How to find the speed of light (Fizeau experiment)

It's the 19th century and we want to find the speed of light.

For this setup, we will need a motorized rotating wheel, which has a large number of teeth and gaps like a gear. It's important that the width of the teeth are exactly the same as the width of the gaps.

The next thing we need is a light source. Let's assume it's a strong laser for simplicity. The same can be achieved with say a light bulb or even a candle with an appropriate set of lenses.

We will also need a semi-reflective mirror. It is a partially glazed glass which allows some light to pass through it and some would get reflected. We'll need to keep it tilted for reasons which will be obvious very soon.

And lastly, we need a regular mirror on the other side of the wheel.

With the setup as shown, when we switch on our laser, the light passes through the semi-glazed mirror. Some of the light would get reflected as well, but we'll ignore that as it is not relevant to this experiment.

The light is pointed towards the outer rim of the wheel where there are teeth. If the wheel rotation is such that the light encounters a gap, it will pass through it and hit the mirror on the other side.

The light beam then gets reflected and traces its path through the gap between the teeth again and hits the semi-glazed mirror. Some of the light passes through it, but again we'll ignore that part. The remaining light will get reflected and here is where we will observe the light.

Now, let's start rotating the wheel. Doing this will cause the light to flicker at our observation point.

We'll keep increasing the speed of rotation gradually. As we do that, at one point we'll observe that the light is completely blocked and no light can reach our observer. Let's see what causes this to happen.

This will only happen when the light passing through the gap between the teeth, by the time it reaches back to the wheel, the wheel has moved exactly one tooth distance. And so it blocks the reflected light beam.

As it has moved exactly one tooth distance, all the light passing through the gap will get blocked. The light from the rightmost edge of the gap will hit the rightmost part of the adjacent tooth. Light passing through the middle of the gap will hit the middle and light passing through the left edge of the gap will hit the left edge of the tooth. So all the light is blocked from the observer's point of view.

From this we can see that the time taken by the light to travel from the rim of the wheel to the mirror and back is same as the time it takes for the wheel to rotate by the width of one tooth. With this information, we can figure out the speed of light if we know the speed at which the wheel is rotating. Let's do some math and find out the speed.

If the number of teeth on the wheel is say 720 and the speed of the wheel at which light blocks is 13 revolutions per second, which means time it takes to rotate one full circle is 1/13 seconds. We can now find out the time it takes for the wheel to move the distance of one tooth. There are 720 teeth and 720 gaps of the same length. So if we divide the speed of the full rotations by 720 + 720, we get the time it takes to rotate one gap length.

Now, the distance between the wheel and the mirror is 8 kilometers or 8000 meters.

Yeah, it has to be that large. So the distance light travels from the wheel to the mirror and back is twice of that, that is 16 kilometers or 16,000 meters.

If we denote C as the speed of light, the time taken by the light to travel 16,000 meters is 16,000 divided by C. Now this time must exactly match the time it takes to rotate the wheel by one tooth distance to effectively block the light completely, which gives us speed of light to be approximately 3 times 10 raised to 8 meters per second.