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

[zell99]

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

 

[zell99]

Has anyone got any ideas? I should have said theta is very small, so small angle approximations are fine where appropriate.
Thanks

 

[m2003]

Thank you for sharing your proposed solution. It seems like you are on the right track with your approach. To solve this problem, you will need to use Snell's law and consider the change in refractive index as the light travels from ground level to height h. Here is a possible solution:

1. Draw a diagram to visualize the problem. Label the known angle \theta, the height h, and the refractive index of air at ground level (n1).

2. Use Snell's law to set up an equation relating the angle of incidence (\theta1) and the angle of refraction (\theta2) at ground level, given by n1 sin(\theta1)=n2 sin(\theta2).

3. Since the man can see the mirage at angles less than \theta to the horizontal, this means that the light must have undergone total internal reflection at some point along its path. This occurs when the angle of incidence is equal to the critical angle, given by \theta_c = sin^-1(n2/n1).

4. Use the known angle \theta and the critical angle \theta_c to solve for the refractive index at height h. This can be done by setting \theta1 = \theta and \theta2 = \theta_c in the Snell's law equation and solving for n2.

5. Once you have n2, you can use the equation you proposed in your attempt at a solution to integrate and find the refractive index at any height h.

I hope this helps guide you in the right direction. Remember to always carefully consider the given information and think about what equations and concepts can be applied to solve the problem.