Optics 7 Question 6
6. 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 $\operatorname{rod} S _1$ 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
(2015 Adv.)
(a) $60 cm$
(b) $70 cm$
(c) $80 cm$
(d) $90 cm$
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Solution:
- $R=10 cm$
Applying $\frac{\mu _2}{v}-\frac{\mu _1}{u}=\frac{\mu _2-\mu _1}{R}$ two times
$$ \begin{aligned} \frac{1}{v}-\frac{1.5}{-50} & =\frac{1-1.5}{-10} \\ \frac{1}{v}+\frac{1.5}{50} & =\frac{0.5}{10} \\ \frac{1}{v}=\frac{0.5}{10}-\frac{1.5}{50} & =\frac{2.5-1.5}{50} \Rightarrow v=50 \\ M N & =d, M I _1=50 cm, \\ N I _1 & =(d-50) cm \end{aligned} $$
$$ \text { Again, } \begin{aligned} \frac{1.5}{\infty}-\frac{1}{-(d-50)} & =\frac{1.5-1}{10} \\ \frac{1}{d-50} & =\frac{1}{20} \\ d & =70 \end{aligned} $$
