Fermat’s Principle

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What is Fermat's Principle?

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 the 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 wavefront is growing from point A, then it will sweep all the possible rays radiating from A.  However, if the wavefront reaches the point B, then it will not only sweep all the ray paths with the same endpoints but also infinitude of nearby paths. 

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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 reach 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 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 dt refers to the time difference taken along the path, that is; 

dt/ds = 0

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The diagram shows that the ray of light passing from point A that is in medium 1 to point B that is in medium 2 covers the shortest distance possible.

Fermat’s Principle- Applications

  • From Fermat's principle, it is found that in homogenous medium rays of light are rectilinear. It shows that light travels in a straight line in a medium having a constant index.

  • The angle of incidence is equal to the angle of reflection, which is often called the Law of Reflection. 

  • Fermat’s principle refers to the fundamental law of optics that are used in the derivation of other laws of geometrical optics.

  • One can make some useful observations about the conic surfaces on the basis of Fermat’s principle. 

For Example:

Two conjugate points are chosen that are perfect images of each other. The points are chosen in such a way that the optical path length of all paths connecting them is equal. In the case of an ellipse, the point source is located at one focus and is an image of the point located at another focus. However, in the case of the parabola, the point on one focus should be infinite.

FAQ (Frequently Asked Questions)

1. What is the Basis of Fermat’s Law?

The transmission of the light is in between the two points when it follows a path to flow. Fermat’s Principle explains that the angle of reflection is equal to the angle of incidence of the light ray. This principle has given rise to Snell’s law of light. According to the basic definition of Fermat’s law, light always tries to follow the shortest path for traveling. 

2. What is Snell’s Law Equation of Light?

Suppose if n1 and n2 are the two incident rays of refraction on two different media. Also, suppose a1 and a2 are the angles of incidence and refraction. Now if the ray R makes a perpendicular line with the normal on the surface of incidence, then, 

Snell’s law is given by n1/n2 = sin a2/sina1