Question

An optical arrangement consists of two concave mirrors M1 and M2, and a convex lens L with a common principal axis, as shown in the figure. The focal length of L is 10 cm. The radii of curvature of M1 and M2 are 20 cm and 24 cm, respectively. The distance between L and M2 is 20 cm. A point object S is placed at the mid-point between L and M2 on the axis. When the distance between L and M1 is $\frac{n}{7}$ cm, one of the images coincides with S. The value of 𝑛 is _______.

Answer 1

For the reflection from M2,
$\frac{1}{v}=\frac{1}{(-10)}+\frac{1}{(-12)}$
$\frac{1}{v}=\frac{1}{10}-\frac{1}{12}$
v=+60 cm
For refraction from L,
$\frac{1}{v}-\frac{1}{u}=\frac{1}{f}$
$\frac{1}{v}-\frac{1}{-80}=\frac{1}{10}$
$v=+\frac{80}{7}$(for l2)
This image should be at focus of M1
$\therefore \frac{20}{2}+\frac{80}{7}=\frac{n}{7}$
n =150
If l2 is formed at the pole of M1,
Then, $\frac{n}{7}=\frac{80}{7}$
n = 80
And if l2 is formed at the center of curvature of M1,
$\frac{n}{7}=\frac{80}{7}+20$
$\therefore n=220$