Question

Let the refractive index of a denser medium with respect to a rare medium be $N_{12}$ and its critical angle ${\theta }_C$ . At an angle of incidence A when light is travelling from denser medium to rarer medium, a part of the light is reflected and the rest is refracted and the angle between reflected and refracted rays is ${90}^o$. Angle A is given by :

• A. ${\tan }^{-1}\left(\sin {\theta }_C\right)$
• B. $\frac{1}{{\tan }^{-1}\left(\sin {\theta }_C\right)}$
• C. $\frac{1}{{\cos }^{-1}\left(\sin {\theta }_C\right)}$
• D. ${\cos }^{-1}\left(\sin {\theta }_C\right)$

Answer 1

The correct option is A ${\tan }^{-1}\left(\sin {\theta }_C\right)$

$n_{12}=\frac{n_d}{n_r}$

$\frac{\sin \theta }{\sin 90}=\frac{n_2}{n_3}$

$\frac{\sin i}{\sin R}=\frac{n_2}{n_1}$

$\sin i=\sin r$

$\frac{\sin r}{\sin R}=\frac{n_2}{n_1}=\frac{\sin {\theta }_c}{\sin 90}$

$\frac{\sin r}{\cos r}=\sin {\theta }_c$

$\cos r$

$\tan r=\sin {\theta }_c$

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