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Question:

A monochromatic light is incident from air on a refracting surface of a prism of angle 75° and refractive index n₀=3. The other refracting surface of the prism is coated by a thin film of material of refractive index n as shown in figure. The light suffers total internal reflection at the coated prism surface for an incidence angle of θ≤60°. The value of n is ____.

1.5

Solution:

Correct option is A. 1.5
For T.I.R at coating
sin c = n/3
Applying Snell's law at first surface
sin θ = 3sin(75 - c)
for limiting condition, at θ=60°
sin 60 = 3sin(75 - c)
√3/2 = 3sin(75 - c)
1/2 = sin(75 - c) ⇒ sin 30 = sin(75 - c)
30 = 75 - c ⇒ c = 45
n/3 = 1/2 ⇒ n = 3/2 = 1.50