A ray of light is incident grazingly on face AB to a right angled prism as shown.  It emerges out of face AC, as shown, e=angle of emergence.  Refractive indices of different media have been shown in the diagram.  Choose the correct alternatives.

By Snell’s law,

$\frac{{\mu }_2}{{\mu }_1}=\frac{\sin {90}^0}{\sin r_1}$

Or      $\sin r_1=\left(\frac{{\mu }_1}{{\mu }_2}\right)$

Similarly, $\frac{\sin e}{\sin r_2}={\mu }_2$

Or  $\sin r_2=\frac{\sin e}{{\mu }_2}$

Or  $\sin \left({90}^0-r_1\right)=\frac{\sin e}{{\mu }_2}$

or  $\cos r_1=\frac{\sin e}{{\mu }_2}$

or   $\sqrt{1-{\sin }^2{r}_1}=\frac{\sin e}{{\mu }_2}$

or  $\sqrt{1-{\left(\frac{{\mu }_1}{{\mu }_2}\right)}^2}=\frac{\sin e}{{\mu }_2}$

$\therefore$ ${\mu }_1^2+{\sin }^2={\mu }_2^2$

Also $\sin e=\frac{1}{{\mu }_2}=\frac{1}{\sqrt{3}}$

Now    $r_1={90}^0-r_2={90}^0-e$

or  $\sin r_1=\sin \left({90}^0-e\right)$

$=\cos e$

$=\sqrt{1-{\sin }^{2}e}$

$=\sqrt{\frac{2}{3}}$