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(D) The system consists of two lenses and a mirror. Let the focal length of each lens be $f_l$ and the radius of curvature of the surfaces be $R$. The

Vedclass · Accepted Answer

$12$

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(D) The system consists of two lenses and a mirror. Let the focal length of each lens be $f_l$ and the radius of curvature of the surfaces be $R$. The mirror is formed by the rear surface of the second lens,which acts as a concave mirror with focal length $f_m = R/2$.Case $1$: Air between lenses.The effective focal length $F$ of the silvered lens system is given by $1/F = 2/f_l + 1/f_m$. Since the image of an object at infinity is formed at $20 \, cm$,$F = 20 \, cm$. For a thin lens,$1/f_l = (\mu_g - 1)(2/R)$. For the mirror,$f_m = R/2$. Given $F = 20 \, cm$,we have $1/20 = 2(2(\mu_g - 1)/R) + 2/R = (8(\mu_g - 1) + 2)/R$. This determines the system parameters.Case $2$: Water between lenses.When water is filled,the system acts as two convex lenses with air,a water lens in the middle,and the mirror. The power of the system is $P_{eq} = 2P_l + P_w + P_m$. Using the lens maker's formula and the equivalent focal length calculation,the new focal length $F'$ is calculated. With water $(\mu_w = 4/3)$ replacing air,the effective focal length becomes $12 \, cm$.

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