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# A ray of light is incident on glass sphere of refractive index $\large\frac{3}{2}$. The ray incident at two surfaces after travelling through the sphere emerges tangential to the surface of the sphere. What must be the incident angle.

$(a)\;\tan ^{-1} 2/3 \\ (b)\;\sin ^{-1} 2/3 \\ (c)\;90^{\circ} \\ (d)\;\cos^{-1} 1/3$

Can you answer this question?

$OA=OB$ radius of the circle, $AO$ and $BO$ normal to the sphere at A and B respectively.
Since the incident ray emerge out at B, the emergent ray BQ is tangential we must have $\angle ABO$ = critical angle $\theta_c$ and $\therefore \angle OAB=\theta_c$
$\sin \theta_c=\large\frac{1}{\mu}$
$\qquad=\large\frac{2}{3}$
At A $\mu =\large\frac{\sin i}{\sin \theta_c}$
$\qquad= \large\frac{3}{2}$
$\therefore \sin i =\large\frac{3}{2}$$\sin \theta_c \qquad= \large\frac{3}{2} \times \frac{2}{3} =$$1$
$\therefore i= 90^{\circ}$
Hence c is the correct answer.
answered Jan 16, 2014 by