Speed of Light Demonstration by the Foucault Method
PARTICLE Summer Workshop
In the mid-19th century, French physicist Leon Foucault made the most accurate measurement to-date of the speed of light using a laboratory-sized apparatus consisting of rotating and fixed mirrors. This method was subsequently improved upon in technique and scale by Albert Michelson in the 1920s, resulting in the most precise “mechanical” measurement of the speed of light, with an error of only ±4km/sec! An apparatus inspired by Foucault’s original technique is accessible as a classroom demonstration or laboratory, and makes an excellent way to illustrate the finite (but large) speed of light.
The basic technique is to send a beam of on a path so that it bounces between a rotating mirror, a fixed mirror and back to the rotating mirror for a total distance 2D, as shown in the above illustration. The time of flight of the light on this path implies that the rotating mirror will have turned very slightly between the two times of arrival of the light. This small rotation will deflect the beam of light through a small angle θ from its original path, producing a measurable effect proportional to the mirror and inversely proportional to the speed of light.
If the rotational frequency (revolutions per unit time) of the mirror is given by υ, then the deflection angle is given by
where the second factor angular frequency of rotation and the last is the time to travel back and forth between the mirrors.
For our demonstration, we used a He-Ne laser as the source (a much better source than Foucault had!). At these distances, the laser acts essentially as a point source of light, so it is necessary to focus the beam so that the points of focus are the fixed mirror (M) and source and observation point (S). Let D’ be the distance between the rotating mirror and the source/observation point, G be the distance between the lens (L) and fixed mirror (M) and G’ be the distance the light travels from the lens to the source point (LR+RS). Then the lens makers’ equation tells us that the focal length, f, of the lens must be:
For our setup, we used a lens with a focal length of 5m, and set G and G’ to be 10m. We set the distance D to be approximately 13m and D’ to be 7m. In order to observe the deflection of the beam, we inserted a thin (to minimize internal reflections) microscope slide beam splitter just before the source. Our rotating mirror is based on a Bosch router motor, with a peak speed of 27000rpm at 120V AC.
Alignment is slightly tricky. I suggest establishing the beam to the rotating mirror first, and then moving the rotating mirror to point approximately at the desired location of the fixed mirror. The beam should go to the center of the lens and the fixed mirror, and then (harder) the spot returning from the fixed mirror must be aligned on the rotating mirror. At this point, the returned beam should be visible after the beam splitter! Chalk dust and water mist make great diagnostics for finding the beam and small variations in the lens position sometimes can compensate for small mirror misalignments. This takes some patience and good clamps for the lens and fixed mirrors!
One you have established a view of the beam after the beam splitter, I suggest measuring the beam spot position at high and low frequency. The deviation of the beam spot, δx, will be
where δυ is the difference between the high and low frequencies. For our setup, we found deviations of about 2-3mm with δυ of about 300Hz.