Optics 4 Question 7
7. 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 ’ $e$ ’ enters the rectangular block at normal incidence. Depending upon the relationships between $\mu _1, \mu _2$ and $\mu _3$, it takes one of the four possible paths ’ $e f$ ‘, ’ $e g$ ‘, ’ $e h$ ’ or ’ $e i$ ‘.
(2013 Adv.)
Match the paths in Column I with conditions of refractive indices in Column II and select the correct answer using the codes given below the lists.
| Column I | Column II | ||
|---|---|---|---|
| P. | $e \rightarrow f$ | 1. | $\mu _1>\sqrt{2} \mu _2$ |
| Q. | $e \rightarrow g$ | 2. | $\mu _2>\mu _1$ and $\mu _2>\mu _3$ |
| R. | $e \rightarrow h$ | 3. | $\mu _1=\mu _2$ |
| S. | $e \rightarrow i$ | 4. | $\mu _2<\mu _1<\sqrt{2} \mu _2$ and $\mu _2>\mu _3$ |
Codes
P $\quad$ Q $\quad$ R $\quad$ S
(a) 2 2 3013
(b) $1 \quad 2 \quad 4 \quad 3$
(c) $4 \begin{array}{lll}4 & 2 & 3\end{array}$
(d) $2 \quad 3 \quad 4 \quad 1$
Show Answer
Solution:
- For $e \rightarrow i$
$$ \begin{array}{lrl} \Rightarrow & 45^{\circ} & >\theta _c \\ \Rightarrow & \sin 45^{\circ} & >\sin \theta _c \\ \Rightarrow & \frac{1}{\sqrt{2}} & >\frac{\mu _2}{\mu _1} \\ \Rightarrow & \mu _1 & >\sqrt{2} \mu _2 \end{array} $$
For $e \rightarrow f$
angle of refraction is lesser than angle of incidence, so $\mu _2>\mu _1$ and then $\mu _2>\mu _3$
$$ \begin{aligned} & \text { For } e \rightarrow g, \quad \mu _1=\mu _2 \\ & \text { for } e \rightarrow h, \mu _2<\mu _1<\sqrt{2} \mu _2 \text { and } \mu _2>\mu _3 \end{aligned} $$
