Find the maximum angle (in degree) at which a ray must be incident on a glass prism of refracting angle A, so that the ray suffers total internal reflection at second face. Refractive index of glass is $\mu$. (This is called condition of no emergence)
$\left(\text{given 0 to}si{n}^{-1}\right[sinA\sqrt{{\mu }^2}-1-cosA]={30}^{\circ })$

1. 30

The correct option is A 30


Let us first consider the critical condition (i.e., the ray is incident at second face at critical angle C)

By Snell's law at Q
$\mu \times sinC=1\times sin{90}^{\circ }$
$\Rightarrow sinC=\frac{1}{\mu }...\left(i\right)$
By Snell's law at P
$1\times sini=\mu sin{r}_1$
$sini=\mu sin(A-C)(\because A=r_1+r_2=r_1+c)$
$sini=\mu [sinAcosC-sinCcosA]$
$sini=\mu [sinA\sqrt{1-si{n}^{2}C}-sinCcosA]$
$\therefore sini=[sinA\sqrt{{\mu }^2-1}-cosA]$
$or,i=si{n}^{-1}[sinA\sqrt{{\mu }^2-1}-cosA]$
If we decrease angle of incidence, $r_1$ decreases as $r_1+r_2=A$
$\Rightarrow r_2\text{increases and it exceeds critical angle of incidence.}$
$\therefore \text{For angle of incidence from}$
$0tosi{n}^{-1}[sinA\sqrt{{\mu }^2-1}-cosA],$ there will be no emergence.