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

Correct Answer: 220

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,
\(\frac{n}{7}=\frac{80}{7}+20\)
\(\therefore n=220\)