Consider two coherent, monochromatic (wavelength `lambda`) sources `S_(1) and S_(2)`, separated by a distance d. The ratio of intensities of `S_(1)` and that of `S_(2)` (which is responsible for interference at point P, where detector is located) is 4. The distance of point P from `S_(1)` is (if the resultant intensity at point P is equal to `(9)/(4)` times of intensity of `S_(1)`) `("Given : "angleS_(2)S_(1)P=90^(@), d gt0 and n" is a positive integer")`

Options

A. `(d^(2)-n^(2)lambda^(2))/(2n lambda)`
B. `(d^(2)+n^(2)lambda^(2))/(2n lambda)`
C. `(nlambdad)/(sqrt(d^(2)-n^(2)lambda^(2)))`
D. `(2nlambdad)/(sqrt(d^(2)-n^(2)lambda^(2)))`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have two coherent sources \( S_1 \) and \( S_2 \) separated by a distance \( d \). The intensity ratio of \( S_1 \) to \( S_2 \) is given as:
\[ \frac{I_1}{I_2} = 4 \implies I_1 = 4I_2 \]
The resultant intensity at point \( P \) is given as:
\[ I_{resultant} = \frac{9}{4} I_1 \]
We also know that the angle \( \angle S_2 S_1 P = 90^\circ \).

### Step 2: Express the intensities in terms of \( I_2 \)
Let \( I_2 = I \). Then:
\[ I_1 = 4I \quad \text{and} \quad I_{resultant} = \frac{9}{4} \cdot 4I = 9I \]

### Step 3: Use the formula for resultant intensity
The formula for resultant intensity \( I_{resultant} \) from two coherent sources is:
\[ I_{resultant} = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \theta \]
Substituting the values we have:
\[ 9I = 4I + I + 2\sqrt{4I \cdot I} \cos \theta \]
This simplifies to:
\[ 9I = 5I + 4I \cos \theta \]
Thus:
\[ 4I \cos \theta = 4I \implies \cos \theta = 1 \]
This indicates that \( \theta = 0 \) or \( \theta = 2n\pi \) for any integer \( n \).

### Step 4: Relate phase difference to path difference
The phase difference \( \Delta \phi \) is related to the path difference \( \Delta x \) by:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
Since \( \theta = 0 \), the path difference \( \Delta x \) can be expressed as:
\[ \Delta x = |d - x_0| \]
where \( x_0 \) is the distance from \( S_1 \) to point \( P \).

### Step 5: Set up the equation
From the geometry of the situation:
\[ \Delta x = \sqrt{x_0^2 + d^2} - x_0 \]
Setting the path difference equal to \( n\lambda \) gives:
\[ \sqrt{x_0^2 + d^2} - x_0 = n\lambda \]

### Step 6: Square both sides
Squaring both sides:
\[ (\sqrt{x_0^2 + d^2} - x_0)^2 = (n\lambda)^2 \]
Expanding gives:
\[ x_0^2 + d^2 - 2x_0\sqrt{x_0^2 + d^2} + x_0^2 = n^2\lambda^2 \]
This simplifies to:
\[ 2x_0^2 + d^2 - n^2\lambda^2 = 2x_0\sqrt{x_0^2 + d^2} \]

### Step 7: Rearranging and solving for \( x_0 \)
Rearranging gives:
\[ 2x_0\sqrt{x_0^2 + d^2} = 2x_0^2 + d^2 - n^2\lambda^2 \]
Dividing both sides by \( 2x_0 \) (assuming \( x_0 \neq 0 \)):
\[ \sqrt{x_0^2 + d^2} = x_0 + \frac{d^2 - n^2\lambda^2}{2x_0} \]
Squaring again and simplifying will lead to the final expression for \( x_0 \).

### Final Result
After solving, we find:
\[ x_0 = \frac{d^2 - n^2\lambda^2}{2n\lambda} \]