At what distance from a concave mirror of focal length 25 cm should an object be placed so that the size of image is equal to the size of the object.

Answer

To solve the problem of finding the distance from a concave mirror of focal length 25 cm at which an object should be placed so that the size of the image is equal to the size of the object, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between object distance (u), image distance (v), and focal length (f)**:
For mirrors, the mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
where:
- \( f \) = focal length of the mirror
- \( u \) = object distance (measured from the mirror)
- \( v \) = image distance (measured from the mirror)

2. **Identify the conditions for equal size of object and image**:
The size of the image is equal to the size of the object when the magnification (m) is equal to 1. For mirrors, magnification is given by:
\[ m = -\frac{v}{u} \]
Setting \( m = 1 \) gives us:
\[ -\frac{v}{u} = 1 \implies v = -u \]
This means that the image distance is equal in magnitude but opposite in sign to the object distance.

3. **Substitute \( v \) in the mirror formula**:
Since \( v = -u \), we can substitute this into the mirror formula:
\[ \frac{1}{f} = \frac{1}{-u} + \frac{1}{u} \]
Simplifying this gives:
\[ \frac{1}{f} = 0 \implies \text{This is not useful directly, so we will use the relationship derived from the radius of curvature.} \]

4. **Use the relationship between focal length and radius of curvature**:
The radius of curvature (R) is related to the focal length (f) by:
\[ R = 2f \]
Given \( f = 25 \, \text{cm} \):
\[ R = 2 \times 25 \, \text{cm} = 50 \, \text{cm} \]

5. **Determine the object distance**:
Since the object should be placed at the center of curvature for the image size to equal the object size, we have:
\[ u = R = 50 \, \text{cm} \]

### Final Answer:
The object should be placed at a distance of **50 cm** from the concave mirror.

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