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

Answer 2

For v₁,
\([\frac{1}{v_{1}}+\frac{1}{-10}=\frac{1}{12}=\frac{1}{v}=-\frac{1}{12}+\frac{1}{10}=\frac{-10+12}{120}=\frac{2}{120}=\frac{1}{60}]\)
v₁=60cm

For v₂,
\(\frac{1}{v_{1}}+\frac{1}{-80}=\frac{1}{10}=\frac{1}{v_{2}}=\frac{1}{10}-\frac{1}{80}=\frac{8-1}{80}=\frac{7}{80}\)

\(v_{2}=\frac{80}{7}cm\)

Reflection from M1

\(\frac{1}{v_{1}}+\frac{1}{-10}=\frac{1}{f_{1}}=\frac{1}{12}\)

\(\frac{1}{v}+\frac{1}{-10}=\frac{1}{f}=\frac{1}{12}=\frac{1}{v_{1}}+\frac{1}{-80}\)

\(\frac{1}{v_{2}}=\frac{1}{10}=\frac{1}{v}-\frac{1}{80}\)

For lens
\(v=\frac{uf}{u+f}=\frac{(-80)10}{-80+10}=\frac{80}{7}cm\)

For an image to form at S itself Ray after reflection from mirror 'M2 must be PARALLEL to the lens
L must form an image at the focus of 'M2'.

\(\frac{n}{7}=10+\frac{80}{7}=\frac{150}{7}\)
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,
\(\frac{n}{7}=\frac{80}{7}+20\)
\(\therefore n=220\)

So, the answer is 80 or 150 or 220