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 placed on the optic axis so that the image gets formed on the object itself.

• A. \( \frac{R}{\mu} \)
• B. \( \frac{R}{2(\mu-3)} \)
• C. \( \mu R \)
• D. \( \frac{R}{2(\mu-1)} \)

Hint:

For a convex lens with one polished surface, the focal length can be calculated by considering both surfaces and using the lens maker's formula.

Answer 1

The Correct Option is D

To find the distance from the lens where an object must be placed so that the image forms on the object itself, we use the formula for a lens with one side completely polished, acting as a combination of a lens and a mirror.

When light travels through a lens and reflects back from a polished surface, the effective focal length (\( f \)) can be calculated using the lens maker's formula combined with the mirror equation.

First, consider the lens maker's formula for a thin lens:

\( \frac{1}{f} = (\mu - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \)

Given that one side is polished, \( R_2 = -R \). Since it's acting as a mirror, its focal length will be:

\( \frac{1}{f} = (\mu - 1) \left(\frac{1}{R} - \frac{1}{-R}\right) = (\mu - 1) \left(\frac{2}{R}\right) \)

Simplifying gives:

\( f = \frac{R}{2(\mu - 1)} \)

For the image to form at the position of the object, the object distance (\( u \)) must equal the image distance (\( v \)). Thus, when derived, the position where the object needs to be placed is:

\( u = \frac{R}{2(\mu - 1)} \)

Therefore, the correct answer is:

\( \frac{R}{2(\mu - 1)} \)