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A thin biconvex lens is prepared from the glass ($$\mu = 1.5$$) both curved surfaces of which have equal radii of 20 cm each. Left side surface of the lens is silvered from outside to make it reflecting. To have the position of image and object at the same place, the object should be placed, from the lens at a distance of __________ cm.
First find focal length of the lens:
$$\frac{1}{f}=(μ-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)=0.5\left(\frac{1}{20}+\frac{1}{20}\right)=\frac{1}{20}$$so f=20 cm
Now system: lens → reflection (silvered surface) → lens again
This behaves like a combination equivalent to a mirror.
For image to coincide with object, the object must be at the centre of curvature of equivalent mirror.
For this system, effective focal length becomes:
$$f_{eq}=\frac{f}{2}=10\text{ cm}$$
So centre of curvature:
$$R=2f_{eq}=20\text{ cm}$$
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