A double convex lens made of glass of refractive index 1.5 and radii of curvature 20 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:

Correct Answer:

Explanation and Step-by-Step Derivation:

1. Understanding Lens Maker's Formula:
When a lens of refractive index n_g is immersed in a liquid of refractive index n_L, its power P is given by the Lens Maker's Formula:

$P=\left(\frac{n_g}{n_L}-1\right)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$

2. Substituting the Given Values:
- Refractive index of glass lens, n_g = 1.5
- For a double convex lens, using Cartesian sign conventions:
R₁ = +20 cm = +0.2 m
R₂ = -20 cm = -0.2 m
Therefore, the curvature factor is:

$\left(\frac{1}{R_1}-\frac{1}{R_2}\right)=\left(\frac{1}{0.2}-\frac{1}{-0.2}\right)=5+5=10{\text{m}}^{-1}$

3. Power as a Function of n_L:
Substituting these values back into the power equation:

$P=10\left(\frac{1.5}{n_L}-1\right)$

4. Analyzing Key Features of the Graph P vs n_L:
- When n_L = 1 (in air):
P = 10(1.5 - 1) = +5 D. Thus, the graph must start at a positive power when n_L = 1.
- When n_L = n_g = 1.5:
P = 10(1 - 1) = 0 D. The power becomes zero, so the graph crosses the n_L-axis at n_L = 1.5.
- When n_L > 1.5:
The fraction 1.5 / n_L < 1, which makes P negative (the double convex lens acts as a diverging lens in a denser medium). As n_L increases further, P approaches -10 D asymptotically.
- Shape of the curve:
Since P is inversely proportional to n_L (with a negative shift), the function represents a rectangular hyperbola curve with a negative slope that decreases continuously as n_L increases.

Conclusion:
The plot that correctly displays a hyperbola starting at P > 0 for n_L = 1, crossing zero at n_L = 1.5, and becoming negative for n_L > 1.5 is shown in the chosen option.