A plano-convex lens acts like a concave mirror of focal length 28 cm when its plane surface is silvered and like a concave mirror of focal length 10 cm when its curved surface is silvered. The refractive index of the material of the lens is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 1.55
A lens with one face silvered acts as a mirror: the light is refracted by the lens, reflected by the silvered face and refracted by the lens again. Its power is
\( \displaystyle P = 2P_{L} + P_{M}, \qquad F = \frac{1}{P} \)
For a plano-convex lens with curved face of radius \( R \), the lens power is \( P_L = (\mu - 1)/R \).
Plane surface silvered: a plane mirror has \( P_M = 0 \).
\( \displaystyle P = \frac{2(\mu - 1)}{R} \;\Rightarrow\; F_1 = \frac{R}{2(\mu - 1)} = 28\,\mathrm{cm} \)
Curved surface silvered: the mirror of radius \( R \) has \( P_M = 2/R \).
\( \displaystyle P = \frac{2(\mu - 1)}{R} + \frac{2}{R} = \frac{2\mu}{R} \;\Rightarrow\; F_2 = \frac{R}{2\mu} = 10\,\mathrm{cm} \)
Dividing the two focal lengths, \( R \) cancels:
\( \displaystyle \frac{F_1}{F_2} = \frac{\mu}{\mu - 1} = \frac{28}{10} = 2.8 \)
\( \mu = 2.8\mu - 2.8 \;\Rightarrow\; 1.8\mu = 2.8 \)
\( \displaystyle \mu = \frac{2.8}{1.8} = \frac{14}{9} \approx 1.556 \)
This is closest to the option 1.55.