Ray Optics

JEE Main level, single correct, medium

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Question 121 of 295

A thin biconvex lens is prepared from the glass (μ=1.5\mu=1.5) both curved surfaces of which have equal radii of 2020 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.

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Answer: (A) 1010

When one surface of a lens is silvered, the system behaves like an equivalent mirror. The light refracts through the lens, reflects off the silvered surface, and refracts through the lens again, so the equivalent power is:
Peq=PL+PM+PL=2PL+PMP_{eq} = P_L + P_M + P_L = 2P_L + P_M

Power of the biconvex lens:
For a thin biconvex lens with R1=+20 cmR_1 = +20\text{ cm}, R2=−20 cmR_2 = -20\text{ cm} and μ=1.5\mu = 1.5, the lens maker's formula gives:
1fL=(μ−1)(1R1−1R2)\dfrac{1}{f_L} = (\mu - 1)\left(\dfrac{1}{R_1} - \dfrac{1}{R_2}\right)

1fL=(1.5−1)(120−1−20)=0.5⋅220=120 cm−1\dfrac{1}{f_L} = (1.5 - 1)\left(\dfrac{1}{20} - \dfrac{1}{-20}\right) = 0.5 \cdot \dfrac{2}{20} = \dfrac{1}{20}\text{ cm}^{-1}

So, PL=120 cm−1P_L = \dfrac{1}{20}\text{ cm}^{-1}.

Power of the silvered surface (acting as a concave mirror):
The back surface has radius of curvature R=20 cmR = 20\text{ cm}, so its focal length is:
fM=R2=10 cmf_M = \dfrac{R}{2} = 10\text{ cm}

PM=1fM=110 cm−1P_M = \dfrac{1}{f_M} = \dfrac{1}{10}\text{ cm}^{-1}

Equivalent focal length:
Peq=2PL+PM=2⋅120+110=110+110=15 cm−1P_{eq} = 2P_L + P_M = 2 \cdot \dfrac{1}{20} + \dfrac{1}{10} = \dfrac{1}{10} + \dfrac{1}{10} = \dfrac{1}{5}\text{ cm}^{-1}

Feq=1Peq=5 cmF_{eq} = \dfrac{1}{P_{eq}} = 5\text{ cm}

Since Peq>0P_{eq} \gt 0, the system behaves as a converging (concave) equivalent mirror of focal length 5 cm5\text{ cm}.

Object position for object and image to coincide:
For a concave mirror, the image coincides with the object when the object is placed at the centre of curvature, i.e., at a distance 2Feq2F_{eq} from the mirror:
u=2Feq=2×5=10 cmu = 2F_{eq} = 2 \times 5 = 10\text{ cm}

Hence, the correct option is (1) 10(1)\ 10.

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