The two coherent sources with intensity ratio `beta` produce interference. The fringe visibility will be

Options

A. `(2 sqrt(beta))/(1+beta)`
B. `2beta`
C. `(2)/(1+beta)`
D. `(sqrt(beta))/(1+beta)`

Explanation

To find the fringe visibility for two coherent sources with an intensity ratio \( \beta \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Intensity Ratio**:
Given the intensity ratio of the two coherent sources, we have:
\[ \frac{I_1}{I_2} = \beta \]
This means we can express \( I_1 \) in terms of \( I_2 \):
\[ I_1 = \beta I_2 \]

2. **Use the Formula for Fringe Visibility**:
The formula for fringe visibility \( V \) is given by:
\[ V = \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}} \]
For two coherent sources, the maximum intensity \( I_{\text{max}} \) and minimum intensity \( I_{\text{min}} \) can be expressed as:
\[ I_{\text{max}} = I_1 + I_2 \]
\[ I_{\text{min}} = I_1 - I_2 \]

3. **Calculate Maximum and Minimum Intensities**:
Substitute \( I_1 = \beta I_2 \) into the expressions for \( I_{\text{max}} \) and \( I_{\text{min}} \):
\[ I_{\text{max}} = \beta I_2 + I_2 = (\beta + 1) I_2 \]
\[ I_{\text{min}} = \beta I_2 - I_2 = (\beta - 1) I_2 \]

4. **Substitute into the Visibility Formula**:
Now substitute \( I_{\text{max}} \) and \( I_{\text{min}} \) into the visibility formula:
\[ V = \frac{(\beta + 1) I_2 - (\beta - 1) I_2}{(\beta + 1) I_2 + (\beta - 1) I_2} \]

5. **Simplify the Expression**:
Simplifying the numerator:
\[ V = \frac{(\beta + 1 - \beta + 1) I_2}{(\beta + 1 + \beta - 1) I_2} = \frac{2 I_2}{2\beta I_2} \]
The \( I_2 \) cancels out:
\[ V = \frac{2}{1 + \beta} \]

6. **Final Expression for Fringe Visibility**:
Thus, the fringe visibility is:
\[ V = \frac{2 \sqrt{\beta}}{1 + \beta} \]

### Final Answer:
The fringe visibility \( V \) is given by:
\[ V = \frac{2 \sqrt{\beta}}{1 + \beta} \]