A rectangular glass slab ABCD of refractive index n₁ is immersed in water of refractive index n₂ (n₁ > n₂). A ray of light is incident at the surface AB of the slab as shown. The maximum value of angle of incidence α_max,  such that the ray comes out only from the other surface CD, is given by

$\sin ^{-1}\left[\frac{n_1}{n_2} \cos \left(\sin ^{-1} \frac{n_2}{n_1}\right)\right]$

The incident ray PQ will emerge from the surface CD only if it is width reflected at surface AD. For this $r_2 \geq C$

As $r_1+r_2=90^{\circ}$

or $r_1=90-r_2$

or $r_1 \leq 90-C$

From Snell's law $\frac{\sin \alpha}{\sin r_1}=\frac{n_1}{n_2}$

For maximum $\alpha, \quad r_1=90-C$

∴ $\frac{\sin \alpha_{\max }}{\sin (90-C)}=\frac{n_1}{n_2}$

$\Rightarrow \sin \alpha_{\max }=\frac{n_1}{n_2} \cos C\left(\text { As } C=\sin ^{-1} \frac{n_1}{n_2}\right)$

$\Rightarrow \alpha_{\max }=\sin ^{-1}\left[\frac{n_1}{n_2} \cos \left(\sin ^{-1} \frac{n_2}{n_1}\right)\right]$