Question

A transparent solid cylindrical rod (refractive index \( \frac{2}{\sqrt{3}} \)) is kept in air. A ray of light incident on its face travels along the surface of the rod, as shown in the figure. Calculate the angle \( \theta \).

Hint:

For light to travel along the surface, the angle of incidence must be such that the refracted light reaches the boundary at \( 90^\circ \), which is the condition for total internal reflection.

Answer 1

For light to travel along the surface of the cylindrical rod, the angle of incidence \( \theta \) must ensure refraction along the surface. Applying Snell's law at the rod-air interface: \[ n_{\text{air}} \sin \theta = n_{\text{rod}} \sin 90^\circ \] With \( n_{\text{air}} = 1 \) and \( n_{\text{rod}} = \frac{2}{\sqrt{3}} \): \[ \sin \theta = \frac{n_{\text{rod}}}{n_{\text{air}}} = \frac{2}{\sqrt{3}} \] Consequently: \[ \theta = \sin^{-1} \left( \frac{2}{\sqrt{3}} \right) \] The angle \( \theta \) is therefore \( \sin^{-1} \left( \frac{2}{\sqrt{3}} \right) \).