Find the nature of image when an object is placed at 2f from the pole of a convex mirror of focal length f

Options

A. Real, inverted and of same size
B. Virtual, erect and of `1/3` times
C. Virtual, erect and 3 times
D. Real, inverted and `1/3` times

Answer

To determine the nature of the image formed by a convex mirror when an object is placed at a distance of 2f from the pole of the mirror (where f is the focal length), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Focal length of the convex mirror, \( f \) (positive for convex mirrors).
- Object distance, \( u = -2f \) (negative as per sign convention for mirrors).

2. **Use the Mirror Formula:**
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
where:
- \( f \) is the focal length,
- \( v \) is the image distance,
- \( u \) is the object distance.

3. **Substitute the Values:**
Substituting the values into the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{-2f} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{2f} \]

4. **Combine the Fractions:**
To combine the fractions on the right side:
\[ \frac{1}{v} = \frac{2}{2f} + \frac{1}{2f} = \frac{3}{2f} \]

5. **Calculate Image Distance \( v \):**
Taking the reciprocal gives:
\[ v = \frac{2f}{3} \]
Since \( v \) is positive, this indicates that the image is formed on the same side as the object.

6. **Determine the Nature of the Image:**
- The image distance \( v \) is positive, indicating that the image is virtual.
- Since it is a convex mirror, the image is always erect.
- To find the size of the image, we can calculate the magnification \( m \):
\[ m = -\frac{v}{u} = -\frac{\frac{2f}{3}}{-2f} = \frac{1}{3} \]
This means the image is diminished (one-third the size of the object).

7. **Conclusion:**
The image formed is virtual, erect, and diminished.

### Final Answer:
The nature of the image is virtual, erect, and diminished.