Question

A ray of light from a denser medium strikes at the interface of a rarer medium at an angle of incidence $i$. If the reflected and refracted rays are mutually perpendicular to each other, then the critical angle is

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

Answer 1

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

Since, reflected and refracted rays are perpendicular.
$r^{\prime }={90}^{\circ }-r$
$[$from geometry$]$
Also, $i=r$
$[$from laws of reflection$]$


From Snell's law,
${\mu }_1\sin i={\mu }_2\sin r^{\prime }$
$\Rightarrow {\mu }_1\sin i={\mu }_2\sin ({90}^{\circ }-r)$
$\Rightarrow {\mu }_1\sin i={\mu }_2\cos r$
$\Rightarrow {\mu }_1\sin i={\mu }_2\cos i$
$\Rightarrow \tan i=\frac{{\mu }_2}{{\mu }_1}$ ...........$\left(1\right)$
From formula,
$i_c={\sin }^{-1}\left(\frac{{\mu }_r}{{\mu }_d}\right)={\sin }^{-1}\left(\frac{{\mu }_2}{{\mu }_1}\right)$
$\Rightarrow i_c={\sin }^{-1}\left(\tan i\right)$
$[$from $\left(1\right)]$

Why this question? It intends to test your understanding of "application of reflection and refraction simultaneously at an interface"