Step 1: Understanding the Concept:
The minimum distance of distinct vision for a normal human eye is 25 cm.
If a person's near point is further away (in this case, 50 cm), they are suffering from Hypermetropia (long-sightedness).
To correct this, a convex lens is used. This lens must form a virtual image of an object placed at the standard reading distance (25 cm) at the person's actual near point (50 cm).
By using the lens formula, we can calculate the focal length required, and subsequently, the power of the lens.
Step 2: Key Formula or Approach:
Lens formula: \[\frac{1}{v} - \frac{1}{u} = \frac{1}{f}\]
Power of a lens (in Diopters): \[P = \frac{100}{f(\text{in cm})}\]
Sign convention: Object distance \(u\) is negative, and image distance \(v\) for a virtual image in this case is also negative.
Step 3: Detailed Explanation:
The person wants to read a book at a distance of 25 cm. Therefore, the object distance is:
\[u = -25 \text{ cm}\]
The lens should create a virtual image at the person's minimum distance of distinct vision, which is 50 cm. Therefore, the image distance is:
\[v = -50 \text{ cm}\]
Now, substitute these values into the lens formula to find the focal length \(f\):
\[\frac{1}{-50} - \frac{1}{-25} = \frac{1}{f}\]
\[-\frac{1}{50} + \frac{1}{25} = \frac{1}{f}\]
To find a common denominator:
\[\frac{-1 + 2}{50} = \frac{1}{f}\]
\[\frac{1}{50} = \frac{1}{f} \implies f = +50 \text{ cm}\]
The positive sign confirms that the lens required is a convex lens.
Now, calculate the power \(P\):
\[P = \frac{100}{f} = \frac{100}{50}\]
\[P = +2 \text{ D}\]
This corresponds to option (C).
Step 4: Final Answer:
The person requires a convex lens of power +2 D to shift the near point from 50 cm back to 25 cm.