An object of height 0.5 m is placed in front of convex mirror. Distance of object from the mirror is equal to the focal length of the mirror. Height of the image of the object is

Options

A. infinite
B. 1 m
C. 0.25 m
D. 0.5 m

Explanation

To find the height of the image formed by a convex mirror when an object is placed in front of it, we can follow these steps:

### Step 1: Understand the given data
- Height of the object (h_o) = 0.5 m
- Distance of the object from the mirror (u) = -f (since the object distance is taken as negative in mirror conventions)
- The distance of the object from the mirror is equal to the focal length of the mirror, so we can denote this distance as f.

### Step 2: Use the mirror formula
The mirror formula for a convex mirror is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Where:
- f = focal length of the mirror (positive for convex mirrors)
- v = image distance (to be determined)
- u = object distance (negative value)

Since the object distance is equal to the focal length, we have:
\[ u = -f \]

### Step 3: Substitute values into the mirror formula
Substituting u into the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{f} \]

Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{f} = \frac{2}{f} \]

### Step 4: Solve for v
Taking the reciprocal gives:
\[ v = \frac{f}{2} \]

### Step 5: Calculate the magnification
The magnification (m) of the image is given by the formula:
\[ m = -\frac{v}{u} \]
Substituting the values:
\[ m = -\frac{\frac{f}{2}}{-f} = \frac{1}{2} \]

### Step 6: Calculate the height of the image
The height of the image (h_i) can be calculated using the magnification:
\[ m = \frac{h_i}{h_o} \]
Substituting the known values:
\[ \frac{1}{2} = \frac{h_i}{0.5} \]

### Step 7: Solve for h_i
Multiplying both sides by 0.5:
\[ h_i = 0.5 \times \frac{1}{2} = 0.25 \text{ m} \]

### Final Answer
The height of the image of the object is **0.25 m**.

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