A long sighted person has a least distance of distinct vision of 50 cm. He wants to reduce to 25 cm. He should use a

Options

A. concave lens of focal length 50 cm
B. convex of focal length 25 cm
C. convex lens of focal length 50 cm
D. concave lens of focal length 25 cm

Explanation

To solve the problem of a long-sighted person who wants to reduce their least distance of distinct vision from 50 cm to 25 cm, we can use the lens formula and the properties of lenses. Here’s a step-by-step solution:

### Step 1: Understand the Problem
The person has a least distance of distinct vision (D) of 50 cm and wants to reduce it to 25 cm. This means we need to find a lens that allows them to see objects clearly at a distance of 25 cm.

### Step 2: Identify the Lens Type
Since the person is long-sighted (hyperopic), they need a convex lens. Convex lenses converge light rays and help in focusing on nearby objects.

### Step 3: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Where:
- \( f \) = focal length of the lens
- \( v \) = image distance (the least distance of distinct vision)
- \( u \) = object distance (the distance at which the person wants to see clearly)

### Step 4: Assign Values
For a long-sighted person:
- The least distance of distinct vision (D) = 50 cm (this is the distance at which they can see clearly without a lens).
- The new least distance they want (D') = 25 cm.
- The object distance (u) for the new least distance will be -25 cm (the negative sign indicates that the object is on the same side as the incoming light).

### Step 5: Calculate Image Distance
Since the person can see clearly at 50 cm without a lens, we can assume the image distance (v) for the lens will be +50 cm when using the lens to see the object at 25 cm.

### Step 6: Substitute in the Lens Formula
Now substituting the values into the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
\[ \frac{1}{f} = \frac{1}{50} - \frac{1}{-25} \]
\[ \frac{1}{f} = \frac{1}{50} + \frac{1}{25} \]
\[ \frac{1}{f} = \frac{1}{50} + \frac{2}{50} = \frac{3}{50} \]

### Step 7: Calculate Focal Length
Now, taking the reciprocal to find the focal length:
\[ f = \frac{50}{3} \approx 16.67 \text{ cm} \]

### Step 8: Conclusion
Since the focal length is positive, it confirms that a convex lens is required. The focal length of approximately 16.67 cm is suitable for the person to reduce their least distance of distinct vision from 50 cm to 25 cm.

### Final Answer
The person should use a **convex lens with a focal length of approximately 16.67 cm**.

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