An object is placed at a distance of `f/2` from a convex lens of focal length f. The image will be

Options

A. At one of the foci, virtual and double its size
B. Is greater than 1.5 but less than 2.0
C. At 2f , virtual and erect
D. None of the above

Answer

To solve the problem, we need to determine the characteristics of the image formed by a convex lens when an object is placed at a distance of \( \frac{f}{2} \) from the lens, where \( f \) is the focal length of the lens.

### Step 1: Identify the given parameters
- Focal length of the lens, \( f \)
- Object distance, \( u = -\frac{f}{2} \) (negative as per sign convention)

### Step 2: Use the lens formula
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Where:
- \( f \) is the focal length,
- \( v \) is the image distance,
- \( u \) is the object distance.

### Step 3: Substitute the values into the lens formula
Substituting \( u = -\frac{f}{2} \) into the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{-\frac{f}{2}} \]
This simplifies to:
\[ \frac{1}{f} = \frac{1}{v} + \frac{2}{f} \]

### Step 4: Rearranging the equation
Rearranging the equation to isolate \( \frac{1}{v} \):
\[ \frac{1}{v} = \frac{1}{f} - \frac{2}{f} \]
\[ \frac{1}{v} = -\frac{1}{f} \]

### Step 5: Solve for \( v \)
Taking the reciprocal gives:
\[ v = -f \]

### Step 6: Analyze the result
The negative sign indicates that the image is formed on the same side as the object, which means it is a virtual image. The magnitude of the image distance is equal to the focal length.

### Step 7: Determine the magnification
The magnification \( m \) is given by the formula:
\[ m = \frac{v}{u} \]
Substituting the values of \( v \) and \( u \):
\[ m = \frac{-f}{-\frac{f}{2}} = \frac{f}{\frac{f}{2}} = 2 \]

### Step 8: Conclusion about the image
Since the magnification is positive and equal to 2, this means:
- The image is virtual,
- The image is erect,
- The image size is double that of the object.

### Final Answer
The image will be virtual, erect, and double the size of the object.

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