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.
B.
C.
D.

Answer

To solve the problem of determining the correct graph between the magnitudes of object distance (u) and image distance (v) for a concave mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Mirror Formula**:
The mirror formula for a concave mirror is given by:
\[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]
where \( f \) is the focal length, \( u \) is the object distance, and \( v \) is the image distance.

2. **Rearranging the Formula**:
We can rearrange the mirror formula to express \( u \) in terms of \( v \):
\[ \frac{1}{u} = \frac{1}{f} - \frac{1}{v} \]
This can be rewritten as:
\[ u = \frac{fv}{v - f} \]

3. **Differentiating the Equation**:
To analyze the relationship between \( u \) and \( v \), we differentiate both sides:
\[ \frac{du}{dv} = \frac{d}{dv}\left(\frac{fv}{v - f}\right) \]
Using the quotient rule, we find:
\[ \frac{du}{dv} = \frac{f(v - f) - fv}{(v - f)^2} = \frac{-f^2}{(v - f)^2} \]

4. **Analyzing the Slope**:
The derivative \( \frac{du}{dv} \) is negative, indicating that as \( v \) increases, \( u \) decreases. This means the graph of \( u \) versus \( v \) will have a negative slope.

5. **Identifying the Graph**:
Since the relationship is non-linear (as indicated by the squared term in the denominator), the graph will not be a straight line. Therefore, we look for a graph that shows a decreasing trend that is not linear.

6. **Conclusion**:
Based on the analysis, the correct graph that represents the relationship between the magnitudes of \( u \) and \( v \) for a concave mirror is the one that shows a decreasing curve. From the options provided, we conclude that option C is the correct answer.