When light from two sources (say slits `S_(1)` and `S_(2)`) interfere, they form alternate dark and bright fringes. Bright fringe is formed at all point where the path difference is an odd multiple of half wavelength. At the condition of equal amplitudes, `A_(1) = A_(2) = a`, the maximum intensity will be `4 a^(2)` and the visibility improves, The resultant intensity can also be indicated with phase factor as `I = 2 a^(2) cos^(2) (phi // 2)`.Using this passage, answer the following questions.
If the path difference between the slits `S_(1)` and `S_(2)` is `(lambda)/(2)` the central fringe will have an intensity of

Options

A. 0
B. `a^(2)`
C. `2 a^(2)`
D. `4 a^(2)`

Explanation

To find the intensity of the central fringe when the path difference between the slits \( S_1 \) and \( S_2 \) is \( \frac{\lambda}{2} \), we can follow these steps:

### Step 1: Understand the Concept of Path Difference
The path difference between the two sources (slits \( S_1 \) and \( S_2 \)) determines the phase difference between the waves coming from the two sources. A path difference of \( \frac{\lambda}{2} \) corresponds to a phase difference of \( 180^\circ \) (or \( \pi \) radians).

### Step 2: Apply the Intensity Formula
The intensity \( I \) of the resultant wave can be expressed using the formula:
\[ I = 2a^2 \cos^2\left(\frac{\phi}{2}\right) \]
where \( \phi \) is the phase difference.

### Step 3: Calculate the Phase Difference
Since the path difference is \( \frac{\lambda}{2} \), the phase difference \( \phi \) is:
\[ \phi = \frac{2\pi}{\lambda} \cdot \text{(path difference)} = \frac{2\pi}{\lambda} \cdot \frac{\lambda}{2} = \pi \]

### Step 4: Substitute the Phase Difference into the Intensity Formula
Now, substituting \( \phi = \pi \) into the intensity formula:
\[ I = 2a^2 \cos^2\left(\frac{\pi}{2}\right) \]

### Step 5: Evaluate the Cosine Function
We know that:
\[ \cos\left(\frac{\pi}{2}\right) = 0 \]
Thus,
\[ I = 2a^2 \cdot 0^2 = 0 \]

### Conclusion
The intensity of the central fringe when the path difference is \( \frac{\lambda}{2} \) is:
\[ \text{Intensity} = 0 \]