Question

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`.

Answer 1

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)
`sin I = mu sin r_(1)` ...(ii)
but `r_(1) + r_(2) = A :. r_(1) = A - r_(2)`
`:.` form (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` ,br. 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]`