Learning from Light

Picture yourself as a cop. You are in your jeep (which is stationary) on a patch of sand which happens to be one one of the sides of a cement road. On the other side of the cement road, there is an identical patch of sand on which there is a notorious criminal in his vehicle. You have been assigned to capture the criminal and bring him to justice.   

To chase him, you need to approach him in your vehicle, but there is a catch. As soon as you start, the criminal will hear the sound of your engine. As of now, he/she doesn’t know where you are but the hum of your engine will give yourself away. You need to find the fastest path to the culprit and remember your jeep is much faster on the road than it is on the dust track.

The question here is, — what will the fastest path look like?

The answer may surprise you at first. The fastest way to get to the criminal will be by meeting the road at a 42 degree angle.

This is because of Fermat’s Principle. Here we introduce light, the entity from which we derive this answer to the posed question. 

By Fermat’s Principle, light takes the fastest and mind you, not necessarily the shortest (by distance) path from two points. 

When light travels from one medium to another it takes the fastest possible path, — which need not be a straight line. When light travels from one medium to another at an angle of incidence of 42 degrees, it moves along the boundary between the two mediums. Here, the edge of the straight cement road resembles the boundary. The greatest angle at which a ray of light can strike a boundary between two different mediums is known as the critical angle. An angle smaller than the critical angle will cause the ray of light to refract. If the angle is greater, the ray of light will reflect off the surface. Note that the critical angle is different in different substances. 42 degrees is the critical angle of light in glass. With this we can say that the 42 degree angle will be the best angle at which the cop should incident. After the cop continues on the road, he/she should leave the road at 42 degrees to reach the criminal in the shortest amount of time. This is due to the reversibility of light.

Now, you can catch the criminal with the help of some basic physics and bring him/her to justice, all with the help of light!

-- Vedang Mahajan