Question

A ray of light incident normally on the horizontal face of the slab and just fail to emerge from the diagonal face of the prism. If prism angle is ${30}^{\circ }$, the refractive index of the prism can be :

• A. $\sqrt{2}$
• B. $\sqrt{3}$
• C. slightly greater than $2$
• D. slightly less than $2$

Answer 1

The correct option is C slightly greater than $2$

Given that, light does not emerge out of diagonal face of the prism. So, $\text{TIR}$ occurs at the diagonal face $\text{AC}$

Light ray will pass the surface $\text{AB}$ without bending since it is incident normally.

It strikes the surface $\text{AC}$ at an angle of incidence $i={30}^{\circ }$.

For required condition , angle of incidence $i$ on the face $\text{AC}$ should be greater than the critical angle $C$ i.e $i>C$

$\Rightarrow {30}^{\circ }>C$ $($ From figure$)$

Substituting , $C={\sin }^{-1}\left(\frac{1}{\mu }\right)$, we get

$\Rightarrow$ $\sin {30}^{\circ }>\frac{1}{\mu }$

$\Rightarrow$$\frac{1}{2}>\frac{1}{\mu }$

$\therefore \mu >2$

Hence, option $\left(c\right)$ is the correct answer.