Principle of Least Time Question

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Discussion Overview

The discussion revolves around Fermat's principle of least time in the context of optics, specifically examining how light behaves when passing through different media, such as a rectangular prism. Participants explore the implications of this principle and its application to real-world scenarios involving refractive indices.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant references Feynman's definition of Fermat's principle, suggesting that light takes the path requiring the shortest time between two points.
  • Another participant questions the application of this principle, noting that a right angle at the entry and exit points of a prism does not occur in practice, indicating a potential misunderstanding of the principle.
  • It is proposed that while a perpendicular crossing minimizes the time spent in the glass, it may extend the path through air, thus affecting total travel time.
  • A later reply suggests that minimizing time in the slower medium does not necessarily equate to minimizing total travel time, raising the question of how to minimize a function of two variables.

Areas of Agreement / Disagreement

Participants express differing views on the application of Fermat's principle, with some suggesting that the principle does not lead to the expected outcomes in real-life scenarios. The discussion remains unresolved regarding the correct interpretation and application of the principle.

Contextual Notes

There are limitations in the assumptions made about light paths and the definitions of the media involved. The discussion highlights the complexity of applying Fermat's principle to real-world optics, particularly in relation to refractive indices and the geometry of light paths.

EternusVia
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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:
refraction.jpg
 
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EternusVia said:
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.
 
EternusVia said:
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.

.
 
Last edited:
EternusVia said:
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?
 

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