What focal length should the reading spectacles have for a person whose near point is `50 cm` ?

Options

A. 25 cm
B. 50 cm
C. `-50 cm`
D. `-25 cm`

Answer

To find the focal length of the reading spectacles for a person whose near point is 50 cm, we can follow these steps:

### Step 1: Understand the Problem
The near point of a person is the closest distance at which they can see objects clearly. In this case, the person's near point is 50 cm, which means they cannot see anything closer than this distance. The normal near point for a person with good vision is typically 25 cm.

### Step 2: Define the Variables
- The near point of the person (V) = -50 cm (negative because it is a virtual image for the lens)
- The normal near point (U) = -25 cm (also negative for the same reason)

### Step 3: Use the Lens Formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{V} - \frac{1}{U} \]
Where:
- \( f \) is the focal length of the lens,
- \( V \) is the image distance,
- \( U \) is the object distance.

### Step 4: Substitute the Values
Substituting the values we have:
\[ \frac{1}{f} = \frac{1}{-50} - \frac{1}{-25} \]
This simplifies to:
\[ \frac{1}{f} = -\frac{1}{50} + \frac{1}{25} \]

### Step 5: Find a Common Denominator
The common denominator for 50 and 25 is 50. Thus, we can rewrite the second term:
\[ \frac{1}{f} = -\frac{1}{50} + \frac{2}{50} \]

### Step 6: Combine the Fractions
Now, combine the fractions:
\[ \frac{1}{f} = \frac{-1 + 2}{50} = \frac{1}{50} \]

### Step 7: Solve for Focal Length
Taking the reciprocal gives us the focal length:
\[ f = 50 \text{ cm} \]

### Final Answer
The focal length of the reading spectacles should be **50 cm**.

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