Question

Two identical glass rods $S_1$ and $S_2$ (refractive index = 1.5) have one convex end of radius of curvature 10 cm. They are placed with the curved surfaces at a distance d as shown in the figure, with their axes (shown by the dashed line) aligned. When a point source of light P is placed inside rod $S_1$ on its axis at a distance of 50 cm from the curved face, the light rays emanating from it are found to be parallel to the axis inside $S_2$. The distance d is

• A. 60 cm
• B. 70 cm
• C. 80 cm
• D. 90 cm

Answer 1

Answer: (B)

This question is of multiple event.
First event on curved surface of $S_1$ & Second on curved surface of $S_2$
Event $- 1\quad n_{2} = 1$
$n_{1}= 1.5$
$\frac{1}{v} - \frac{1.5}{u} = \frac{1-1.5}{R}$
$\frac{1}{v}- \frac{1.5}{-50} = \frac{-0.5}{-10}$
$\frac{1}{v} + \frac{1.5}{50} = \frac{1}{20}$
$\frac{1}{v} = \frac{1}{20}-\frac{1.5}{50}$
$\frac{1}{v} = \frac{5-3}{100} = \frac{2}{100}\quad\quad∴ v = 50 \,cm$
for second event $\quad u = - \left(d - 50\right)$
$v = ∞$
$R = + 10\, cm$
$\frac{1.5}{v}-\frac{1}{u} = \frac{1.5-1}{R}$
$\frac{1.5}{\infty} - \frac{1}{-\left(d-50\right)} = \frac{0.5}{10}$
$\frac{1}{d-50} = \frac{1}{20}$
$20 = d - 50$
$d = 70 \,cm$