The distance v of the real image formed by a convex lens is measured for various object distance u. A graph is poltted between v and u, which one of the following graphs is correct

Options

A.
B.
C.
D.

Answer

To solve the problem, we need to analyze the relationship between the object distance (u) and the image distance (v) for a convex lens using the lens formula:

1. **Write down the lens formula**:
\[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \]
where \( f \) is the focal length of the lens.

2. **Rearranging the lens formula**:
We can rearrange the equation to express \( v \) in terms of \( u \):
\[ \frac{1}{v} = \frac{1}{u} + \frac{1}{f} \]
This indicates that \( v \) is related to \( u \) in a non-linear way.

3. **Understanding the nature of the graph**:
Since the relationship is non-linear, we can conclude that the graph of \( v \) versus \( u \) will not be a straight line. Therefore, options that suggest a linear graph can be eliminated.

4. **Analyzing specific cases**:
- When the object is at the focal point (i.e., \( u = f \)), substituting this into the lens formula gives:
\[ \frac{1}{v} = \frac{1}{f} + \frac{1}{f} = \frac{2}{f} \implies v = \frac{f}{2} \]
- As \( u \) approaches infinity (the object is very far away), \( v \) approaches \( f \) (the image is formed at the focal point).

5. **Identifying the correct graph**:
- The graph should show that as \( u \) increases, \( v \) also increases but at a decreasing rate, indicating that the graph is asymptotic. The correct graph will show that \( v \) approaches a maximum value as \( u \) increases.

6. **Conclusion**:
Based on the analysis, the correct graph that represents the relationship between \( v \) and \( u \) for a convex lens is option D.