A person cannot see the object beyond 3 m distinctly. Find focal length of the lens required to correct this defect of vision.

Options

A. 25 cm
B. - 3 m
C. 3 m
D. `infty`

Answer

To solve the problem of finding the focal length of the lens required to correct the vision defect of a person who cannot see objects beyond 3 meters distinctly, we can follow these steps:

### Step 1: Understand the Vision Defect
The person has a near point at 3 meters, meaning they can see objects clearly only up to this distance. Beyond this distance, objects appear blurry. The far point for a person with normal vision is considered to be at infinity.

### Step 2: Identify the Lens Formula
The lens formula relates the object distance (U), the image distance (V), and the focal length (F) of the lens. The formula is given by:
\[ \frac{1}{F} = \frac{1}{V} - \frac{1}{U} \]

### Step 3: Assign Values
In this scenario:
- The object distance (U) is -3 m (the negative sign indicates that the object is on the same side as the incoming light).
- The image distance (V) is at infinity, which we can denote as +∞.

### Step 4: Substitute Values into the Lens Formula
Substituting the values into the lens formula:
\[ \frac{1}{F} = \frac{1}{\infty} - \frac{1}{-3} \]

### Step 5: Simplify the Equation
Since \( \frac{1}{\infty} = 0 \), the equation simplifies to:
\[ \frac{1}{F} = 0 + \frac{1}{3} \]
\[ \frac{1}{F} = \frac{1}{3} \]

### Step 6: Calculate Focal Length
Taking the reciprocal gives us:
\[ F = 3 \text{ m} \]

### Step 7: Determine the Sign of Focal Length
Since the lens is used to correct a vision defect (myopia), it must be a diverging lens. Therefore, the focal length is negative:
\[ F = -3 \text{ m} \]

### Final Answer
The focal length of the lens required to correct the defect of vision is:
\[ F = -3 \text{ m} \]

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