A convex lens forms an image at a distance of 16 cm from the lens when an object is placed on the opposite side of the lens at a distance of 20 cm from its optical centre. Using the lens formula and the definition of power, find the power of the lens in diopters.

Introduction / Context:
Convex lenses are converging lenses that can form real or virtual images depending on the position of the object. The power of a lens is a measure of how strongly it converges or diverges light and is widely used in optics, particularly in eyeglasses and optical instruments. This question asks you to find the power of a convex lens given the object distance and image distance using basic lens formulas.

Given Data / Assumptions:

• Object distance magnitude = 20 cm.
• Image distance magnitude = 16 cm.
• The lens is convex and forms a real image on the opposite side.
• We use the standard sign convention for thin lenses and the lens formula.

Concept / Approach:
The thin lens formula is: 1 / f = 1 / v - 1 / u where f is focal length, v is image distance, and u is object distance. Using the usual sign convention for a convex lens, we take u as negative for an object placed on the incoming side and v as positive for a real image on the outgoing side. Once we find f in meters, the power P of the lens is given by: P = 1 / f (in meters) and is expressed in diopters.

Step-by-Step Solution:
Step 1: Assign signs. For a convex lens, take object distance u = -20 cm and image distance v = +16 cm. Step 2: Use the lens formula: 1 / f = 1 / v - 1 / u. Step 3: Substitute values: 1 / f = 1 / 16 - 1 / (-20) = 1 / 16 + 1 / 20. Step 4: Calculate 1 / 16 + 1 / 20. Use a common denominator 80: 1 / 16 = 5 / 80 and 1 / 20 = 4 / 80. So 1 / f = (5 + 4) / 80 = 9 / 80. Step 5: Therefore, f = 80 / 9 cm, which is approximately 8.89 cm. Step 6: Convert focal length to meters: f ≈ 8.89 cm = 0.0889 m. Step 7: Compute power P = 1 / f ≈ 1 / 0.0889 ≈ 11.25 diopters. Step 8: Since the lens is convex and converging, the power is positive, so the correct answer is 11.25 diopters.

Verification / Alternative check:
We can quickly check the result by estimating. A lens with focal length around 10 cm has power around 10 diopters. Our calculated focal length is slightly less than 9 cm, so a power just above 11 diopters is reasonable. Also, convex lenses always have positive power values in the usual convention, so negative options can be discarded for a converging lens in this context.

Why Other Options Are Wrong:

• -3.75 diopters: The magnitude is much smaller than calculated and the negative sign would correspond to a diverging lens, not a convex converging lens.
• -11.25 diopters: The magnitude matches our calculation but the sign is wrong. A convex lens has positive power, so a negative value is not correct here.
• 3.75 diopters: This is too small in magnitude compared to the value obtained from the correct focal length.

Common Pitfalls:
Common mistakes include forgetting to use the correct sign convention for u and v, mixing up centimeters and meters when computing power, and inverting the power formula. Some students directly use focal length in centimeters in P = 1 / f, which gives the wrong numerical value because power must be calculated from f in meters. Carefully converting units and applying the formula step by step helps avoid these errors.

Final Answer:
The power of the convex lens is 11.25 diopters.