Focal length of convex lens is f. If this lens is cut along parallel to its length in two equal parts then focal length of its half part will be:

Options

A. `f//2`
B. `f `
C. `(3f )/(2)`
D. `2f`

Answer

To solve the problem of finding the focal length of a half part of a convex lens when the original lens has a focal length of \( f \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Lens Maker's Formula**:
The lens maker's formula is given by:
\[ \frac{1}{f} = \left(\frac{\mu_2}{\mu_1} - 1\right) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]
where \( f \) is the focal length of the lens, \( \mu_1 \) is the refractive index of the medium from which light is coming (air, \( \mu_1 = 1 \)), \( \mu_2 \) is the refractive index of the lens material, and \( R_1 \) and \( R_2 \) are the radii of curvature of the lens surfaces.

2. **Applying the Formula to the Original Lens**:
For the original convex lens with focal length \( f \):
\[ \frac{1}{f} = (\mu - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]

3. **Cutting the Lens in Half**:
When the lens is cut in half parallel to its length, one part will have a curved surface and the other will have a plane surface. Let’s denote the focal length of the half lens as \( f' \).

4. **Analyzing the Half Lens**:
For the half lens:
- The first surface remains curved with radius \( R \).
- The second surface becomes a plane surface, which can be treated as having an infinite radius of curvature (\( R_2 = \infty \)).

5. **Using the Lens Maker's Formula for the Half Lens**:
The lens maker's formula for the half lens becomes:
\[ \frac{1}{f'} = (\mu - 1) \left(\frac{1}{R} - \frac{1}{\infty}\right) = (\mu - 1) \frac{1}{R} \]

6. **Relating the Focal Lengths**:
From the original lens, we have:
\[ \frac{1}{f} = (\mu - 1) \left(\frac{1}{R_1} - \frac{1}{R_2}\right) \]
If we assume \( R_1 = R \) and \( R_2 = -R \) for a symmetric lens, we can express:
\[ \frac{1}{f} = (\mu - 1) \left(\frac{1}{R} + \frac{1}{R}\right) = 2(\mu - 1) \frac{1}{R} \]
Thus, we can relate \( \frac{1}{R} \) to \( \frac{1}{f} \):
\[ \frac{1}{R} = \frac{1}{2f} (\mu - 1) \]

7. **Substituting Back**:
Substituting \( \frac{1}{R} \) back into the equation for \( f' \):
\[ \frac{1}{f'} = (\mu - 1) \frac{1}{R} = (\mu - 1) \cdot \frac{1}{2f} = \frac{(\mu - 1)}{2f} \]
Thus, we can conclude:
\[ f' = 2f \]

### Conclusion:
The focal length of the half part of the convex lens is \( 2f \).