One side of an equilateral prism is coated with a transparent material of refractive index n₂. The refractive index of the prism is 1.6.

Find the minimum value of n₂ required so that total internal reflection occurs from the painted face.

The prism is equilateral, so each angle is:

60°

The incident ray enters normally to one face. Therefore there is no deviation at the first surface.

The ray reaches the coated surface making an angle of incidence:

i = 60°

For total internal reflection:

i ≥ C

where C is the critical angle.

For the limiting condition:

i = C = 60°

Using:

sin C = n₂/n₁

where:

• n₁ = 1.6
• n₂ = required refractive index

Therefore:

n₂ = 1.6 sin60°

= 1.6 × √3/2

= 0.8√3

Final Answer

n₂ = 0.8√3