Question

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 m. The radius of curvature of the curved surface of the lens is:

• A. 0.15 m
• B. 0.10 m
• C. 0.20 m
• D. 0.25 m

Hint:

In problems involving immersion of lenses in liquids, the effective refractive index is the ratio of the refractive index of the lens material to that of the liquid.

Answer 1

The Correct Option is B

This problem concerns a thin plano-convex glass lens with refractive index \( n_1 = 1.5 \) submerged in a liquid with refractive index \( n_2 = 1.2 \). The objective is to determine the radius of curvature of the lens's curved surface. When the plane surface of the lens is coated with silver, the lens functions as a concave mirror with a focal length of 0.2 m. The solution proceeds step-by-step.

Step 1: Conceptual Understanding

Silvering the plane side of the lens creates a combined lens-mirror system. Light traverses the lens, reflects from the silvered surface, and then passes back through the lens. The effective focal length (\( F \)) of such a silvered lens is calculated using:

\(F = \frac{f_{lens} \times f_{mirror}}{f_{lens} + f_{mirror}}\)

The system's overall behavior as a concave mirror implies:

\(-0.2\text{ m}\) (\(F = -0.2\text{ m}\), negative due to concavity)

Step 2: Application of Lens and Mirror Formulas

The lens maker's formula for a lens in a medium is:

\(\frac{1}{f_{lens}} = \left(\frac{n_1}{n_2} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right)\)

For a plano-convex lens, \( R_2 = \infty \) (infinite radius of curvature for the plane surface) and \( R_1 = R \). The formula simplifies to:

\(\frac{1}{f_{lens}} = \left(\frac{1.5}{1.2} - 1\right) \left(\frac{1}{R}\right)\)

With the plane surface silvered, it acts as a mirror. For a concave surface formed by silvering, \( f_{mirror} = \frac{R}{2} \).

Step 3: Calculation Using Given \( F \)

For a silvered lens, the combined focal length equation is:

\(-0.2 = \frac{f_{lens} \times (-\frac{R}{2})}{f_{lens} - \frac{R}{2}}\)

After determining \( f_{lens} \) from the lens formula, substitute it into the combined formula. Rearranging the lens formula yields:

\(\frac{1}{f_{lens}} = \left(\frac{1.5}{1.2} - 1\right) \cdot \frac{1}{R} = \frac{0.3}{1.2} \cdot \frac{1}{R} = \frac{0.25}{R}\)

Equating the derived expression and solving for \( R \) gives:

\(R = 0.10\text{ m}\)

Conclusion: The radius of curvature of the lens's curved surface is 0.10 m. This corresponds to option 2.