a rotating mirror (left) toward a stationary mirror (top). Right panel: The reflected light from the stationary mirror bounces from the rotating mirror that has advanced an angle θ during the transit of the light. The telescope at an angle 2θ from the source picks up the reflected beam from the rotating mirror.
Figure 2: Schematic of the Fizeau apparatus. The light passes on one side of a tooth on the way out, and the other side on the way back, assuming the cog rotates one tooth during transit of the light.

The Fizeau–Foucault apparatus (1850) (Figure 1) was designed by the French physicists Hippolyte Fizeau and Léon Foucault for measuring the speed of light. The apparatus involves light reflecting off a rotating mirror, toward a stationary mirror some 20 miles (35 kilometers) away. As the rotating mirror will have moved slightly in the time it takes for the light to bounce off the stationary mirror (and return to the rotating mirror), it will thus be deflected away from the original source, by a small angle.[1] If the distance between mirrors is h, the time between the first and second reflections on the rotating mirror is 2h/c (c = speed of light). If the mirror rotates at a known constant angular rate $\omega$, the angle θ is swept in the same time as the light roundtrip, so:

$\frac{\theta}{\omega} = \frac {2h}{c}=t \ .$

In other words the speed of light is calculated from the observed angle θ, known angular speed and measured distance h as

$c = \frac {2 \omega h}{\theta } \ .$

The detector is at an angle 2θ from the source direction because the normal to the rotating mirror rotates by θ, decreasing by θ both the angle of incidence of the beam and its angle of reflection.

Foucault based his apparatus on an earlier experiment by Fizeau (Figure 2) who, in 1849, used two fixed mirrors about 8 km apart, one partially obscured by a rotating cogwheel with over 100 teeth rotating a few hundred times a second.[2] Fizeau's value for light's speed was about 5% too high.

The Fizeau experiment to measure the speed of light in water has been viewed as "driving the last nail in the coffin" of Newton's corpuscle theory of light when it showed that light travels more slowly through water than through air.[3] Newton predicted refraction as a pull of the medium upon the light, implying an increased speed of light in the medium. However, Foucault showed the speed of light in water to be less than in air, not more, by inserting a tube of water in the light path.[4]

## References

1. ^ Ralph Baierlein (2001). Newton to Einstein: the trail of light : an excursion to the wave-particle duality and the special theory of relativity. Cambridge University Press. p. 44; Figure 2.6 and discussion. ISBN 0-521-42323-6.
2. ^ Abdul Al-Azzawi (2006). Photonics: principles and practices. CRC Press. p. 9. ISBN 0-8493-8290-4.
3. ^ David Cassidy, Gerald Holton, James Rutherford (2002). Understanding Physics. Birkhäuser. ISBN 0-387-98756-8.
4. ^ Bruce H Walker (1998). Optical Engineering Fundamentals. SPIE Press. p. 13. ISBN 0-8194-2764-0.