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# Light is incident at $30^{\circ}$ on one side of a right angled isosceles prism of $\mu =\large\frac{3}{2}$. What will be the angle of emergence.

$(a)\;60^{\circ} \\ (b)\;45^{\circ} \\ (c)\;no\;emergence\;takes\;place \\ (d)\;the \;light\; retraces \; its\;path$

$i_1$= angle of incidence $=30^{\circ}$
$\large\frac{\sin i_1}{\sin r_1}=\frac{3}{2}$
$\large\frac{\Large\frac{1}{2}}{\sin r_1}=\frac{3}{2}$
$\sin r_1 =\large\frac{3}{2}$
Since $r_1 +r_2 =90^{\circ}$
$r_2 =90^{\circ}-r_1$
$\therefore \cos r_2 =\large\frac{1}{3}$
$\sin r_2 =\sqrt {1- \cos ^2 r_2}$
$\sin r_2=\sqrt {1- \bigg(\frac{1}{3}\bigg)^2}$
$\qquad= \large\frac{2 \sqrt 2}{3}$-----(1)
Also,
$\sin c= \large\frac{1}{\mu}$
$\sin c =\large\frac{2}{3}$----(2)
From (1) and (2) we infer that
$r_2 >c$
$\therefore$ No refraction takes place at 2nd face total internal reflection takes place.
$\therefore$ no emergent ray exists.
Hence c is the correct answer.

edited Jul 15, 2014