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 given that the magnification produced is \( \frac{1}{3} \), we can follow these steps:

### Step 1: Understand the Magnification Formula
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 magnification is positive, so we can write:
\[ M = \frac{1}{3} \]

### Step 2: Relate Image Distance to Object Distance
From the magnification formula, we can express the image distance \( V \) in terms of the object distance \( U \):
\[ \frac{1}{3} = -\frac{V}{U} \]
This can be rearranged to find \( V \):
\[ V = -\frac{U}{3} \]

### Step 3: Use the Mirror Formula
The mirror formula for a convex mirror is given by:
\[ \frac{1}{f} = \frac{1}{U} + \frac{1}{V} \]
Substituting \( V = -\frac{U}{3} \) into the mirror formula gives:
\[ \frac{1}{f} = \frac{1}{U} - \frac{3}{U} \]
This simplifies to:
\[ \frac{1}{f} = \frac{-2}{U} \]

### Step 4: Rearranging for Object Distance
Now, we can rearrange this equation to find \( U \):
\[ U = -2f \]

### Step 5: Conclusion
Since the object distance \( U \) is negative (as per the sign convention for mirrors), the distance of the object from the mirror is:
\[ |U| = 2f \]
Thus, the distance of the object from the convex mirror is \( 2f \).

### Summary
The distance of the object from the convex mirror is \( 2f \), where \( f \) is the focal length of the convex mirror.

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