Question

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

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

Answer 1

The correct option is B ${sin}^{-1}\left(tani\right)$

In the figure given in the question, we may apply Snell's law for the refraction of incident beam of light.

$\mu \sin i=\sin r^{\prime }$

But $r+{90}^{\circ }+r^{\prime }={180}^{\circ }$

Therefore $r^{\prime }={90}^{\circ }-r$

Thus $\mu \sin i=\cos r$

But from Law of Reflection, $i=r$

Therefore, $\mu =\cot i$

Now the critical angle is the angle of incidence at which the refracted ray goes parallel to the surface,

$\mu \sin {\theta }_c=1$

Hence $\sin {\theta }_c=\frac{1}{\mu }=\tan i$

${\theta }_c=si{n}^{-1}\left(\tan i\right)$