A transparent, solid cylindrical rod has a refractive index of 2/ $\sqrt{3}$ . It is surrounded by air. A light ray is incident at the midpoint of one end of the rod as shown in the figure. The incident angle $θ$ for which the light ray grazes along the wall of the rod is

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$\sin^{- 1} (\frac{2}{\sqrt{3}})$

$\sin^{- 1} (\frac{1}{\sqrt{3}})$

$\sin^{- 1} (\frac{1}{2})$

$\sin^{- 1} (\frac{\sqrt{3}}{2})$

answer is A.

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Detailed Solution

$\begin{array}{l} at A: μ_{1} {sinθ}_{1} = μ_{2} {sinθ}_{2} \\ \frac{2}{\sqrt{3}} {sini}_{c} = 1 \sin 90^{\circ} \\ ∴ i_{c} = 60^{\circ} \\ ∴ r = 30^{\circ} \\ At B: {(μ_{1} {sinθ}_{1})}_{air} = {(μ_{2} {sinθ}_{2})}_{rod} \\ 1 \times sinθ = \frac{2}{\sqrt{3}} \sin 30^{\circ} \\ ∴ θ = \sin^{- 1} (\frac{1}{\sqrt{3}}) \end{array}$