The Speed of Light: Foucault–Michelson Rotating Mirror
Aim: to determine the fundamental speed of light in vacuum \(c\) using a high-speed rotating mirror and folded long-distance optical baseline, measuring beam angular shift \(\Delta\theta\) and micrometer displacement \(s\).
Optical Breadboard Layout
Eyepiece Crosshair View (Micrometer Displacement \(s\))
Calibration Graph: Displacement \(s\) vs Frequency \(f\)
Observation Table
Theory of the Rotating Mirror Method
Pioneered by Léon Foucault (1862) and perfected on a grand scale by Albert A. Michelson (1926 between Mount Wilson and Mount San Antonio), this terrestrial method measures the round-trip transit time of light over an accurately surveyed baseline \(D\).
- A collimated laser beam strikes one facet of an octagonal or flat mirror rotating at frequency \(f\) (revolutions per second).
- The reflected beam travels to a stationary distant concave mirror at distance \(D\) and reflects back along its path. The round-trip transit time is: $$ \Delta t = \frac{2D}{c} $$
- During this transit time \(\Delta t\), the rotating mirror has turned through an angle: $$ \Delta\theta = \omega \Delta t = 2\pi f \left(\frac{2D}{c}\right) = \frac{4\pi f D}{c} $$
- By the law of reflection, turning a mirror by \(\Delta\theta\) deflects the reflected beam by \(2\Delta\theta\). At a focal distance \(D'\) to the micrometer eyepiece: $$ s \approx 2 D' \Delta\theta = \frac{8\pi f D D'}{c} $$
Rearranging for the speed of light:
$$ c = \frac{8\pi f D D'}{s} $$Experimental Precision & Systematic Corrections
In Michelson's 1926 Mount Wilson experiment, the baseline \(D = 35,373.21\) meters was surveyed by the U.S. Coast and Geodetic Survey to an accuracy of 1 part in 1,000,000.
- Air Refractive Index: Light travels in atmospheric air at speed \(v = c / n_{\text{air}}\), where \(n_{\text{air}} \approx 1.000277\) at STP. The vacuum speed of light is recovered by multiplying the measured value by \(n_{\text{air}}\).
- Stroboscopic Tachometer: The rotational speed \(f\) was regulated by tuning an air turbine against an electrically driven tuning fork calibrated to astronomical observatory clocks.
Viva Questions
Why does the deflection angle equal \(2\Delta\theta\) rather than \(\Delta\theta\)?
When a plane reflecting surface tilts by angle \(\Delta\theta\), the angle of incidence increases by \(\Delta\theta\), and by the law of reflection (\(\theta_i = \theta_r\)) the angle of reflection increases by another \(\Delta\theta\), causing a total beam deflection of \(2\Delta\theta\).
Why use an octagonal (8-sided) mirror rather than a simple 2-sided mirror?
If an 8-sided mirror rotates by exactly \(1/8\) of a turn (\(45^\circ\)) during the light transit time, the returning beam reflects off the next adjacent face into the eyepiece at the zero position. This eliminates the need for measuring large linear displacements and yields extreme precision.