Where should an object be placed so that a real and inverted image of the same size as the object is obtained using a convex lens ?

Options

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

Explanation

To determine where an object should be placed in front of a convex lens to obtain a real and inverted image of the same size, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Lens Formula**: The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens. It is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]

2. **Identify Image Characteristics**: We need a real and inverted image that is the same size as the object. For a convex lens, this occurs when the object is placed at a distance equal to twice the focal length (2f).

3. **Set Object Distance**: Let the object distance (u) be -2f (the negative sign indicates that the object is on the same side as the incoming light).

4. **Calculate Image Distance**: Substitute the value of u into the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{(-2f)} \]
Rearranging gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{2f} \]
Finding a common denominator:
\[ \frac{1}{v} = \frac{2 + 1}{2f} = \frac{3}{2f} \]
Therefore:
\[ v = \frac{2f}{3} \]

5. **Determine Size of the Image**: The magnification (m) of the lens is given by:
\[ m = \frac{h'}{h} = -\frac{v}{u} \]
Since we want the image to be the same size as the object (h' = h), we set:
\[ -\frac{v}{u} = -1 \]
This implies:
\[ v = u \]
Since we have u = -2f, we find that v = 2f, confirming that the image is real, inverted, and of the same size.

6. **Conclusion**: The object should be placed at a distance equal to twice the focal length (2f) of the lens.

### Final Answer:
The object should be placed at a distance equal to twice the focal length of the lens (2f).