Question:

Let the refractive index of a denser medium with respect to a rare medium be N12 and its critical angle θC. At an angle of incidence A when light is travelling from denser medium to rarer medium, a part of the light is reflected and the rest is refracted and the angle between reflected and refracted rays is 90°. Angle A is given by :

cos⁻¹(sinθC)

tan⁻¹(sinθC)

1/tan⁻¹(sinθC)

1/cos⁻¹(sinθC)

n12 = nd/nr

sin i / sin r = n2/n1

sin i = sin r (n2/n1)

At critical angle, sin i = 1 and sin r = sin θC

1/sin θC = n2/n1 = n12

Angle between reflected and refracted rays is 90°.

r + i = 90°

r = 90° - i

sin i = n12 sin r

sin i = n12 sin(90° - i)

sin i = n12 cos i

tan i = n12

At critical angle, n12 = 1/sin θC

tan i = 1/sin θC

i = tan⁻¹(1/sin θC)