Chapter 9

# Fermat's principle

Take 10 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

Pierre de Fermat (French mathematician and physicist, - ) postulated  that the light rays met a very general principle that the path taken by the light to travel from one point to another was one for which the travel time was minimum (indeed, an extreme that can be minimum or maximum).

A swimming coach, located at a point beach, wants to apply this principle to rescue as quickly as possible a vacationer (located in ) about to drown close to the beach.

We note and velocity vectors (supposed constant) of the swimming coach on the beach (when running) and in water (where he swims).

## Question

What should be the path followed by the swimming coach in order to have Fermat's principle verified and the vacationer safe ?

Deduce the expression of the law of refraction in optics.

### Solution

We choose a frame of reference that simplifies the problem : let pass the x-axis through the straight line that separates the beach from the sea and the y-axis through point , initial position of the swimming coach.

In such a frame of reference, points and have the coordinates and .

The swimming coach's trajectory will consist of two straight portions and , where is the point where the swimming coach starts swimming.

One can notice that the distance will be greater than the distance since the swimming coach will certainly run faster than current swimming !

The time taken by the swimming coach to get from to is :

By developing values of and , we obtain the following dependence of as a function of the abscissa of  :

The extremum of is reached when the derivative with respect to is zero.

Yet :

Noting that (see figure for the definitions of the angles) :

And :

The condition of extremum time made by the light is then expressed as (where the angles and , by analogy with optics, see figure above, can be called angle of incidence and angle of refraction) :

It is obvious that this time extremal corresponds to a minimum ; indeed, the distance and thus the time can easily be made very large if the swimming coach, then certainly lacking professional conscience, decided to go shopping for example before rescueing the poor vacationer !

#### Complément : Case of light and laws of Snell-Descartes

Consider two mediums and of the respective refractive indexes and .

Given two points and respectively located in the medium of index (point ) and in the medium of index (point ).

Fermat's principle unables to assert that the path taken by the light to get from to is such that the time taken by light is extreme (usually, minimum).

By applying this principle, similar reasoning to that performed in the case of the path followed by the swimming coach, is used to demonstrate the stated law of refraction, around 1620, by physicists Snell and Descartes :

Where and are respectively the angles of incidence and refraction.

Remember that the index of a medium used to find the velocity of light in this medium as a function of that in vacuum is :

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