The place faces of two identical plano-convex lenses, each with focal length f are pressed against each other using an optical glue to from a usual convex lens .The distance from the optical centre at which an object must be placed to obtain the image same as the size of the object is

Options

A. `f/4`
B. `f/2`
C. f
D. 2f

Answer

To solve the problem, we need to determine the distance from the optical center at which an object must be placed to obtain an image that is the same size as the object when using two identical plano-convex lenses glued together.

### Step-by-Step Solution:

1. **Understanding the System**:
- We have two identical plano-convex lenses, each with a focal length \( f \).
- When these lenses are pressed together, they form a new lens system.

2. **Finding the Focal Length of the Combined Lens**:
- The formula for the focal length \( F \) of two lenses in contact is given by:
\[ \frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} \]
- Since both lenses have the same focal length \( f \):
\[ \frac{1}{F} = \frac{1}{f} + \frac{1}{f} = \frac{2}{f} \]
- Therefore, the focal length \( F \) of the combined lens is:
\[ F = \frac{f}{2} \]

3. **Using the Lens Formula**:
- The lens formula relates the object distance \( u \), image distance \( v \), and focal length \( F \):
\[ \frac{1}{F} = \frac{1}{v} - \frac{1}{u} \]
- Rearranging gives:
\[ \frac{1}{u} = \frac{1}{v} - \frac{1}{F} \]

4. **Condition for Same Size Image**:
- For the image to be the same size as the object, the magnification \( m \) must be 1:
\[ m = \frac{h'}{h} = \frac{v}{u} = 1 \implies v = u \]
- Therefore, we can substitute \( v \) with \( u \) in the lens formula:
\[ \frac{1}{F} = \frac{1}{u} - \frac{1}{u} \implies \frac{1}{F} = 0 \]
- This means we need to find the distance \( u \) where the image distance \( v \) equals \( u \).

5. **Substituting the Focal Length**:
- From the earlier calculation, we have \( F = \frac{f}{2} \):
\[ \frac{1}{u} = \frac{1}{u} - \frac{2}{f} \]
- This leads to:
\[ \frac{1}{u} = \frac{2}{f} \]
- Thus, solving for \( u \):
\[ u = \frac{f}{2} \]

### Final Answer:
The distance from the optical center at which the object must be placed to obtain an image the same size as the object is:
\[ u = f \]