An object infront of a concave mirror of focal length f. A virtual image is formed with a magnification of 2. To obtain a real image of same magnification the object has to be moved by a distance

Options

A. f
B. f/2
C. 3f/2
D. 2f/3

Answer

To solve the problem step by step, we will follow the concepts of concave mirrors, magnification, and the mirror formula.

### Step 1: Understand the given information
- We have a concave mirror with a focal length \( f \).
- A virtual image is formed with a magnification \( m = 2 \).

### Step 2: Use the magnification formula
For a concave mirror, the magnification \( m \) is given by:
\[ m = -\frac{v}{u} \]
Where:
- \( v \) = image distance
- \( u \) = object distance

Since the magnification is given as \( 2 \), we can write:
\[ 2 = -\frac{v}{u} \]
From this, we can express \( v \) in terms of \( u \):
\[ v = -2u \quad \text{(Equation 1)} \]

### Step 3: Use the mirror formula
The mirror formula is given by:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
Substituting \( v \) from Equation 1 into the mirror formula:
\[ \frac{1}{f} = \frac{1}{-2u} + \frac{1}{u} \]
Finding a common denominator:
\[ \frac{1}{f} = \frac{-1 + 2}{2u} = \frac{1}{2u} \]
Thus, we can rearrange to find \( u \):
\[ u = \frac{f}{2} \quad \text{(Equation 2)} \]

### Step 4: Finding the new object distance for a real image
Now, we need to find the new object distance \( u' \) when the image is real and has the same magnification \( m' = -2 \). For a real image, the magnification formula becomes:
\[ m' = -\frac{v'}{u'} \]
Setting \( m' = -2 \):
\[ -2 = -\frac{v'}{u'} \]
Thus, we have:
\[ v' = 2u' \quad \text{(Equation 3)} \]

### Step 5: Substitute into the mirror formula for real image
Using the mirror formula again:
\[ \frac{1}{f} = \frac{1}{v'} + \frac{1}{u'} \]
Substituting \( v' \) from Equation 3:
\[ \frac{1}{f} = \frac{1}{2u'} + \frac{1}{u'} \]
Finding a common denominator:
\[ \frac{1}{f} = \frac{1 + 2}{2u'} = \frac{3}{2u'} \]
Rearranging gives:
\[ u' = \frac{3f}{2} \quad \text{(Equation 4)} \]

### Step 6: Calculate the distance moved
The distance moved by the object is:
\[ \text{Distance moved} = u' - u \]
Substituting \( u' \) from Equation 4 and \( u \) from Equation 2:
\[ \text{Distance moved} = \frac{3f}{2} - \frac{f}{2} = \frac{2f}{2} = f \]

### Final Answer
The object has to be moved by a distance of \( f \).

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