[Physics Class Notes] on Fermat’s Principle Pdf for Exam

There was no simple relationship determined between the angles of refraction and incidence since 1621. Dutch investigator Willebrord Snell in 1621 claimed that it is the sines of the angles of refraction and incidence that uphold a constant ratio. Though the statement claimed was correct; however, the observation has not addressed any cause.  However, Fermat was able to determine the cause of the refractive behavior of light. In 1662, Fermat demonstrated that a beam of light follows a path of the fastest time instead of a smaller distance.  For the refraction and reflection at the plane surfaces, Fermat’s Principle of Least Time holds. It means that a ray of light passing from one point to another chooses a certain path along which the time taken is minimum. However, for the spherical surfaces, the time taken by the light ray is either maximum or minimum. 

Who Proposed the Fermat’s Principle?

Fermat’s principle was stated by Pierre de Fermat in 1662 and is suitable to study optical devices. Initially, the principle was a controversial statement as it appeared to assign knowledge of nature’s intent. Until the 19th century, nobody was aware that alternative paths traveling from one point to another are a fundamental property of waves. 

When two points A and B are specified, and a wave front is growing from point A, then it will sweep all the possible rays radiating from A.  However, if the wave front reaches point B, then it will not only sweep all the ray paths with the same endpoints but also the infinitude of nearby paths. 

()

The diagram shows that any light rays passing from one medium to another medium bend depending on the rarer and denser medium. 

Fermat’s principle demonstrates that any ray that occurs from one point to another point will cover the smallest path possible. 

State Fermat’s Principle

According to Fermat’s principle, light traveling between two points pursues a path such that the optical length between the points is equal. The principle is the link between wave optics and ray optics. 

One of the great ways to state Fermat’s principle is that the path taken by light rays in traveling between two points involves either a maximum or minimum time. It means that two light rays deviating from a distant object will have the same optical path lengths. According to Fermat’s principle of Least Time, the path traveled by light rays between the given two points is the actual path that can be traversed in the least time. 

An Example of Fermat’s Principle

Consider a beam of light traveling from point A to point B. Here, point A is in the air, and point B is in the glass. Fermat presented that the path of light rays is specified by the Principle of Least Time. Fermat’s Principle of Least Time states that a light ray going from A to B will take a shorter time. 

As the speed of light is identical everywhere along all the possible paths, the shortest path is the one that involves the shortest distance. 

According to Fermat’s principle, the entire path made by light to travel from A to B should satisfy the boundary conditions along with the condition that ray takes the path that requires the least amount of time. 

Let the path taken by ray to travel from A to B is equal to ds. Here, ds refer to the small difference of the first order and it refers to the time difference taken along the path, that is;