Question

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

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

Answer 1

Answer: (D)

$r+r^{\prime}+90^{\circ}=180^{\circ} $
$\Rightarrow r^{\prime}=90-r=90-i$
$n_{1} \sin i=n_{2} \sin r^{\prime}=n_{2} \sin (90-i)$
$n_{1} \sin i=n_{2} \cos i $
$\Rightarrow \tan i=\frac{n_{2}}{n_{1}}$
Now $\sin C=\frac{n_{2}}{n_{1}}=\tan i$
$\Rightarrow C=\sin ^{-1}(\tan i)=\sin ^{-1}(\tan r)$