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 \( \theta \).
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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.
The light travels along the surface of the cylindrical rod. For this to happen, the angle of incidence \( \theta \) must be such that the light is refracted along the surface. Using Snell's law at the interface between the rod and air: \[ n_{\text{air}} \sin \theta = n_{\text{rod}} \sin 90^\circ \] Given \( n_{\text{air}} = 1 \) and \( n_{\text{rod}} = \frac{2}{\sqrt{3}} \), we have: \[ \sin \theta = \frac{n_{\text{rod}}}{n_{\text{air}}} = \frac{2}{\sqrt{3}} \] Thus: \[ \theta = \sin^{-1} \left( \frac{2}{\sqrt{3}} \right) \] Therefore, the angle \( \theta \) is \( \sin^{-1} \left( \frac{2}{\sqrt{3}} \right) \).
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