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 against each other, they effectively form a new lens system.

2. **Finding the Equivalent Focal Length**:
For two thin lenses in contact, the formula for the equivalent focal length \( f_{eq} \) is given by:
\[ \frac{1}{f_{eq}} = \frac{1}{f_1} + \frac{1}{f_2} \]
Since both lenses have the same focal length \( f \):
\[ \frac{1}{f_{eq}} = \frac{1}{f} + \frac{1}{f} = \frac{2}{f} \]
Therefore, the equivalent focal length \( f_{eq} \) is:
\[ f_{eq} = \frac{f}{2} \]

3. **Condition for Image Size**:
To obtain an image that is the same size as the object, the magnification \( m \) must be equal to 1. The magnification for a lens is given by:
\[ m = \frac{v}{u} = 1 \]
where \( v \) is the image distance and \( u \) is the object distance. This implies:
\[ v = -u \]

4. **Using the Lens Formula**:
The lens formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f_{eq} \):
\[ \frac{1}{f_{eq}} = \frac{1}{v} - \frac{1}{u} \]
Substituting \( f_{eq} = \frac{f}{2} \) and \( v = -u \):
\[ \frac{2}{f} = \frac{1}{-u} - \frac{1}{u} \]
Simplifying the right side:
\[ \frac{2}{f} = -\frac{2}{u} \]

5. **Solving for Object Distance \( u \)**:
Rearranging the equation:
\[ \frac{2}{f} = -\frac{2}{u} \implies u = -f \]
Since we are interested in the magnitude of the object distance, we take:
\[ u = f \]

### 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 \( f \).