Magnification produced by a convex mirror is `1/3`, then distance of the object from mirror is

Options

A. `f/3`
B. `(2f)/3`
C. 1f
D. 2f

Answer

To find the distance of the object from a convex mirror when the magnification is given as \( \frac{1}{3} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Magnification**:
The magnification \( m \) produced by a mirror is given by the formula:
\[ m = -\frac{v}{u} \]
where \( v \) is the image distance and \( u \) is the object distance. For a convex mirror, the image is virtual, upright, and reduced in size.

2. **Substituting the Given Magnification**:
We know from the problem that:
\[ m = \frac{1}{3} \]
Therefore, we can set up the equation:
\[ \frac{1}{3} = -\frac{v}{u} \]

3. **Rearranging the Equation**:
From the magnification equation, we can express \( v \) in terms of \( u \):
\[ v = -\frac{1}{3} u \]

4. **Using the Mirror Formula**:
The mirror formula for a convex mirror is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Here, \( f \) is the focal length of the convex mirror, which is positive.

5. **Substituting for \( v \)**:
Substitute \( v = -\frac{1}{3} u \) into the mirror formula:
\[ \frac{1}{f} = \frac{1}{-\frac{1}{3}u} + \frac{1}{u} \]
Simplifying the first term:
\[ \frac{1}{f} = -\frac{3}{u} + \frac{1}{u} \]
Combine the fractions:
\[ \frac{1}{f} = -\frac{3}{u} + \frac{1}{u} = -\frac{2}{u} \]

6. **Rearranging to Find \( u \)**:
Now, rearranging gives:
\[ \frac{1}{f} = -\frac{2}{u} \implies u = -2f \]

7. **Conclusion**:
The distance of the object from the convex mirror is:
\[ u = -2f \]
The negative sign indicates that the object is located on the same side as the incoming light, which is the convention for object distances in optics.