Two identical sources each of intensity `I_(0)` have a separation `d = lambda // 8`, where `lambda` is the wavelength of the waves emitted by either source. The phase difference of the sources is `pi // 4` The intensity distribution `I(theta)` in the radiation field as a function of `theta` Which specifies the direction from the sources to the distant observation point P is given by

Options

A. `I (theta) = I_(0) cos^(2) theta`
B. `I (theta) = (I_(0))/(4) cos^(2) ((pi theta)/(8))`
C. `I (theta) = 4 I_(0) cos^(2) [(pi)/(8) (sin theta + 1)]`
D. `I (theta) = I_(0) sin^(2) theta`

Explanation

To solve the problem, we need to derive the intensity distribution \( I(\theta) \) in the radiation field as a function of the angle \( \theta \) for two coherent sources with given conditions. Here’s a step-by-step solution:

### Step 1: Understand the parameters
We have two identical sources, each with intensity \( I_0 \). The separation between the sources is given by \( d = \frac{\lambda}{8} \), where \( \lambda \) is the wavelength of the emitted waves. The phase difference between the two sources is \( \Delta \phi = \frac{\pi}{4} \).

### Step 2: Calculate the path difference
The path difference \( \Delta x \) between the two waves reaching a point \( P \) at an angle \( \theta \) can be expressed as:
\[ \Delta x = d \sin \theta \]
Substituting \( d \):
\[ \Delta x = \frac{\lambda}{8} \sin \theta \]

### Step 3: Relate path difference to phase difference
The phase difference \( \Delta \phi \) due to the path difference can be calculated using the formula:
\[ \Delta \phi = \frac{2\pi}{\lambda} \Delta x \]
Substituting \( \Delta x \):
\[ \Delta \phi = \frac{2\pi}{\lambda} \left( \frac{\lambda}{8} \sin \theta \right) = \frac{\pi}{4} \sin \theta \]

### Step 4: Total phase difference
The total phase difference \( \phi \) at point \( P \) is the sum of the initial phase difference and the phase difference due to path difference:
\[ \phi = \Delta \phi + \text{initial phase difference} = \frac{\pi}{4} + \frac{\pi}{4} \sin \theta \]

### Step 5: Calculate the intensity
The intensity \( I(\theta) \) at point \( P \) for two coherent sources can be expressed as:
\[ I(\theta) = 4 I_0 \cos^2\left(\frac{\phi}{2}\right) \]
Substituting for \( \phi \):
\[ I(\theta) = 4 I_0 \cos^2\left(\frac{1}{2} \left( \frac{\pi}{4} + \frac{\pi}{4} \sin \theta \right)\right) \]
This simplifies to:
\[ I(\theta) = 4 I_0 \cos^2\left(\frac{\pi}{8} (1 + \sin \theta)\right) \]

### Final Expression
Thus, the intensity distribution \( I(\theta) \) is given by:
\[ I(\theta) = 4 I_0 \cos^2\left(\frac{\pi}{8} (1 + \sin \theta)\right) \]