Two concave mirrors of the same focal lenght f are kept facing each other. A point source is kept symmetrically between them. What should be the distance between them so that only one image is formed at the source?

Options

A. f
B. 2f
C. 3f
D. 4f

Answer

To solve the problem of determining the distance between two concave mirrors facing each other, where a point source is placed symmetrically between them, we need to analyze the conditions under which only one image is formed at the source.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two concave mirrors with the same focal length \( f \).
- A point source is placed symmetrically between the two mirrors.

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

3. **Case 1: Object at the Center of Curvature**:
- The center of curvature \( C \) is at a distance \( r = 2f \) from the mirror (where \( r \) is the radius of curvature).
- If the point source is placed at the center of curvature of one mirror, the image will also form at the center of curvature of the other mirror.
- Therefore, if the distance between the mirrors is \( d \), we have:
\[ d = 2r = 2(2f) = 4f \]

4. **Case 2: Object at the Focus**:
- If the point source is placed at the focus of one mirror, the rays will diverge and appear to come from the focus of the other mirror.
- The distance between the two mirrors in this case would be:
\[ d = f + f = 2f \]

5. **Conclusion**:
- To ensure that only one image is formed at the source, the distance between the two mirrors can be either \( 4f \) (when the source is at the center of curvature) or \( 2f \) (when the source is at the focus).
- Therefore, the possible distances between the mirrors are \( 2f \) and \( 4f \).

### Final Answer:
The distance between the two concave mirrors should be either \( 2f \) or \( 4f \) for only one image to be formed at the source.