In an experiment for finding the focal length of a thin convex lens using two - pin method the position of the image `.upsilon.` is recorded for various positrons .u. of the object. Which of the following best represents object distance u versus image distance `upsilon- `graph.

Options

A.
B.
C.
D.

Answer

To find the correct representation of the object distance \( u \) versus image distance \( v \) graph for a thin convex lens using the two-pin method, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Lens Formula**:
The relationship between the object distance \( u \), image distance \( v \), and focal length \( f \) of a convex lens is given by the lens formula:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
This formula indicates how \( u \) and \( v \) are related to the focal length.

2. **Analyzing Object and Image Distances**:
- When the object is placed at a very large distance (approaching infinity), the image distance \( v \) approaches the focal point \( f \) of the lens.
- As the object moves closer to the lens (decreasing \( u \)), the image distance \( v \) increases and can theoretically approach infinity when the object is at the focal point of the lens.

3. **Identifying Key Points**:
- At \( u = \infty \), \( v \) approaches \( f \).
- At \( u = 2f \), \( v = 2f \) (the object is at the center of curvature).
- As \( u \) decreases from \( 2f \) to \( f \), \( v \) increases from \( 2f \) to infinity.

4. **Graph Representation**:
- The graph of \( v \) versus \( u \) is not a straight line but rather a curve.
- The curve starts from a point where \( u \) is very large (and \( v \) is close to \( f \)) and rises steeply as \( u \) decreases, indicating that \( v \) increases rapidly.

5. **Choosing the Correct Graph**:
- Among the given options, the graph that best represents this relationship is a curve that starts from a low value of \( v \) when \( u \) is large and rises steeply as \( u \) decreases.

### Final Conclusion:
The correct representation of the object distance \( u \) versus image distance \( v \) graph for a thin convex lens is a curve that starts from a low value of \( v \) when \( u \) is large and increases steeply as \( u \) decreases.