Given a thin convex lens (refractive index $$\mu_{2}$$), kept in a liquid (refractive index $$\mu_{1},\mu_{1}<\mu_{2}$$) having radii of curvatures $$|R_{1}|$$ and $$|R_{2}|$$ . Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?

A thin convex lens with refractive index $$\mu_2$$ is immersed in a liquid of refractive index $$\mu_1$$ ($$\mu_1 < \mu_2$$), and its second surface is silvered, effectively forming a combination of a lens and a mirror. We seek the object distance for which the image coincides with the object position.

In this silvered lens system light passes through the lens, reflects off the silvered surface, and passes through the lens again, so the system acts as an equivalent mirror whose power is given by $$P_{eq} = 2P_{lens} + P_{mirror}$$.

For the thin lens in the surrounding medium, the lens maker’s formula yields

$$\frac{1}{f_{lens}} = \left(\frac{\mu_2}{\mu_1} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) = \left(\frac{\mu_2}{\mu_1} - 1\right)\left(\frac{1}{|R_1|} + \frac{1}{|R_2|}\right) = \frac{(\mu_2 - \mu_1)}{\mu_1}\cdot\frac{|R_1| + |R_2|}{|R_1|\cdot |R_2|}.$$

The silvered surface behaves as a concave mirror of radius $$|R_2|$$, so

$$\frac{1}{f_{mirror}} = \frac{2}{|R_2|}.$$

Therefore, the equivalent focal length satisfies

$$\frac{1}{f_{eq}} = \frac{2}{f_{lens}} + \frac{1}{f_{mirror}} = \frac{2(\mu_2 - \mu_1)(|R_1| + |R_2|)}{\mu_1|R_1|\cdot |R_2|} + \frac{2}{|R_2|} = \frac{2(\mu_2 - \mu_1)(|R_1| + |R_2|) + 2\mu_1|R_1|}{\mu_1|R_1|\cdot |R_2|} = \frac{2\mu_2|R_1| + 2\mu_2|R_2| - 2\mu_1|R_1| - 2\mu_1|R_2| + 2\mu_1|R_1|}{\mu_1|R_1|\cdot |R_2|} = \frac{2\mu_2(|R_1| + |R_2|) - 2\mu_1|R_2|}{\mu_1|R_1|\cdot |R_2|}.$$

For the image to coincide with the object, the object must be placed at the center of curvature of the equivalent mirror, i.e., at $$R_{eq} = 2f_{eq}$$. Hence the required object distance is

$$R_{eq} = 2f_{eq} = \frac{2\mu_1|R_1|\cdot |R_2|}{2[\mu_2(|R_1| + |R_2|) - \mu_1|R_2|]} = \frac{\mu_1|R_1|\cdot |R_2|}{\mu_2(|R_1| + |R_2|) - \mu_1|R_2|},$$

which matches Option A.