Radius of concave mirror is 40 cm and the size of image (real) is twice as that of object, then the object distance is

Options

A. 60 cm
B. 20 cm
C. 40 cm
D. 30 cm

Answer

To solve the problem, we need to find the object distance (u) for a concave mirror given the radius of curvature (R) and the magnification (m).

### Step-by-Step Solution:

1. **Identify the given values:**
- Radius of curvature (R) = 40 cm
- Magnification (m) = 2 (since the image size is twice the object size)

2. **Calculate the focal length (f) of the concave mirror:**
The focal length (f) is related to the radius of curvature (R) by the formula:
\[ f = -\frac{R}{2} \]
Substituting the value of R:
\[ f = -\frac{40 \, \text{cm}}{2} = -20 \, \text{cm} \]

3. **Use the magnification formula:**
The magnification (m) for mirrors is given by:
\[ m = \frac{h'}{h} = \frac{v}{u} \]
where \( h' \) is the height of the image, \( h \) is the height of the object, \( v \) is the image distance, and \( u \) is the object distance. Given that \( m = 2 \):
\[ 2 = \frac{v}{u} \]
From this, we can express the image distance (v) in terms of object distance (u):
\[ v = 2u \]

4. **Substitute into the mirror formula:**
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( f = -20 \, \text{cm} \) and \( v = 2u \):
\[ \frac{1}{-20} = \frac{1}{2u} + \frac{1}{u} \]

5. **Combine the fractions on the right side:**
The right side can be simplified:
\[ \frac{1}{2u} + \frac{1}{u} = \frac{1 + 2}{2u} = \frac{3}{2u} \]
Thus, we have:
\[ \frac{1}{-20} = \frac{3}{2u} \]

6. **Cross-multiply to solve for u:**
Cross-multiplying gives:
\[ -20 \cdot 3 = 2u \]
\[ -60 = 2u \]
Dividing both sides by 2:
\[ u = -30 \, \text{cm} \]

7. **Conclusion:**
The object distance (u) is -30 cm, indicating that the object is placed in front of the concave mirror.

### Final Answer:
The object distance is \( u = -30 \, \text{cm} \).

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