1. Refraction at the end face

   Let the ray enter the cylindrical rod through the flat circular end face.

   • Refractive index of air: $n_1 = 1$
   • Refractive index of rod: $n_2 = \dfrac{2}{\sqrt{3}}$

   Suppose after entering the rod, the ray makes an angle $r$ with the axis of the rod. Since the axis is normal to the end face, $r$ is also the angle of refraction at the first surface.

   By Snell's law at the end face, $n_1 \sin \theta = n_2 \sin r$ $\sin \theta = \frac{2}{\sqrt{3}} \sin r$

2. Condition for grazing along the wall

   For the ray to just graze the cylindrical wall, the angle of incidence at the curved surface must be equal to the critical angle.

   Let $i$ be the angle of incidence at the wall. Since the ray makes angle $r$ with the axis, and the normal to the cylindrical wall is perpendicular to the axis, $i = 90^\circ - r$

   The critical angle $c$ for rod-to-air is given by $\sin c = \frac{1}{\mu} = \frac{1}{2/\sqrt{3}} = \frac{\sqrt{3}}{2}$ $c = 60^\circ$

   For grazing emergence, $i = c$ $90^\circ - r = 60^\circ$ $r = 30^\circ$

3. Now use Snell's law

   Substitute $r = 30^\circ$ into $\sin \theta = \frac{2}{\sqrt{3}} \sin r$

   $\sin \theta = \frac{2}{\sqrt{3}} \cdot \sin 30^\circ$ $\sin \theta = \frac{2}{\sqrt{3}} \cdot \frac{1}{2} = \frac{1}{\sqrt{3}}$

   Therefore, $\theta = \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$

4. Check options

   • A: $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)$ → incorrect
   • B: $\sin^{-1}\left(\frac{2}{\sqrt{3}}\right)$ → impossible since argument $>1$
   • C: $\sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$ → correct
   • D: $\sin^{-1}\left(\frac{1}{2}\right)$ → incorrect

Hence, the correct answer is Option C.