Light from a point source falls on a screen. If the separation between the source and the screen is increased by 1%, the illuminance will decrease (nearly) by

Options

A. 0.005
B. 0.01
C. 0.02
D. 0.04

Answer

To solve the problem, we will use the relationship between illuminance (E) and distance (r) from a point light source. The illuminance is given by the formula:

\[ E = \frac{I}{r^2} \]

where:
- \( E \) is the illuminance,
- \( I \) is the intensity of the light source,
- \( r \) is the distance from the light source to the screen.

### Step 1: Understand the relationship between illuminance and distance
From the formula, we see that illuminance is inversely proportional to the square of the distance. If the distance increases, the illuminance decreases.

### Step 2: Increase the distance by 1%
Let the initial distance be \( r \). If the distance is increased by 1%, the new distance \( r' \) will be:

\[ r' = r + 0.01r = 1.01r \]

### Step 3: Calculate the new illuminance
Using the new distance \( r' \), the new illuminance \( E' \) can be calculated as:

\[ E' = \frac{I}{(r')^2} = \frac{I}{(1.01r)^2} = \frac{I}{1.0201r^2} \]

### Step 4: Find the change in illuminance
Now, we can find the ratio of the new illuminance \( E' \) to the original illuminance \( E \):

\[ \frac{E'}{E} = \frac{\frac{I}{1.0201r^2}}{\frac{I}{r^2}} = \frac{1}{1.0201} \]

### Step 5: Calculate the percentage decrease in illuminance
To find the percentage decrease in illuminance, we can calculate:

\[ \text{Percentage decrease} = \left(1 - \frac{E'}{E}\right) \times 100\% \]

Substituting the ratio we found:

\[ \text{Percentage decrease} = \left(1 - \frac{1}{1.0201}\right) \times 100\% \]

Calculating this gives:

\[ \text{Percentage decrease} \approx \left(1 - 0.9804\right) \times 100\% \approx 1.96\% \]

### Step 6: Conclusion
Thus, the illuminance decreases by approximately 2%. The closest answer to this is 0.02, which corresponds to option C.

### Summary of Steps:
1. Understand the relationship between illuminance and distance.
2. Increase the distance by 1%.
3. Calculate the new illuminance using the new distance.
4. Find the change in illuminance.
5. Calculate the percentage decrease in illuminance.
6. Conclude with the closest answer.