A thin plano-convex lens made of glass of refractive index $1.5$ is immersed in a liquid of refractive index $1.2$. When the plane side of the lens is silver coated for complete reflection, the lens immersed in the liquid behaves like a concave mirror of focal length $0.2\ \text{m}$. The radius of curvature of the curved surface of the lens is
Answer: (D) $0.10\ \text{m}$
Light passes through the lens, reflects from the silvered plane face and passes back through the lens. The plane mirror has zero power, so
$$P = 2P_L, \qquad \frac{1}{F} = \frac{2}{f_L}$$
Focal length of the lens in the liquid:
$$\frac{1}{f_L} = \left(\frac{1.5}{1.2} - 1\right)\frac{1}{R} = \frac{0.25}{R} \Rightarrow f_L = 4R$$
$$F = \frac{f_L}{2} = 2R = 0.2\ \text{m} \Rightarrow R = 0.10\ \text{m}$$
Solution by Sreeraj P, M.Sc Physics