Question

A ray of light from a denser medium strikes a rarer medium at angle of incidence $i$. The reflected and refracted rays make an angle of $90^{\circ}$ with each other. The angles of reflection and refraction are $r$ and $r'$ respectively. The critical angle is

• A. $\sin ^{-1}\left(\tan r'\right)$
• B. $\sin ^{-1}(\tan r)$
• C. $\tan ^{-1}\left(\tan r'\right)$
• D. $\tan ^{-1}(\tan i)$

Answer 1

Answer: (B)

At the critical angle, $\sin C=\frac{1}{\mu}=\frac{1}{\sin r' / \sin i}=\frac{\sin i}{\sin r'}$
As is clear as shown in figure

$\angle C B D=90^{\circ}$
$\therefore 90^{\circ}-r+90^{\circ}-r'=90^{\circ}$
or $r'=90^{\circ}-r$
$\therefore \sin C =\frac{\sin i}{\sin \left(90^{\circ}-r\right)}=\frac{\sin i}{\cos r} $
$=\frac{\sin i}{\cos i}=\tan i (\because i=r) $
$ C =\sin ^{-1}(\tan i)=\sin ^{-1}(\tan r) .$