In an experiment to find focal length of a concave mirror, a graph is drawn between the magnitudes of (u) and (v). The graph looks like.

Options

A. a)
B. b)
C. c)
D. d)

Answer

To determine the correct graph representing the relationship between the magnitudes of \( u \) (object distance) and \( v \) (image distance) in an experiment to find the focal length of a concave mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mirror Formula**:
The focal length \( f \) of a concave mirror is given by the mirror formula:
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

2. **Rearranging the Formula**:
Rearranging the formula to express \( \frac{1}{u} \):
\[ \frac{1}{u} = \frac{1}{f} - \frac{1}{v} \]

3. **Differentiate the Equation**:
To find the relationship between \( u \) and \( v \), we differentiate the rearranged equation:
\[ -\frac{du}{u^2} = -\frac{dv}{v^2} \]

4. **Express the Slope**:
Rearranging gives us:
\[ \frac{dv}{du} = -\frac{v^2}{u^2} \]
This indicates that the slope \( \frac{dv}{du} \) is negative, meaning that as \( u \) increases, \( v \) decreases.

5. **Analyze the Graph Options**:
Since the slope \( \frac{dv}{du} \) is negative, we look for graphs that show a decreasing trend. Options B and C both show decreasing slopes.

6. **Determine the Nature of the Graph**:
The derived equation \( \frac{dv}{du} = -\frac{v^2}{u^2} \) suggests a non-linear relationship, indicating that the graph cannot be a straight line. Thus, we reject option B, which shows a straight line.

7. **Select the Correct Option**:
Option C, which shows an exponentially decreasing curve, is the correct representation of the relationship between the magnitudes of \( u \) and \( v \).

### Conclusion:
The correct graph representing the relationship between the magnitudes of \( u \) and \( v \) in the experiment is option C.

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