The interference pattern is obtained with two coherent light sources of intensity ratio n. In the interference patten, the ratio `(I_(max)-I_(min))/(I_(max)+I_(min))` will be

Options

A. `sqrtn/(n+1)`
B. `(2sqrtn)/(n+1)`
C. `sqrtn/(n+1^2)`
D. `(2sqrtn)/(n+1^2)`

Answer

To solve the problem, we need to find the ratio \((I_{max} - I_{min}) / (I_{max} + I_{min})\) for two coherent light sources with an intensity ratio \(n\).

### Step-by-Step Solution:

1. **Define Intensities**: Let the intensities of the two coherent light sources be \(I_1\) and \(I_2\). Given that the intensity ratio is \(n\), we can express this as:
\[ \frac{I_1}{I_2} = n \quad \text{or} \quad I_1 = n I_2 \]

2. **Calculate Maximum Intensity**: The maximum intensity \(I_{max}\) in an interference pattern is given by:
\[ I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
Substituting \(I_1\) and \(I_2\):
\[ I_{max} = (\sqrt{n I_2} + \sqrt{I_2})^2 = (\sqrt{I_2}(\sqrt{n} + 1))^2 = I_2(n + 1 + 2\sqrt{n}) \]

3. **Calculate Minimum Intensity**: The minimum intensity \(I_{min}\) is given by:
\[ I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2 \]
Substituting \(I_1\) and \(I_2\):
\[ I_{min} = (\sqrt{n I_2} - \sqrt{I_2})^2 = (\sqrt{I_2}(\sqrt{n} - 1))^2 = I_2(n - 1 - 2\sqrt{n}) \]

4. **Calculate the Difference and Sum of Intensities**:
- The difference \(I_{max} - I_{min}\):
\[ I_{max} - I_{min} = I_2(n + 1 + 2\sqrt{n}) - I_2(n - 1 - 2\sqrt{n}) = I_2(2 + 4\sqrt{n}) \]
- The sum \(I_{max} + I_{min}\):
\[ I_{max} + I_{min} = I_2(n + 1 + 2\sqrt{n}) + I_2(n - 1 - 2\sqrt{n}) = I_2(2n) \]

5. **Form the Ratio**:
Now we can form the ratio:
\[ \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{I_2(2 + 4\sqrt{n})}{I_2(2n)} = \frac{2 + 4\sqrt{n}}{2n} \]
Simplifying this gives:
\[ = \frac{2(1 + 2\sqrt{n})}{2n} = \frac{1 + 2\sqrt{n}}{n} \]

### Final Answer:
The ratio \(\frac{I_{max} - I_{min}}{I_{max} + I_{min}}\) is:
\[ \frac{1 + 2\sqrt{n}}{n} \]