A ray of light the face `AB` of a glass prism of refractive index `mu` at an angle of incidence `i`. Find the value of `i` such that no ray emerges from the face `AC` of the prism. Given angle of prism is `A`.

Explanation

As no ray emerges from face `AC` of prism, therefore, refracted ray `QR` must be undergoing total internal reflection at `R`.
`:. r_2 = C`
As `sin C = (1)/(mu) = sin r_2`
`:. mu sin r_2 = 1`…(i)
Also, `(sin i)/(sin r_1) = mu` ….(ii)
But `r_1 + r_2 = A :. r_1 = A - r_2`
`:.` from (ii), `sin i = mu sin (A - r_2) = mu (sin A cos r_2 - cos A sin r_2)`
`sin i = mu sin A cos r_2 - mu sin r_2 cos A`
using (i), `sin i = mu sin A cos r_2 - cos A`....(iii)
From (i), `sin r_2 = (1)/(mu)`
`cos r_2 = sqrt(1 - sin^2 r_2) = sqrt((1 - (1)/(mu^2))`
From (iii), `sin i = mu sin A sqrt((1 - (1)/(mu^2)) - cos A`
Hence, `i = sin^-1 [mu sin A sqrt(1 - (1)/(mu^2)) - cos A ]`.