Principle of Least Time Question

[EternusVia]

Hi everyone,

I'm taking my second semester of calc based physics this coming spring. I have a question in optics stemming from my reading of Feynman's lectures on physics. He gives a rough definition of Fermat's principle of least time as follows:

"... out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time." (Feynman's emphasis)

My question is this. We know that there exist certain refractive indices for light as it passes through various media. Say we have a rectangular prism. Light is hitting one of the long edges at an angle. According to my understanding of Fermat's PoLT as stated above, the light should travel through the prism forming a right angle at the entry point and a right angle at the exit point (on the inside of the prism). This would minimize the amount of time that the light is in the slow medium, the glass. The total travel time for light would also be minimized, I think.

Of course, this doesn't happen in real life. So my understanding of Fermat's PoLT must be lacking. Could someone explain where I'm going wrong with light and the principle of least time?

Something like this happens in real life:

 

[Stephen Tashi]

He gives a rough definition of Fermat's principle of least time as follows:

"... out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time." (Feynman's emphasis)

Applying that principle requires being given two points that the light "wants" to go through.

 

[NascentOxygen]

This would minimize the amount of time that the light is in the slow medium, the glass. The total travel time for light would also be minimized, I think.

Of course, this doesn't happen in real life. So my understanding of Fermat's PoLT must be lacking. Could someone explain where I'm going wrong with light and the principle of least time?

A perpendicular crossing does minimize the time spent in the glass, but it extends the path through air of that emergent ray.

.

 

[Dale]

This would minimize the amount of time that the light is in the slow medium, the glass. The total travel time for light would also be minimized, I think.

The time in the slower medium would indeed be minimized, but not the total time.

Do you know how to minimize a function of two variables?

 

[PJC01]

Hello,

First of all, it's great that you're already exploring and questioning concepts in physics, especially in optics! The principle of least time, also known as Fermat's principle, is a fundamental concept in optics that helps us understand the behavior of light in different media.

You are correct in your understanding that light takes the path that requires the shortest time to travel from one point to another. However, this principle does not necessarily mean that the light will always take a straight line path. In fact, as you mentioned, in a rectangular prism, the light will not take a straight path but will instead bend at the entry and exit points.

This is because, in addition to minimizing the time spent in a slower medium, the light also follows the law of refraction, which states that the angle of incidence is equal to the angle of refraction. In other words, the light bends at the interface between two media in order to maintain this relationship.

Therefore, in the case of a rectangular prism, the light will take a path that minimizes the total travel time while also following the law of refraction. This may not be a straight path, but it will still be the path that takes the shortest time.

I hope this explanation helps clarify your understanding of Fermat's principle and its application in real-life scenarios. Keep up the curiosity and critical thinking in your studies of physics!