What is the angle θ of the prism?

Question

Here is my diagram: https://i.stack.imgur.com/WYSnE.png

A parallel beam of light is incident from air at an angle a on the side PQ of a right angled triangular prism of refractive index n=√2. Light undergoes total internal reflection in the prism at the face PR when a has a minimum value of 45°. What is the angle θ of the prism?

 

I'm a bit confused right now. However, I think that we need to use Snell's law. 

 

Kindest Regards!

Answer 1

Use Snell's law both at PQ and at PR.

Medium 1 is air, medium 2 is the glass.

At PQ:

n₁sinθ₁ = n₂sinθ₂

n₁ ≅ 1
θ₁ = α = 45°
n₂ = √2

sinθ₂ = (n1/n2)sinθ₁

sinθ₂ = (1/√2)sin45° = 1/2

 

θ₂ = 30°

 

At PR:

 

ρ = angle between incident ray and normal to PR

 

n₂sinρ = n₁sin90°  [internal reflection at PR]

 

√2 sinρ = 1

 

sinρ = 1/√2

 

ρ = 45º

 

Consider the triangle with corners at

 

(1) point P

(2) the point of incidence of the ray entering the glass at α = 45°

(3) the point where the ray inside the glass is reflected internally

 

(I know this is hard to see without a diagram, but I cannot upload a diagram.)

 

At (1), the angle is θ.

At (2), the angle is 90º + θ₂ = 90º + 30° = 120°

At (3), the angle is 180° - (90° + ρ) = 180° - (90° + 45°) = 45°

 

Angles (1)+(2)+(3) = 180°

 

θ + 120° + 45° = 180°

 

θ = 15º