Question

A light ray entering a right-angled prism undergoes refraction at the face AC, as shown in Fig. 1.

                  Fig. 1

1. What is the refractive index of the material of the prism in Fig. 1?
2.    
   1. If the side AC of the above prism is now surrounded by a liquid of refractive index `2/sqrt3`, as shown in Fig. 2, determine if the light ray continues to graze along the interface AC or undergoes total internal reflection, or undergoes refraction into the liquid.
      
                            Fig. 2
   2. Draw the ray diagram to represent the path followed by the incident ray with the corresponding angle values.
      `["Given,"  sin^-1(sqrt2/sqrt3 ) = 54.6°]`

Answer 1

I. Since the light ray enters perpendicular to the face AB, the angle of incidence on the face AC will be 45°.

sinθ_C = `1/n`

`sin 45"°" = 1/n = 1/sqrt2`

So, n = `sqrt2`

II.

a. In Fig. 2, the face AC of the prism is surrounded by a liquid so,

`n = (ng)/n_1 = sqrt2/((2/sqrt3)) = sqrt3/sqrt2`

`sintheta_C = 1/n = sqrt2/sqrt3theta_C`

= `sin^-1(sqrt2/sqrt3)`

= 54.6°

Since the angle of incidence on the surface AC is 45°, which is less than the critical angle for the pair of media (glass and the liquid), the ray neither undergoes grazing along surface AC nor does it suffer total internal reflection.

b.

Instead, it passes through the surface AC and undergoes refraction into the liquid.

For refracting interface AC, n₁ sin i = n₂ sin r

`n_1*sin45"°" = (2/sqrt3) sinr`

`sinr = sqrt3/2`

∴ r = 60°