For a convex lens, the image is formed at a distance of 25 cm from the lens when the object is placed on the other side of the lens at 12 cm from the optical centre. What is the power of this convex lens (in diopters)?

Introduction / Context:
Lenses are widely used in spectacles, cameras, microscopes and many optical instruments. Two important quantities that describe a lens are its focal length and its power. The power of a lens is the reciprocal of its focal length in metres and is measured in diopters. This numerical problem uses the lens formula to find the focal length of a convex lens from given object and image distances, and then converts that focal length into power.

Given Data / Assumptions:

• The lens is a convex (converging) lens.
• Image distance from the lens, v = +25 cm (real image on the side opposite to the object).
• Object distance from the optical centre, u = -12 cm (object placed in front of the lens by the usual sign convention).
• Lens formula: 1 / f = 1 / v - 1 / u, with distances measured from the optical centre.
• Power P of a lens: P = 1 / f in metres, measured in diopters (D).

Concept / Approach:
We first apply the lens formula, taking care of the signs for object and image distances. For a convex lens, conventional sign convention takes distances measured opposite to the direction of incident light as negative. Thus, the object in front of the lens gives a negative u, and a real image on the other side gives positive v. Once we calculate focal length f in centimetres, we convert it to metres and then compute the power P as 1 / f. A positive power indicates a converging or convex lens, which matches the given lens type.

Step-by-Step Solution:
Step 1: Write the lens formula: 1 / f = 1 / v - 1 / u.Step 2: Substitute the values using sign convention: v = +25 cm, u = -12 cm.Step 3: Compute 1 / f = 1 / 25 - 1 / (-12) = 1 / 25 + 1 / 12.Step 4: Find a common denominator: 1 / 25 = 12 / 300 and 1 / 12 = 25 / 300, so 1 / f = (12 + 25) / 300 = 37 / 300.Step 5: Therefore f = 300 / 37 cm, which is approximately 8.108 cm.Step 6: Convert focal length to metres: f = 8.108 cm = 0.08108 m approximately.Step 7: Calculate power P = 1 / f in diopters: P ≈ 1 / 0.08108 ≈ 12.33 diopters, positive for a convex lens.

Verification / Alternative check:
The focal length of about 8 cm is reasonable for a fairly strong convex lens. A lens with f around 10 cm would have a power of roughly +10 diopters, so a slightly shorter focal length near 8 cm gives a slightly higher power near +12 diopters. Checking the options, only 12.33 diopters matches the positive value we expect for a converging lens. Both negative power values would correspond to concave lenses, which contradicts the given information. The smaller positive power 4.33 diopters would imply a much longer focal length of about 23 cm, inconsistent with the object and image distances calculated through the lens formula.

Why Other Options Are Wrong:
-4.33 diopters and -12.33 diopters both imply a diverging lens with negative focal length, which conflicts with the problem statement that the lens is convex. A convex lens must have positive power in the usual sign convention. The value 4.33 diopters is too small; it would correspond to f around 0.23 m, which does not satisfy the given object and image distances when substituted into the lens formula. Therefore, these options do not satisfy the correct optics relationships.

Common Pitfalls:
Students often mix up the sign convention, forgetting to take u as negative for an object in front of the lens, which can lead to incorrect focal lengths. Another mistake is converting centimetres to metres incorrectly or forgetting this step when computing power in diopters. Some learners also confuse the idea that higher power means shorter focal length; remembering P = 1 / f in metres helps avoid this confusion. Always pay attention to units, signs and the given type of lens when solving such problems.

Final Answer:
The power of the convex lens is approximately +12.33 diopters.