Question

A ray of light from a denser medium strikes a rarer medium at angle of incidence $i$ as shown in the figure. The reflected and refracted rays make an angle of ${90}^{\circ }$ with each other. The angles of reflection and refraction are $r$ and $r^{\prime }$ respectively. The critical angle is

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

Answer 1

The correct option is A

${\sin }^{-1}\left(\tan r\right)$

$i=r$ [Law of reflection]

$r^{\prime }={90}^{\circ }-r$ [Given]

Now, refractive index, $n=\frac{\sin r^{\prime }}{\sin r}=\frac{sin({90}^{\circ }-r)}{\sin r}$ [Snell's Law]
$\Rightarrow n=\frac{\cos r}{\sin r}=\frac{1}{\tan r}$
We know that, $n=\frac{1}{\sin C}$ where $C=$ critical angle
$\therefore \frac{1}{\sin C}=\frac{1}{\tan r}$
$\Rightarrow C={\sin }^{-1}\left(\tan r\right)$