A ray of light enters at grazing angle of incidence into an assembly of three isosceles right-angled prisms having refractive indices ${\mu }_1=\sqrt{2},{\mu }_2=\sqrt{x}$ and  ${\mu }_3=\sqrt{3}$ . If finally emergent light ray also emerges at grazing angle then calculate x :

Apply Snell's law on various surfaces one by one :

1 sin $90^{\circ }$ $={\mu }_1$ sin $r_1\Rightarrow$ sin $r_1=\frac{1}{\sqrt{2}}\Rightarrow r_1=45^{\circ }$ 

We have the incident angle for the second interface given using the relation sin$i_2$=cos$r_1$
Thus we get

${\mu }_1$ cos $r_1$= ${\mu }_2$ sin $r_2\Rightarrow$ sin $r_2$ = $\frac{1}{{\mu }_2}$(cos$r_2$=$\frac{1}{\sqrt{2}}$)

${\mu }_2$ cos $r_2={\mu }_3$ sin $r_3$ $\Rightarrow$ sin $r_3$ $=\frac{{\mu }_2\sqrt{1-si{n}^2{r}_2}}{\sqrt{3}}$

${\mu }_3$ cos $r_3=1$ = $\frac{{\mu }_2^2-1}{\sqrt{3}}$

sin ${}^2{r}_3$ + cos ${}^2{r}_3=1$

$\Rightarrow \frac{{\mu }_2^2-1}{3}+\frac{1}{3}=1\Rightarrow {{\mu }_2}^2=3\Rightarrow {\mu }_2=\sqrt{3}$

Thus we get $x=3.$