Question

Two identical glass rods $S_1$ and $S_2$ (refractive index = 1.5) have one convex end of radius of curvature $10cm.$ 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 $50cm$ 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. $60cm$
• B. $70cm$
• C. $80cm$
• D. $90cm$

Answer 1

The correct option is B $70cm$

Applying $(\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R})$ two times, we will get

$\Rightarrow \frac{1}{v}-\frac{1.5}{-50}=\frac{1-1.5}{-10}$

$\Rightarrow \frac{1}{v}=\frac{0.5}{10}-\frac{1.5}{50}=\frac{2.5-1.5}{50}$

$\Rightarrow v=50.$

$\Rightarrow AB=d,$ $A{I}_1=50cm$ $B{I}_1=(d-50)cm$

Again, $\frac{1.5}{\infty }-\frac{1}{-(d-50)}=\frac{1.5-1}{10}$

$\Rightarrow \frac{1}{d-50}=\frac{1}{20}$

$\Rightarrow d=70cm.$

Hence, the answer is $70cm.$