Where should an object be placed in front of a convex lens to get a real to get real image of the size of the object ?

Options

A. At the principal focus of the lens
B. At twice the focal length
C. At infinity
D. Between the optical centre of the lens and its principal focus

Answer

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

### Step-by-Step Solution:

1. **Understanding Magnification**:
- The magnification (m) of a lens is defined as the ratio of the height of the image (h') to the height of the object (h).
- Since we want the image to be of the same size as the object, we have:
\[ \frac{h'}{h} = 1 \quad \Rightarrow \quad m = 1 \]

2. **Relating Magnification to Image and Object Distances**:
- For a lens, magnification can also be expressed in terms of the image distance (v) and object distance (u):
\[ m = \frac{v}{u} \]
- Since we have established that \( m = 1 \), we can write:
\[ \frac{v}{u} = 1 \quad \Rightarrow \quad v = u \]

3. **Considering the Nature of the Image**:
- A real image formed by a convex lens is inverted. Therefore, the magnification will actually be:
\[ m = -1 \]
- This means:
\[ \frac{h'}{h} = -1 \quad \Rightarrow \quad h' = -h \]
- Thus, we can conclude:
\[ v = -u \]

4. **Using the Lens Formula**:
- The lens formula relates the object distance (u), image distance (v), and the focal length (f) of the lens:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
- Substituting \( v = -u \) into the lens formula gives:
\[ \frac{1}{f} = \frac{1}{-u} - \frac{1}{u} \]
- This simplifies to:
\[ \frac{1}{f} = -\frac{1}{u} - \frac{1}{u} = -\frac{2}{u} \]

5. **Solving for Object Distance (u)**:
- Rearranging the equation gives:
\[ -\frac{2}{u} = \frac{1}{f} \quad \Rightarrow \quad u = -2f \]

6. **Conclusion**:
- The negative sign indicates that the object is placed on the same side as the incoming light. Therefore, the object should be placed at a distance of \( 2f \) (which is the center of curvature) in front of the convex lens to obtain a real image of the same size as the object.

### Final Answer:
The object should be placed at the center of curvature of the lens, which is at a distance of \( 2f \) from the lens.

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