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.
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Options

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

Answer

A

Explanation

In Fig., applying Snell's Law at `L`,
`mu = (sin prop)/(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 `Delta PLM, (theta + 90^@) + r_1 + (90 - theta_c) = 180^@`
`theta + r_1 - theta_c = 0` or `theta = theta_c - r_1 = 45^@ - 30^@ = 15^@`.
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