A point source of light moves ijn a straight line paralel to a plane table. Consider a small portion of the table directly below the line of movement of the source. The illuminance at this portion varies with its distasnce r from the source as

Options

A. `Iprop1/r`
B. `Iprop1/r^2`
C. `Iprop1/r^3`
D. `Iprop1/r^4`

Answer

To solve the problem, we need to analyze how the illuminance (E) at a point on the table varies with the distance (r) from the point source of light.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have a point source of light moving in a straight line parallel to a plane table.
- We consider a small portion of the table directly below the line of movement of the source.

2. **Defining Variables**:
- Let \( I \) be the intensity of the light source.
- Let \( r \) be the distance from the source to the point on the table directly below it.
- Let \( L \) be the constant distance from the source to the table.

3. **Using the Illuminance Formula**:
- The illuminance \( E \) at a point is given by the formula:
\[ E = \frac{I \cdot \cos(\theta)}{r^2} \]
- Here, \( \theta \) is the angle between the line from the source to the point on the table and the vertical line from the source to the table.

4. **Relating Angles**:
- From the geometry of the situation, we can express \( \cos(\theta) \) in terms of \( L \) and \( r \):
\[ \cos(\theta) = \frac{L}{\sqrt{L^2 + r^2}} \]

5. **Substituting into the Illuminance Formula**:
- Substitute \( \cos(\theta) \) into the illuminance formula:
\[ E = \frac{I \cdot \frac{L}{\sqrt{L^2 + r^2}}}{r^2} \]
- This simplifies to:
\[ E = \frac{IL}{r^2 \sqrt{L^2 + r^2}} \]

6. **Analyzing the Relationship**:
- As \( r \) increases, the term \( r^2 \) in the denominator increases, which means \( E \) decreases.
- For large distances \( r \) compared to \( L \), we can approximate \( \sqrt{L^2 + r^2} \approx r \), leading to:
\[ E \approx \frac{IL}{r^3} \]

7. **Conclusion**:
- Thus, we find that the illuminance \( E \) is inversely proportional to \( r^3 \):
\[ E \propto \frac{1}{r^3} \]

### Final Answer:
The illuminance at the portion of the table varies with its distance \( r \) from the source as:
\[ E \propto \frac{1}{r^3} \]