A short linear object of length b lies on the axis of a concave mirror of focal length F at a distance u from the pole. The length of the image will be

Options

A. `((f)/(u-f))b`
B. `((u-f)/f)b`
C. `((f)/(u-f))^2b`
D. `((u-f)/f)^2b`

Answer

To find the length of the image of a short linear object of length \( b \) placed at a distance \( u \) from the pole of a concave mirror with focal length \( F \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Setup**:
- We have a concave mirror with a focal length \( F \).
- The object is placed at a distance \( u \) from the mirror's pole.
- The object has a length \( b \).

2. **Use the Mirror Formula**:
- The mirror formula is given by:
\[ \frac{1}{v} + \frac{1}{u} = \frac{1}{F} \]
- Here, for a concave mirror, the focal length \( F \) is considered negative. Thus, we can rewrite the equation as:
\[ \frac{1}{v} + \frac{1}{(-u)} = \frac{1}{(-F)} \]

3. **Rearranging the Mirror Formula**:
- Rearranging the formula gives:
\[ \frac{1}{v} = \frac{1}{u} - \frac{1}{F} \]
- This can be simplified to:
\[ v = \frac{uF}{u - F} \]

4. **Determine the Magnification**:
- The magnification \( m \) produced by the mirror is given by:
\[ m = -\frac{v}{u} \]
- Substituting the value of \( v \):
\[ m = -\frac{\frac{uF}{u - F}}{u} = -\frac{F}{u - F} \]

5. **Calculate the Length of the Image**:
- The length of the image \( b' \) can be calculated using the magnification:
\[ b' = m \cdot b = -\frac{F}{u - F} \cdot b \]
- Since we are interested in the magnitude of the image length, we take the absolute value:
\[ |b'| = \frac{F}{u - F} \cdot b \]

### Final Result:
The length of the image is given by:
\[ |b'| = \frac{F \cdot b}{u - F} \]