The relation between the linear magnification `m`, the object distance `u` and the focal length `f` is

Options

A. `m=(f-u)/(f)`
B. `m=(f)/(m-u)`
C. `m=(f+u)/(f)`
D. `m=(f)/(f+u)`

Explanation

To derive the relation between linear magnification \( m \), object distance \( u \), and focal length \( f \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The linear magnification \( m \) is defined as the ratio of the height of the image \( h' \) to the height of the object \( h \), given by:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
where \( v \) is the image distance and \( u \) is the object distance.

2. **Use the Lens Formula**:
- The lens formula relates the object distance \( u \), image distance \( v \), and focal length \( f \):
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]

3. **Express \( v \) in terms of \( m \) and \( u \)**:
- From the magnification formula, we can express \( v \):
\[ v = -m u \]

4. **Substitute \( v \) into the Lens Formula**:
- Substitute \( v = -m u \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{-m u} \]

5. **Simplify the Equation**:
- This can be simplified as follows:
\[ \frac{1}{f} = \frac{1}{u} - \frac{1}{m u} \]
- Combine the fractions:
\[ \frac{1}{f} = \frac{m - 1}{m u} \]

6. **Rearranging for \( m \)**:
- Rearranging gives:
\[ \frac{1}{m} = \frac{u}{f} \cdot \frac{1}{m - 1} \]
- Inverting the equation yields:
\[ m = \frac{f}{f - u} \]

### Final Relation:
The final relation between linear magnification \( m \), object distance \( u \), and focal length \( f \) is:
\[ m = \frac{f}{f - u} \]