Question

If light ray is incident at $i={90}^{\circ },$ find minimum value of prism angle in degrees for which emergence is not possible.

• A. ${15}^{\circ }$
• B. ${30}^{\circ }$
• C. ${45}^{\circ }$
• D. ${60}^{\circ }$

Answer 1

The correct option is D ${60}^{\circ }$

Emergence won't be possible when light gets reflected internally.


Given:
$i={90}^{\circ };{\mu }_1=1;{\mu }_2=2$

Now using Snell's law at face $\text{AB}$

$\Rightarrow {\mu }_1\sin i={\mu }_2\sin r$

$\Rightarrow 1\times \sin {90}^{\circ }=2\sin r$

$\Rightarrow \sin r=\frac{1}{2}$

$\therefore r={30}^{\circ }$

For no emergence

$r^{\prime }>{\theta }_c$ $({\theta }_c=\text{critical angle})$

And we know, $r+r^{\prime }=A$

$or,r^{\prime }=A-r$

$\therefore A-r>{\theta }_c$

$\Rightarrow A>r+{\theta }_c......\left(1\right)$

Using Snell' law at face $\text{AC}$

$\Rightarrow 2\times \sin {\theta }_c=1\times \sin {90}^{\circ }$

$\therefore {\theta }_c={30}^{\circ }$

So, eq.(1) becomes

$A>{30}^{\circ }+{30}^{\circ }={60}^{\circ }$

Hence, option $\left(d\right)$ is correct.