An object is placed at a distance of 30 cm from a concave mirror and its real image is formed at a distance of 30 cm from the mirror. The focal length of the mirror is

Options

A. 15 cm
B. 45 cm
C. 30 cm
D. 20 cm

Explanation

To find the focal length of the concave mirror given that the object distance (u) is -30 cm and the image distance (v) is also -30 cm, we can use the mirror formula:

### Step-by-Step Solution:

1. **Identify the given values:**
- Object distance (u) = -30 cm (negative because the object is in front of the mirror)
- Image distance (v) = -30 cm (negative because the image is real and formed on the same side as the object)

2. **Write the mirror formula:**
The mirror formula for concave mirrors is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

3. **Substitute the values into the mirror formula:**
\[ \frac{1}{f} = \frac{1}{-30} + \frac{1}{-30} \]

4. **Calculate the right-hand side:**
\[ \frac{1}{f} = -\frac{1}{30} - \frac{1}{30} = -\frac{2}{30} \]
Simplifying this gives:
\[ \frac{1}{f} = -\frac{1}{15} \]

5. **Find the focal length (f):**
Taking the reciprocal of both sides, we have:
\[ f = -15 \text{ cm} \]

### Final Answer:
The focal length of the concave mirror is **-15 cm**.