A convex mirror forms an image one-fourth the size of the object. If object is at a distance of 0.5 m from the mirror the focal length of the mirror is

Options

A. 0.16
B. `-1.5 m`
C. 0.4 m
D. `-0.4 m`

Answer

To find the focal length of a convex mirror when the image size is one-fourth the size of the object and the object is placed at a distance of 0.5 m, we can follow these steps:

### Step 1: Understand the relationship between object height and image height
Given that the image size is one-fourth the size of the object, we can express this as:
\[ \text{Magnification} (m) = \frac{h_i}{h_o} = \frac{1}{4} \]
where \( h_i \) is the height of the image and \( h_o \) is the height of the object.

### Step 2: Use the magnification formula
The magnification for mirrors is also given by the formula:
\[ m = -\frac{V}{U} \]
where \( V \) is the image distance and \( U \) is the object distance. Since the object distance for a convex mirror is taken as negative, we have:
\[ m = -\frac{V}{-0.5} = \frac{V}{0.5} \]

### Step 3: Set up the equation for magnification
From the magnification we found earlier:
\[ \frac{1}{4} = \frac{V}{0.5} \]
Cross-multiplying gives:
\[ V = \frac{0.5}{4} = 0.125 \text{ m} \]

### Step 4: Use the mirror formula to find the focal length
The mirror formula is given by:
\[ \frac{1}{F} = \frac{1}{V} + \frac{1}{U} \]
Substituting the values we have:
- \( V = 0.125 \) m
- \( U = -0.5 \) m (since it's a convex mirror)

So we can write:
\[ \frac{1}{F} = \frac{1}{0.125} + \frac{1}{-0.5} \]

### Step 5: Calculate the values
Calculating \( \frac{1}{0.125} \):
\[ \frac{1}{0.125} = 8 \]
Calculating \( \frac{1}{-0.5} \):
\[ \frac{1}{-0.5} = -2 \]
Now substituting these values back into the equation:
\[ \frac{1}{F} = 8 - 2 = 6 \]

### Step 6: Find the focal length
Taking the reciprocal gives:
\[ F = \frac{1}{6} \approx 0.1667 \text{ m} \]

Thus, the focal length of the convex mirror is approximately **0.1667 m**.