Two coherent light sources having intensity in the ratio 2x produce an interference pattern. The ratio `(I_(max)-I_(min))/(I_(max)+I_(min))` will be :

Options

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

Answer

To solve the problem, we need to find the ratio \((I_{max} - I_{min}) / (I_{max} + I_{min})\) given that the intensities of two coherent light sources are in the ratio \(2x\).

### Step-by-Step Solution:

1. **Define Intensities**:
Let the intensity of the second source \(I_2 = I\). Therefore, the intensity of the first source \(I_1 = 2xI\) based on the given ratio.

2. **Calculate Maximum Intensity \(I_{max}\)**:
The formula for maximum intensity when two coherent sources interfere is:
\[ I_{max} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]
Substituting the values:
\[ I_{max} = 2xI + I + 2\sqrt{(2xI)(I)} = (2x + 1)I + 2\sqrt{2x}I \]
Simplifying this:
\[ I_{max} = I(2x + 1 + 2\sqrt{2x}) \]

3. **Calculate Minimum Intensity \(I_{min}\)**:
The formula for minimum intensity is:
\[ I_{min} = I_1 + I_2 - 2\sqrt{I_1 I_2} \]
Substituting the values:
\[ I_{min} = 2xI + I - 2\sqrt{(2xI)(I)} = (2x + 1)I - 2\sqrt{2x}I \]
Simplifying this:
\[ I_{min} = I(2x + 1 - 2\sqrt{2x}) \]

4. **Calculate \(I_{max} - I_{min}\)**:
\[ I_{max} - I_{min} = [I(2x + 1 + 2\sqrt{2x})] - [I(2x + 1 - 2\sqrt{2x})] \]
This simplifies to:
\[ I_{max} - I_{min} = I(4\sqrt{2x}) \]

5. **Calculate \(I_{max} + I_{min}\)**:
\[ I_{max} + I_{min} = [I(2x + 1 + 2\sqrt{2x})] + [I(2x + 1 - 2\sqrt{2x})] \]
This simplifies to:
\[ I_{max} + I_{min} = I(4x + 2) \]

6. **Calculate the Ratio**:
Now, we can find the ratio:
\[ \frac{I_{max} - I_{min}}{I_{max} + I_{min}} = \frac{I(4\sqrt{2x})}{I(4x + 2)} = \frac{4\sqrt{2x}}{4x + 2} \]
Simplifying further:
\[ = \frac{2\sqrt{2x}}{2x + 1} \]

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