The apparatus used by the French scientist Armand Fizeau in...

The apparatus used by the French scientist Armand Fizeau in 1849 for measuring the speed of light is illustrated in $v$ Fig. $26.19 .$ Teeth on a rotating wheel periodically interrupt a beam of light. The flashes of light travel to a plane mirror and are reflected back to an observer. Show that if the wheel is rotated at just the right frequency $f$, the light passing through one gap reaches the mirror and is reflected to the observer through the very next gap. When $f$ is adjusted so this happens, show that the speed of light is given by $c=2 f N L,$ where $N$ is the number of gaps in the wheel, $f$ is the frequency (in revolutions per second) made by the rotating wheel, and $L$ is the distance between the wheel and the mirror.

Key Concepts

Relationship Between Rotational Frequency, Number of Gaps, and Distance

The derivation of the speed of light in this experiment depends on linking the rotational frequency of the wheel, the number of gaps (which translates to the spacing in time between successive open positions), and the distance to the mirror. This connection leads to the formula c = 2fNL, where the round-trip distance (2L) is divided by the effective time delay determined by the product of the rotational frequency and the number of gaps.

Time-of-Flight Measurement

This concept involves determining the speed of light by measuring the time it takes for a light beam to travel a known distance and return to its source. By knowing the distance and the time delay, one can calculate the speed, which in this context utilizes the round-trip distance of the light beam.

Periodic Interruption Using a Rotating Device

A rotating wheel with a series of gaps is used to periodically allow and block the passage of light. This periodic interruption serves as a timing mechanism, enabling the precise control and measurement of when light pulses are emitted and received, which is critical for establishing correspondence between the rotation speed and the light’s travel time.

Synchronization of Optical Signals

The principle of synchronization here is that light passing through one gap of the rotating wheel returns to be observed through the next gap when the wheel rotates at just the right frequency. This alignment of the light's travel time with the intervals between successive gaps allows for an accurate determination of the speed of light by matching the mechanical time intervals with the optical delay.

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