Question

A right angled prism of refractive index ${\mu }_1$ is placed in a rectangular block of refractive index ${\mu }_2$, which is surrounded by a medium of refractive index ${\mu }_3,$ as shown in the figure. A ray of light ${}^{\prime }{e}^{\prime }$ enters the rectangular block at normal incidence. Depending upon the relationship between ${\mu }_1,{\mu }_2$ and ${\mu }_3,$ it takes one of the four possible paths ${}^{\prime }e{f}^{\prime }{,}^{\prime }e{g}^{\prime }{,}^{\prime }e{h}^{\prime }$ or ${}^{\prime }e{i}^{\prime }$.


Match the paths in $\text{List -I}$ with conditions of refractive indices in $\text{List -II}$ and choose the correct answer.
List - I List - II

$\text{(P)}e\to f$	$\left(1\right){\mu }_1>\sqrt{2}{\mu }_2$
$\text{(Q)}e\to g$	$\left(2\right){\mu }_2>{\mu }_1\&{\mu }_2>{\mu }_3$
$\text{(R)}e\to h$	$\left(3\right){\mu }_1={\mu }_2={\mu }_3$
$\text{(S)}e\to i$	$\left(4\right){\mu }_2<{\mu }_1<\sqrt{2}{\mu }_2\&{\mu }_2>{\mu }_1$

• A. $\text{P-2 , Q-3 , R-1 , S-4}$
• B. $\text{P-1 , Q-2 , R-4 , S-3}$
• C. $\text{P-4 , Q-1 , R-2 , S-3}$
• D. $\text{P-2 , Q-3 , R-4 , S-1}$

Answer 1

The correct option is D $\text{P-2 , Q-3 , R-4 , S-1}$

The given figure,



Case $\text{(P})$: When ray travelling along the path ${}^{\prime }\text{ef}{}^{\prime }$ :

At the surface of prism, ray is moving towards normal $\text{MN}$ so, we can conclude that , ${\mu }_2>{\mu }_1$.

At the surface of block, ray moving away from normal $\text{PQ}$ so, ${\mu }_3<{\mu }_2$.

Hence, $\text{P}$ matches with $\left(2\right)$.

Case $\text{(Q})$: When ray travelling along the path ${}^{\prime }\text{eg}{}^{\prime }$

There is no deviation of rays on both surfaces, so it is clearly evident from ray diagram that , ${\mu }_1={\mu }_2={\mu }_3$

Hence, $\text{Q}$ matches with $\left(3\right)$

Case $\text{(R})$: When ray is travelling along the path ${}^{\prime }\text{eh}{}^{\prime }$

At the prism surface ray moving away from normal $\text{MN}$. so, we can conclude that , ${\mu }_2<{\mu }_1.$

At block surface , ray is moving away from normal $\text{PQ}$. so , we can conclude that , ${\mu }_3<{\mu }_2$ but total internal reflection does not take place on prism surface.
So , ${\mu }_1\sin {45}^{\circ }>{\mu }_2\sin {90}^{\circ }$

$\therefore {\mu }_1>\sqrt{2}{\mu }_2$

Hence, $\text{R}$ matches with $\left(4\right)$.

Case $\text{(S})$:When ray is travelling along the path $\text{'ei'}$

Total internal reflection takes place at the surface of prism . so,

${\mu }_1\sin {45}^{\circ }<{\mu }_2\sin {90}^{\circ }$

$\therefore {\mu }_1>\sqrt{2}{\mu }_2$.

Hence, $\text{S}$ matches with $\left(1\right)$

Thus, option (d) is the correct answer.