A glass prism of refractive index 1.5 is immersed in water $\left(\mu=\frac{4}{3}\right)$. A light beam incident normally on the face AB is totally reflected to reach the face BC if:

$\sin \theta>\frac{8}{9}$

Here, ${ }^{a} \mu_{g}=1.5=\frac{3}{2},{ }^{a} \mu_{w}=\frac{4}{3}$

${ }^{a} \mu_{w} \times{ }^{w} \mu_{g}={ }^{a} \mu_{g}$

∴ ${ }^{w} \mu_{g}=\frac{{ }^{a} \mu_{g}}{{ }^{a} \mu_{w}}=\frac{3 / 2}{4 / 3}=\frac{9}{8}$ 

As $\sin C=\frac{1}{{ }^{w} \mu_{g}}=\frac{1}{9 / 8}=\frac{8}{9}$

$C=\sin ^{-1}\left(\frac{8}{9}\right)$

For total internal reflection, $\theta>C$

$\theta>\sin ^{-1}\left(\frac{8}{9}\right)$  or  $\sin \theta>\frac{8}{9}$