Q121 / 180 · COMEDK 2018 · Physics

A ray of light enters from a rarer to a denser medium. The angle of incidence is $i$. Then, the reflected and refracted rays are mutually perpendicular to each other. The critical angle for the pair of media is

  • A
    $\sin ^{-1}(\tan i)$
  • B
    $\tan ^{-1}(\sin i)$
  • Correct answer
    C
    $\sin ^{-1}(\cot i)$
  • D
    $\cos ^{-1}(\tan i)$

Solution

The angle of incidence $=i$ Let $r$ be the angle of refraction. According to given situation, reflected rays and refracted rays are perpendicular to each other. i.e. $\quad i+r=90^{\circ}$ $\therefore$ According to Snell's law, $$ \begin{aligned} \mu &=\frac{\sin i}{\sin r} \\ &=\frac{\sin i}{\sin (90-i)} \quad[\because \text { From Eq. (i) }] \\ &=\frac{\sin i}{\cos i} \\ \mu &=\tan i \end{aligned} $$ If $i_{C}$ be the critical angle for the pair of media, then $$ \mu=\frac{1}{\sin i_{C}} $$ From Eas. (ii) and (iii), we get $\tan i=\frac{1}{\sin i_{C}}$ $\Rightarrow \quad \sin i_{C}=\cot i$ $\Rightarrow \quad i_{C}=\sin ^{-1}(\cot i)$

Hence, the correct answer is option: 'C'.

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