Solving Snell's Law Problem: Find Refractive Index at Height h

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SUMMARY

The discussion focuses on solving a physics problem involving Snell's Law to determine the refractive index of air at a height h, given a known angle θ and the refractive index at ground level. The user proposes to analyze the air in infinitesimal strips at a constant height, aiming to derive the relationship between the angle and the refractive index through integration. The challenge lies in establishing the initial value of θ and incorporating the height h into the calculations, particularly under the assumption of small angles for simplification.

PREREQUISITES
  • Understanding of Snell's Law: n1 sin(θ1) = n2 sin(θ2)
  • Knowledge of calculus, specifically integration techniques
  • Familiarity with the concept of refractive index in optics
  • Basic principles of optics related to mirages and light refraction
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  • Research the application of small angle approximations in optics
  • Study the integration of functions related to refractive index changes
  • Explore examples of mirage phenomena and their mathematical descriptions
  • Learn about the behavior of light in varying refractive index environments
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Students studying physics, particularly those focusing on optics and light behavior, as well as educators seeking to explain the principles of Snell's Law and refractive index variations.

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Homework Statement


A man, height h, can see a mirage at angles less than a known angle \theta to the horizontal. The refractive index of air is at ground level is known. Find the refractive index of air at height h.

Homework Equations


Snell's law: n1 sin(\theta 1)=n2 sin(\theta2) where angles are measured relative to the normal of the boundary.
I'm assuming it's a normal mirage, i.e. can see an image of the sky in the ground.

The Attempt at a Solution


My plan was to split the air up into infintesimal stips at constant height, find d\theta as a function of d(refractive index) and integrate to find \theta as a function of refractive index. The problem I have is I don't know what the initial value of theta is, and I obviously need to include h somewhere.
If anyone could point me in the right direction I'd really appreciate it.
Thanks
 
Last edited:
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Has anyone got any ideas? I should have said theta is very small, so small angle approximations are fine where appropriate.
Thanks
 

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