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. (a) `(sqrtn)/((n+1)^2)`
B. (b) `(2sqrtn)/((n+1)^2)`
C. (c) `(sqrtn)/(n+1)`
D. (d) `(2sqrtn)/(n+1)`

Explanation

To solve the problem of finding the ratio \((I_{max} - I_{min}) / (I_{max} + I_{min})\) for two coherent light sources with an intensity ratio \(n\), we can follow these steps:

### Step 1: Define the Intensities
Let the intensities of the two coherent sources be \(I_1\) and \(I_2\). According to the problem, the intensity ratio is given as:
\[ \frac{I_1}{I_2} = n \]
This implies:
\[ I_1 = n I_2 \]

### Step 2: Calculate \(I_{max}\) and \(I_{min}\)
The maximum intensity \(I_{max}\) and minimum intensity \(I_{min}\) in an interference pattern can be expressed as:
\[ I_{max} = \left( \sqrt{I_1} + \sqrt{I_2} \right)^2 \]
\[ I_{min} = \left( \sqrt{I_1} - \sqrt{I_2} \right)^2 \]

### Step 3: Substitute the Values
Substituting \(I_1\) and \(I_2\) into these equations:
\[ I_{max} = \left( \sqrt{n I_2} + \sqrt{I_2} \right)^2 = \left( \sqrt{I_2}(\sqrt{n} + 1) \right)^2 = I_2 \left( \sqrt{n} + 1 \right)^2 \]
\[ I_{min} = \left( \sqrt{n I_2} - \sqrt{I_2} \right)^2 = \left( \sqrt{I_2}(\sqrt{n} - 1) \right)^2 = I_2 \left( \sqrt{n} - 1 \right)^2 \]

### Step 4: Calculate \(I_{max} - I_{min}\) and \(I_{max} + I_{min}\)
Now, calculate \(I_{max} - I_{min}\):
\[ I_{max} - I_{min} = I_2 \left( \left( \sqrt{n} + 1 \right)^2 - \left( \sqrt{n} - 1 \right)^2 \right) \]
Using the identity \(a^2 - b^2 = (a-b)(a+b)\):
\[ = I_2 \left( \left( \sqrt{n} + 1 - (\sqrt{n} - 1) \right) \left( \sqrt{n} + 1 + (\sqrt{n} - 1) \right) \right) \]
\[ = I_2 \left( 2 \cdot 1 \cdot (2\sqrt{n}) \right) = 4 I_2 \sqrt{n} \]

Now calculate \(I_{max} + I_{min}\):
\[ I_{max} + I_{min} = I_2 \left( \left( \sqrt{n} + 1 \right)^2 + \left( \sqrt{n} - 1 \right)^2 \right) \]
\[ = I_2 \left( 2n + 2 \right) = 2 I_2 (n + 1) \]

### Step 5: Find the Ratio
Now we can find the ratio:
\[ \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{4 I_2 \sqrt{n}}{2 I_2 (n + 1)} = \frac{2 \sqrt{n}}{n + 1} \]

### Final Answer
Thus, the required ratio is:
\[ \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{2 \sqrt{n}}{n + 1} \]