In the figure given below, for an angle of incidence 45 ° at the top surface, what is the minimum refractive index needed for total internal reflection at the vertical face?

Step-by-step solution

At point A , by Snell's law μ = sin ⁡ 45 o sin ⁡ r ⇒ sin ⁡ r = 1 μ 2 ... (i) At point B , total internal reflection with sin ⁡ i 1 = 1 μ From figure, i 1 = 90 o - r .... (ii) sin ⁡ 90 o - r = 1 μ ⇒ cos ⁡ r = 1 μ .... (iii) Now, cos ⁡ r = 1 - sin 2 ⁡ r = 1 - 1 2 μ 2 = 2 μ 2 - 1 μ 2 From equation (ii) and (iii), 1 μ = 2 μ 2 - 1 2 μ 2 Squaring both sides and then solving, we get μ = 3 2