Question

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

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

Answer 1

Answer: (C)

It is quite clear $r+r=90{}^{\circ }$ or $r=90{}^{\circ }-r$ and $i=r$ The refractive index is given by $\mu =\frac{\sin i}{\sin r}=\frac{\sin i}{\sin (90{}^{\circ }-r)}=\frac{\sin i}{\cos r}$ $=\frac{\sin i}{\cos i}=\tan i$ Now $\frac{1}{\sin C}=\tan i$ $\Rightarrow$ $\sin C={(\tan i)}^{-1}$ so, $C={\sin }^{-1}{(\tan i)}^{-1}$