Question

A double convex lens made of glass of refractive index \(1.5\) and radii of curvature \(20\,\text{cm}\) each is immersed in a liquid of refractive index \(n_L\). The correct plot showing the variation of the power, in the units of diopter \((D)\), as a function of \(n_L\), is:

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

Power of a lens in a medium: \[ P= \left(\frac{n_g}{n_m}-1\right) \left(\frac1{R_1}-\frac1{R_2}\right) \] If surrounding medium has larger refractive index than the lens, the lens may behave like a concave lens.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Power \( P \) is given by Lens Maker's Formula: \( P = \frac{1}{f} = \left(\frac{n_{lens}}{n_{medium}} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \).
Step 3: Detailed Explanation:
1) Substituting values:
\( R_1 = 20 \text{ cm} = 0.2 \text{ m} \), \( R_2 = -20 \text{ cm} = -0.2 \text{ m} \).
\( \frac{1}{R_1} - \frac{1}{R_2} = \frac{1}{0.2} + \frac{1}{0.2} = 10 \text{ m}^{-1} \).
\( P(n_L) = \left(\frac{1.5}{n_L} - 1\right) \times 10 = \frac{15}{n_L} - 10 \).
2) Analyzing the function:
- At \( n_L = 1 \) (air): \( P = 15 - 10 = 5 D \).
- At \( n_L = 1.5 \) (liquid same as lens): \( P = 0 \).
- As \( n_L \to \infty \), \( P \to -10 \).
- Derivative \( P' = -15/n_L^2<0 \) (decreasing).
- Second derivative \( P'' = 30/n_L^3>0 \) (concave upwards).
Step 4: Final Answer:
The graph in option (A) matches these characteristics.