Question

A light ray is travelling from air to glass. The reflected and refracted are perpendicular to each other. If the angle of incidence in air is $i,$ which of the following is a correct relation for refractive index of glass?

• A. ${\mu }_g=\sin i$
• B. ${\mu }_g=\cos i$
• C. ${\mu }_g=\tan i$
• D. ${\mu }_g=\cot i$

Answer 1

The correct option is C ${\mu }_g=\tan i$


Applying laws of reflection at interface,
$\angle i=\angle r$
Since incident ray is perpendicular to the refracted ray,
$\angle r+{90}^{\circ }+\angle r^{\prime }={180}^{\circ }$
$\Rightarrow \angle r+\angle r^{\prime }={90}^{\circ }$
$\Rightarrow \angle r^{\prime }={90}^{\circ }-r={90}^{\circ }-\angle i$
Applying Snell's law at the interface,
$\frac{\sin i}{\sin r^{\prime }}={\mu }_g$
$\Rightarrow \frac{\sin i}{\sin (90-i)}={\mu }_g$
$\Rightarrow \frac{\sin i}{\cos i}={\mu }_g$
$\therefore \tan i={\mu }_g$
Hence, option $\left(c\right)$ is the correct answer.

Why this question? Always remember the relation $tani=\mu$, for the condition of reflected ray is perpendicular to refracted ray in medium.