# Critical Angle

1. Sep 19, 2005

### Romperstomper

In my physics lab, we have to derive an equation. The equation is for a 45, 45, 90 degree triangle. An beam of light hits the surface, then refracts. The refracted beam hits the second surface at the critical angle and refracts along the surface. Here's a crude pic of what I'm talking about.
Pic of triangle

The equation is: √(2)*Sin (σ)+1=√(n^2-1)

The material is in air, so n1=1 and n2=n. So, Sin (σ1)=nSin(σ2). I moved around the upper equations and got Sin(σ)=[√(n^2-1)-1]/√(2)=nSin(σ2). This is where I'm having trouble. I'm not sure where to go from here. I know it has to do with Sin/Cos/Tan bits of the triangles that are made, but I can't figure it out. Any help will be appretiated. Thanks.

2. Sep 19, 2005

### lightgrav

did you write snell's law for the ray as it exits?
Since this is supposed to be at the critical angle,
n sin(theta_glass,exit) = "1" .
Now use the 45 degree angle in the triangle
to relate theta_glass,exit to sigma_2 .

The goal is to write sigma_2 in terms of the "1".

3. Sep 21, 2005

### Romperstomper

Thank you for the help. Can you elaborate on relating critical angle back to sigma_2? I'm having trouble doing it. The triangle that I come up with has the angles sigma_2, the critical angle, and 135. When I try to build a relationship between sigma_2 and the critical angle, I don't get anything close to what the equation is. Basically, I took sigma_2 = (180-135)-critical_angle. Am I using the wrong triangle, or is there another way of doing it?