A parallel beam of light is incident from air at an angle `alpha` on the side of right angled triangular prism of refractive index `mu = sqrt(2)`. Light undergoes total internal reflection in the prism at the face `PR` when `alpha` has a minimum value of `45^@`. The angle `theta` of the prism is.
.

Options

A. `15^(@)`
B. `22.5^(@)`
C. `30^(@)`
D. `45^(@)`

Answer

A

Explanation

In Fig. 6(CF).16, applying Snell's Law at `L`,
`mu = (sin alpha)/(sin r_(1)) = sqrt(2) = (sin 45^(@))/(sin r_(1))`
`sin r_(1) = (sin 45^(@))/(sqrt(2)) = (1//sqrt(2))/(sqrt(2)) = (1)/(2), r_(1) = 30^(@)`
`sin theta_( c) = (1)/(mu) = (1)/(sqrt(2)) :. theta_( c) = 45^(@)`
In `deltaPLM`, `(theta + 90^(@)) + r_(1) + (90 - theta_( c)) = 180^(@)`
`theta + r_(1) - theta_( c) = 0 or theta - theta_( c) - r_(1) = 45^(@) - 30^(@) = 15^(@)`