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 π is _______.
Approach Solution - 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,
\(\frac{n}{7}=\frac{80}{7}+20\)
\(\therefore n=220\)
Approach Solution -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
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Concepts Used:
Ray Optics and Optical Instruments
Optics, deals with the determination of behaviour and the properties of light, along with its interactions with the matter and also with the instruments that are used to detect it.
Ray optics is also known as the geometrical optics and it is a branch of science which describes light propagation.
Reflection is the change in direction of light at an interface in-between two different media so that the wave-front returns into a medium from which it was originated.
Speed of light is the rate at which the light travels in free space.
A phenomenal change in image formed when the light is passed from one medium to another which is called Refraction.
Total Internal Reflection is the reflection of light when the light ray enters into a rarer medium from a denser medium and the angle of incidence is higher than the critical angle of incidence then that light ray will be reflected back to the denser medium.
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