Question

If a ray of light in a denser medium strikes a rarer medium at an angle of incidence $i$, the angles of reflection and refraction are respectively $r$ and $r^{\prime }$. If the reflected and refracted rays are at right angles to each other, the critical angle for the given pair of media is :

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

Answer 1

Answer: (B)

${\sin }^{-1}\left(\tan r\right)$

We know that,
$\frac{I}{\mu }=\frac{\sin i}{\sin r^{\prime }}$
But $r+r^{\prime }={90}^o$ $r^{\prime }={90}^o-r$
$\Rightarrow \sin r^{\prime }=\sin ({90}^o-r)$
$\sin r^{\prime }=\cos r\Rightarrow \sin r^{\prime }=\cos i$
$\frac{1}{\mu }=\frac{\sin i}{\cos i}\Rightarrow \frac{1}{\mu }=\tan i\Rightarrow \sin i_C=\tan i=\tan r$
The critical angle for the pair of media
$i_C={\sin }^{-1}\left(\tan r\right)$