What will be the height of image when an object of 2 mm is placed on the axis of a convex mirror at a distance 20 cm of radius of curvature 40 cm

Options

A. 20 mm
B. 10 mm
C. 6 mm
D. 1 mm

Answer

To find the height of the image formed by a convex mirror when an object of height 2 mm is placed at a distance of 20 cm, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Height of object (\(h_o\)) = 2 mm
- Object distance (\(u\)) = -20 cm (negative because the object is in front of the mirror)
- Radius of curvature (\(R\)) = 40 cm

2. **Calculate the Focal Length (\(f\)):**
- The focal length of a convex mirror is given by the formula:
\[ f = \frac{R}{2} \]
- Substituting the value of \(R\):
\[ f = \frac{40 \text{ cm}}{2} = 20 \text{ cm} \]
- Since it is a convex mirror, the focal length is positive:
\[ f = +20 \text{ cm} \]

3. **Use the Mirror Formula to Find Image Distance (\(v\)):**
- The mirror formula is:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
- Substituting the known values:
\[ \frac{1}{20} = \frac{1}{v} + \frac{1}{-20} \]
- Rearranging gives:
\[ \frac{1}{v} = \frac{1}{20} + \frac{1}{20} = \frac{2}{20} = \frac{1}{10} \]
- Therefore, the image distance is:
\[ v = 10 \text{ cm} \]

4. **Calculate the Magnification (\(m\)):**
- The magnification for mirrors is given by:
\[ m = -\frac{v}{u} \]
- Substituting the values:
\[ m = -\frac{10}{-20} = \frac{1}{2} \]

5. **Calculate the Height of the Image (\(h_i\)):**
- The magnification is also related to the heights of the object and image:
\[ m = \frac{h_i}{h_o} \]
- Rearranging gives:
\[ h_i = m \cdot h_o \]
- Substituting the values:
\[ h_i = \frac{1}{2} \cdot 2 \text{ mm} = 1 \text{ mm} \]

### Final Answer:
The height of the image (\(h_i\)) is **1 mm**.

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