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 $\overrightarrow{OC}$ 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,
$\mu =\frac{\sin i}{\sin r}=\frac{\sin \theta }{\sin \left(90^{\circ }-\theta \right)}$
$=\frac{\sin \theta }{\cos \theta }=\tan \theta$