Question

A ray of light from a denser medium strikes a rarer medium at an angle of incidence $i.$ The reflected and refracted rays make an angle of $\pi /2$ with each other. If the angles of reflection and refraction are $r$ and $r^{\prime },$ then the critical angle will be

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

Answer 1

Answer: (C)

$\sin {\theta }_c=\frac{{\mu }_R}{{\mu }_D}$. Also, Snell's law gives,
${\mu }_D.\sin i={\mu }_R\sin r^{\prime }$
$\Rightarrow \sin {\theta }_c=\frac{\sin i}{\sin r^{\prime }}=\frac{\sin i}{\sin (90-r)}$
As $i=x,\sin {\theta }_c=\frac{\sin i}{\cos r}=\frac{\sin i}{\cos i}=tani$
${\theta }_c={\sin }^{-1}(\tan i)$