# Homework Help: Finding the index of refraction with respect to the vertical axis

1. Dec 30, 2009

### greedo

1. The problem statement, all variables and given/known data
Studying the mirage, we can assume that the index of refraction depends only on the vertical coordinate z. A ray of light starts from {0,0} with angle $$\beta\leq30$$. It's trajectory fits on an ellipse with major axis a and minor axis b.

a) Give $$n(z)=?$$
b) What is the trajectory of a ray of light that starts with angle $$2\beta$$ ?

2. Relevant equations
From functional analysis we derived that in this situation:
$$\frac{n[z]}{\sqrt{1+z'[x]^2}}=\text{constant}$$

3. The attempt at a solution

What I have problems with is expressing u and v in terms of beta,a and b. If I could do that then I could give the equation for the ellipse, order it for z[x] and insert it into the equation in 2. I was supposed to do this in an exam in about half an hour, so it has to have a simple solution that I cannot see even after looking at it for hours.

I have made a drawing

Last edited: Dec 31, 2009
2. Dec 31, 2009

### Redbelly98

Staff Emeritus
Welcome to Physics Forums.

Are u and v the (x,z) coordinates of the center of the ellipse? What about starting with the general equation for an ellipse, and setting z'=tan(β) at the origin?

3. Dec 31, 2009

### greedo

Last edited by a moderator: May 4, 2017