How Can Fermat's Principle and Snell's Law Help You Find a Car Key in Water?

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SUMMARY

This discussion focuses on applying Fermat's Principle and Snell's Law to determine the distance of a car key submerged in water and the time it takes for light to reach the observer's eyes. The key is located at the bottom of a pool with a refractive index of 1.33 and a depth of 3.0 m, while the observer's eye level is 1.5 m above the surface. The discussion includes calculations for light travel time under various conditions, emphasizing the significance of the actual path light takes as per Fermat's Principle and the implications of Snell's Law for minimizing travel time.

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  • Understanding of Snell's Law and its application in optics
  • Familiarity with Fermat's Principle in physics
  • Basic algebra for solving equations related to light travel time
  • Knowledge of refractive indices and their impact on light behavior
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  • Study the derivation and applications of Snell's Law in various mediums
  • Explore Fermat's Principle and its implications in optics and physics
  • Learn about light refraction and reflection in different materials
  • Investigate practical problems involving light travel time in real-world scenarios
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Students and educators in physics, optical engineers, and anyone interested in the practical applications of light behavior in different mediums.

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1. Your car key has fallen to the bottom of a clear pool of water (n = 1.33) with a uniform depth of 3.0 m. You are standing at the edge of the pool and your eyes are a height of 1.5 m above the surface. You can see the key by staring at an angle 60o below horizontal. (A) How far is the key from the edge of the pool? (B) i. How much time does it take light reflecting off the key to reach your eyes? ii. How much time would it take the light to travel to your eyes if it traveled in a straight line (instead of refracting at the surface)? iii. How much time would it take the light to travel if it exited the water vertically, then turned and traveled directly to your eyes? iv. What do your results suggest is special about the actual path the light ray follows? This is known as Fermat’s Principle. (C) Imagine a light ray that reflects off the key, strikes the surface of the water a distance x from the edge of this pool, and then bends to travel towards your eye. How much time would it take the light to travel to your eyes in this general case? (D) Use your result from part C to find the distance x which minimizes the time it takes for the light to reach your eyes. What does this tell you about Snell’s Law?



Homework Equations



Snell's Law


3. The Attempt at a Solution
I can easily solve part a and b using algebra and snells law but I'm wondering about how to approach part c and d. thank you

 
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