The two coherent sources of equal intensity produce maximum intensity of 100 units at a point. If the intensity of one of the sources is reduced by 50% by reducing its width then the intensity of light at the same point will be

Options

A. 90
B. 89
C. 67
D. `72.85`

Answer

To solve the problem, we need to calculate the resultant intensity when one of the coherent sources has its intensity reduced by 50%. Let's break down the solution step by step.

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- We have two coherent sources of equal intensity, producing a maximum intensity of 100 units at a point.
- Let the intensity of each source be \( I_0 \).

2. **Finding the Initial Intensity**:
- The maximum intensity \( I_{\text{max}} \) for two coherent sources is given by:
\[ I_{\text{max}} = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \theta \]
- For maximum intensity, \( \cos \theta = 1 \):
\[ I_{\text{max}} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]
- Since both sources are equal, \( I_1 = I_2 = I_0 \):
\[ I_{\text{max}} = 2I_0 + 2\sqrt{I_0 I_0} = 2I_0 + 2I_0 = 4I_0 \]
- Given \( I_{\text{max}} = 100 \):
\[ 4I_0 = 100 \implies I_0 = 25 \text{ units} \]

3. **Adjusting the Intensity of One Source**:
- One of the sources has its intensity reduced by 50%, so the new intensities are:
- \( I_1 = I_0 = 25 \) units
- \( I_2 = \frac{I_0}{2} = \frac{25}{2} = 12.5 \) units

4. **Calculating the New Resultant Intensity**:
- Using the formula for maximum intensity again:
\[ I_{\text{new}} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]
- Substituting the new values:
\[ I_{\text{new}} = 25 + 12.5 + 2\sqrt{25 \times 12.5} \]
- Calculate \( \sqrt{25 \times 12.5} \):
\[ \sqrt{25 \times 12.5} = \sqrt{312.5} \approx 17.68 \]
- Now substituting back:
\[ I_{\text{new}} = 25 + 12.5 + 2 \times 17.68 \approx 25 + 12.5 + 35.36 \]
- Adding these values:
\[ I_{\text{new}} \approx 25 + 12.5 + 35.36 = 72.86 \text{ units} \]

5. **Final Result**:
- The intensity of light at the same point after reducing the intensity of one source by 50% is approximately **72.85 units**.