The intensity produced by a long cylindrical light source at a small distance r from the source is proportional to

Options

A. `1/r^2`
B. `1/r^3`
C. `1/r`
D. none of these

Answer

To solve the problem regarding the intensity produced by a long cylindrical light source at a small distance \( r \) from the source, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We are dealing with a long cylindrical light source and need to determine how the intensity of light varies with distance \( r \) from the source.

2. **Identify the Geometry**: The light source is cylindrical and emits light uniformly along its length. The wavefronts produced by this source will be cylindrical as well.

3. **Define Intensity**: Intensity \( I \) is defined as the power \( P \) per unit area \( A \):
\[ I = \frac{P}{A} \]

4. **Calculate the Area**: For a cylindrical wavefront at a distance \( r \) from the axis of the cylinder, the area \( A \) through which the power is distributed is the lateral surface area of the cylinder. The formula for the lateral surface area of a cylinder is:
\[ A = 2 \pi r L \]
where \( L \) is the length of the cylinder.

5. **Substitute into the Intensity Formula**: Now, substituting the area into the intensity formula, we get:
\[ I = \frac{P}{2 \pi r L} \]

6. **Analyze the Relationship**: From the equation \( I = \frac{P}{2 \pi r L} \), we can see that intensity \( I \) is inversely proportional to \( r \):
\[ I \propto \frac{1}{r} \]

7. **Conclusion**: Therefore, the intensity produced by a long cylindrical light source at a small distance \( r \) from the source is inversely proportional to \( r \).

### Answer:
The correct option is that the intensity is inversely proportional to \( r \) (Option 3).

---