The equiconvex lens has focal length f. If is cut perpendicular to the principal axis passin through optical centre, then focal length of each half is

Options

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

Explanation

To find the focal length of each half of an equiconvex lens when it is cut perpendicular to the principal axis passing through the optical center, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Lens Configuration**:
- An equiconvex lens has two surfaces with the same radius of curvature (let's denote this radius as \( R \)).
- The lens has a focal length \( f \).

2. **Apply the Lens Maker's Formula**:
- The lens maker's formula is given by:
\[ \frac{1}{f} = \left( \mu - 1 \right) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
- For an equiconvex lens, we have:
- \( R_1 = +R \) (for the first surface)
- \( R_2 = -R \) (for the second surface)

3. **Substituting Values into the Formula**:
- Substituting \( R_1 \) and \( R_2 \) into the lens maker's formula:
\[ \frac{1}{f} = \left( \mu - 1 \right) \left( \frac{1}{R} - \left(-\frac{1}{R}\right) \right) \]
- This simplifies to:
\[ \frac{1}{f} = \left( \mu - 1 \right) \left( \frac{2}{R} \right) \]

4. **Rearranging the Equation**:
- Rearranging gives:
\[ f = \frac{R}{2(\mu - 1)} \]

5. **Cutting the Lens**:
- When the lens is cut perpendicular to the principal axis through the optical center, each half of the lens will still have one curved surface with radius \( R \) and the other surface will be flat (infinite radius of curvature).
- Thus, for each half:
- The new \( R_1 = +R \) (curved surface)
- The new \( R_2 = +\infty \) (flat surface)

6. **Applying the Lens Maker's Formula Again**:
- For each half, the lens maker's formula becomes:
\[ \frac{1}{f'} = \left( \mu - 1 \right) \left( \frac{1}{R} - 0 \right) \]
- This simplifies to:
\[ \frac{1}{f'} = \frac{\mu - 1}{R} \]

7. **Finding the Focal Length of Each Half**:
- Rearranging gives:
\[ f' = \frac{R}{\mu - 1} \]

8. **Relating the New Focal Length to the Original**:
- From the earlier equation \( f = \frac{R}{2(\mu - 1)} \), we can see that:
\[ f' = 2f \]

### Conclusion:
The focal length of each half of the lens after cutting is \( 2f \).