Question

As shown in the figure, light is incident normally on one face of the prism. A liquid of refractive index $\mu$ is placed on the horizontal face $AC$. The refractive index of prism is $3/2$. If total internal reflection taken place on face $AC$, $\mu$ should be less than $\frac{I\sqrt{3}}{4}$, where $I$ is an integer. Find the value of $I$.

Answer 1

Critical angle between glass and liquid.
$\sin {\theta }_C=\frac{\mu }{\left(3/2\right)}=\frac{2\mu }{3}$
Angle of incidence on $AC={60}^{\circ }$
For $TIR,i>{\theta }_C$

$\Rightarrow \sin {60}^{\circ }>\sin {\theta }_c=\frac{2}{3}\mu$

$\mu <\frac{3\sqrt{3}}{4}=\frac{I\sqrt{3}}{4}$ (given)

So, $I=3$.