Question

A ray of light is incident normally on one face of an equilateral glass prism of refractive index \( P \). When the prism is completely immersed in a transparent medium, it is observed that the emergent ray just grazes the adjacent face. Find the refractive index of the medium.

Hint:

The critical angle condition is used to determine the refractive index of the medium when total internal reflection occurs.

Answer 1

Applying Snell’s Law:
- The angle of refraction at the second face of the prism is the critical angle \( C \), so: \[ \sin C = \frac{n_{\text{medium}}}{n_{\text{prism}}} \] - For an equilateral prism, the angle of incidence inside the prism is: \[ r = \frac{A}{2} = \frac{60^\circ}{2} = 30^\circ \] - The critical angle \( C \) is given by: \[ \sin C = \frac{1}{n} \] Since \( C = 60^\circ \): \[ \sin 60^\circ = \frac{n_{\text{medium}}}{n_{\text{prism}}} \] \[ \frac{\sqrt{3}}{2} = \frac{n_{\text{medium}}}{n_{\text{prism}}} \] \[ n_{\text{medium}} = \frac{\sqrt{3}}{2} n_{\text{prism}} \] Thus, the refractive index of the medium is \( \frac{\sqrt{3}}{2} n_{\text{prism}} \).