Two coherent sources of intensity ratio ` beta^2` interfere. Then, the value of `(I_(max)- I_(min))//(I_(max)+I_(min))` is

Options

A. `(1+beta)/(sqrt beta)`
B. `(sqrt(1+beta)/beta)`
C. `(1+beta)/beta`
D. None of these

Explanation

To solve the problem, we need to find the value of the expression \((I_{max} - I_{min}) / (I_{max} + I_{min})\) given that the intensity ratio of two coherent sources is \(\beta^2\).

### Step-by-Step Solution:

1. **Define the Intensities**:
Let the intensity of the first source be \(I_1\) and the intensity of the second source be \(I_2\). Given that the intensity ratio is \(\beta^2\), we can express this as:
\[ \frac{I_1}{I_2} = \beta^2 \]
For simplicity, assume \(I_2 = 1\) unit. Then, we have:
\[ I_1 = \beta^2 \]

2. **Calculate \(I_{max}\) and \(I_{min}\)**:
The maximum intensity \(I_{max}\) and minimum intensity \(I_{min}\) for two coherent sources can be calculated using the following formulas:
\[ I_{max} = ( \sqrt{I_1} + \sqrt{I_2} )^2 \]
\[ I_{min} = ( \sqrt{I_1} - \sqrt{I_2} )^2 \]

Substituting \(I_1\) and \(I_2\):
\[ I_{max} = ( \sqrt{\beta^2} + \sqrt{1} )^2 = ( \beta + 1 )^2 \]
\[ I_{min} = ( \sqrt{\beta^2} - \sqrt{1} )^2 = ( \beta - 1 )^2 \]

3. **Expand \(I_{max}\) and \(I_{min}\)**:
Now, we can expand these expressions:
\[ I_{max} = (\beta + 1)^2 = \beta^2 + 2\beta + 1 \]
\[ I_{min} = (\beta - 1)^2 = \beta^2 - 2\beta + 1 \]

4. **Calculate \(I_{max} - I_{min}\)**:
Now we find \(I_{max} - I_{min}\):
\[ I_{max} - I_{min} = (\beta^2 + 2\beta + 1) - (\beta^2 - 2\beta + 1) \]
Simplifying this gives:
\[ I_{max} - I_{min} = 4\beta \]

5. **Calculate \(I_{max} + I_{min}\)**:
Next, we calculate \(I_{max} + I_{min}\):
\[ I_{max} + I_{min} = (\beta^2 + 2\beta + 1) + (\beta^2 - 2\beta + 1) \]
Simplifying this gives:
\[ I_{max} + I_{min} = 2\beta^2 + 2 \]

6. **Formulate the Final Expression**:
Now we can substitute \(I_{max} - I_{min}\) and \(I_{max} + I_{min}\) into the original expression:
\[ \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{4\beta}{2(\beta^2 + 1)} \]
Simplifying this gives:
\[ \frac{2\beta}{\beta^2 + 1} \]

### Final Answer:
Thus, the value of the expression \((I_{max} - I_{min}) / (I_{max} + I_{min})\) is:
\[ \frac{2\beta}{\beta^2 + 1} \]