Question

Given is a thin convex lens of glass (refractive index $\mu$) and each side having radius of curvature $R$. One side is polished for complete reflection. At what distance from the lens, an object be placed on the optic axis so that the image gets formed on the object itself?

• A. R / $\mu$
• B. R / ($\mu$ - 1)
• C. R / (2$\mu$ - 1)
• D. R / (2$\mu$ - 3)

Answer 1

Answer: (D)

R / (2$\mu$ - 3)

Answer 2

The system consists of a thin convex lens with one surface polished, effectively making it a lens-mirror combination. For the image to be formed on the object itself, the object must be placed at the center of curvature of the equivalent mirror system.

The effective power ($P_{eff}$) of such a silvered lens system is given by the formula: $P_{eff} = 2P_L + P_M$ where:

• $P_L$ is the power of the complete lens (if it were not silvered).
• $P_M$ is the power of the polished surface acting as a mirror.

1. Calculate the power of the lens ($P_L$): The lens is a thin convex lens with both sides having a radius of curvature $R$. For a biconvex lens, the radii of curvature are $R_1 = +R$ and $R_2 = -R$. The focal length of the lens ($f_L$) is given by the lens maker's formula: $\frac{1}{f_L} = (\mu - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)$ $\frac{1}{f_L} = (\mu - 1) \left(\frac{1}{R} - \frac{1}{-R}\right)$ $\frac{1}{f_L} = (\mu - 1) \left(\frac{1}{R} + \frac{1}{R}\right)$ $\frac{1}{f_L} = (\mu - 1) \left(\frac{2}{R}\right)$ So, $P_L = \frac{2(\mu - 1)}{R}$.

2. Calculate the power of the mirror ($P_M$): The polished surface is a convex surface with radius of curvature $R$. It acts as a convex mirror. The focal length of a convex mirror ($f_M$) is $f_M = -R/2$. The power of the mirror is $P_M = \frac{1}{f_M} = \frac{1}{-R/2} = -\frac{2}{R}$.

3. Calculate the effective power ($P_{eff}$) of the system: $P_{eff} = 2P_L + P_M$ $P_{eff} = 2 \left(\frac{2(\mu - 1)}{R}\right) + \left(-\frac{2}{R}\right)$ $P_{eff} = \frac{4(\mu - 1)}{R} - \frac{2}{R}$ $P_{eff} = \frac{4\mu - 4 - 2}{R}$ $P_{eff} = \frac{4\mu - 6}{R}$

4. Calculate the effective focal length ($F_{eff}$) of the system: $F_{eff} = \frac{1}{P_{eff}} = \frac{R}{4\mu - 6}$

5. Determine the object distance for the image to be formed on the object itself: For a mirror system, if the image is formed on the object itself, the object must be placed at the center of curvature of the equivalent mirror. The center of curvature is at a distance equal to twice the focal length ($2F_{eff}$). Let $x$ be the object distance from the lens. $x = 2F_{eff}$ $x = 2 \times \frac{R}{4\mu - 6}$ $x = \frac{2R}{2(2\mu - 3)}$ $x = \frac{R}{2\mu - 3}$

The object should be placed at a distance $R / (2\mu - 3)$ from the lens.