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}} $