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 $70cm$

For $S_1$,
On applying the formula of Refraction for curved surface
$\frac{{\mu }_2}{v}-\frac{{\mu }_1}{u}=\frac{{\mu }_2-{\mu }_1}{R}$ .....(1)

where, ${\mu }_1=1.5,{\mu }_2=1,R=10cm$
on substituting the above values in eq. (1)
$\frac{1}{v}-\frac{1.5}{(-50)}=\frac{1-1.5}{-10}$
or
$\frac{1}{v}+\frac{3}{100}=\frac{1}{20}$
$v=50cm$
This acts as image for $S_2$ with distance from it's aperture as $(d-50)cm$

Given, $v=\infty ,u=-(d-50)cm,R=10cm$
By substituting the above values in eq. (1),

$\frac{1}{\infty }-\frac{1}{-(d-50)}=\frac{1.5-1}{10}$
$0-\frac{1}{-(d-50)}=\frac{1.5-1}{10}$
or
$\frac{1}{d-50}=\frac{1}{20}$
or

$d=70cm$

Hence, option B is correct.