Question

What is the smallest angle θ₁ for which a laser beam will undergo total internal reflection on the hypotenuse of the glass prism in the figure?
Express your answer using two significant figures.
θ₁ = ___ degrees

Answer 1

Rfractive index of the glass prism is n = 1.5

angle of incidence ois θ1  

Let the corresponding angle of refraction be θ2. Let the angle of incidence on the hypotenuse be θ3.

By using snells law

           n1 sinθ1 = n2 sin θ2

                 sin θ2 = n1 sin θ1 / n2

                           = sin θ1 / 1.5

                      θ2 = sin ^-1 ( sin θ1 / 1.5 )

θ3 = θ2 + 30°
Now suppose that θ3 is the critical angle for the glass. Then it must satisfy this equation:

         sin θ3 = 1/1.5
               θ3 = sin^-1 ( 2 / 3) ˜ 41.8°
θ2 + 30 ° = 41.8°

         sin ^-1 ( sin θ1 / 1.5 ) = 11.8 ^o        
                   sin θ1 / 1.5 = sin 11.8

                      θ1   = 17.8631 ^o