Monochromatic light falls on a right-angled prism at an angle of incidence `45^@`. The emergent light is found to slide along the face AC. Find the refractive index of material of prism

Explanation

Since the emergent light slides along the face AC, angle of emergence is `90^@`, as shown.
It implies that angle of incidence of the ray that falls on face AC is equal to the critical angle `theta_C`.
`therefore r_2 = theta_C ...(1)`

From the prism theory, we know
` r_1 +r_2 = A = 90^@ `
` therefore r_2 = 90^@ - r_1 ......(2)`
from equaiton (1) and (2) `,90^@ - r_1 = theta_c `
` therefore sin ( 90^@ -r_1) = sin theta_C (or) cos r_1 = sin theta_C`
But sin ` theta_C = (1)/(mu ) therefore cos r_1 = (1)/(mu)`
Applying snell.s law at the boundary AB , ` 1 xx sin 45^@ = mu sin r_1 = mu sqrt(1- (1)/( mu^2))`
` therefore (1)/( sqrt(2)) = sqrt(mu^2 -1) or mu^2 = 3//2 = 1.5 mu = sqrt(1.5)`