Question

A ray of light strikes a transparent surface from air at an angle $\theta$. If the angle between the reflected and refracted ray is a right angle, the refractive index of the other medium is given by

• A. $\mu=\frac{1}{\tan \theta}$
• B. $\mu=\tan ^{2} \theta$
• C. $\mu=\sin \theta$
• D. $\mu=\tan \theta$

Answer 1

Answer: (D)

Angle of incidence $\angle AON=\theta$.
$\overrightarrow{OB}$ is the reflected ray and $\overrightarrow{O C}$ is the refracted ray. It is given that
$\angle BOC=90^{\circ}$
$\therefore $ From the diagram
$\theta+\angle BOC+r=180^{\circ}$
or $ \theta+90^{\circ}+r=180^{\circ}$
or $ r=90^{\circ}-\theta$
$\therefore $ Refractive index,
$\mu=\frac{\sin\, i}{\sin \,r}=\frac{\sin\, \theta}{\sin \left(90^{\circ}-\theta\right)}$
$=\frac{\sin \theta}{\cos \theta}=\tan \,\theta$