Let `S_(1)` and `S_(2)` be the two slits in Young's double-slit experiment. If central maxima is observed at P and angle `/_ S_(1) P S_(2) = theta`, then fringe width for the light of wavelength `lambda` will be

Options

A. `lambda // theta`
B. `lambda theta`
C. `2 lambda // theta`
D. `lambda // 2 theta`

Explanation

To find the fringe width in Young's double-slit experiment, we can follow these steps:

### Step 1: Understand the Setup
In Young's double-slit experiment, we have two slits \( S_1 \) and \( S_2 \). The central maxima is observed at point P, and the angle \( \angle S_1 P S_2 = \theta \).

### Step 2: Relate the Angle to the Fringe Width
For small angles, we can use the approximation:
\[ \cos \left( \frac{\theta}{2} \right) \approx 1 - \frac{\theta^2}{8} \quad \text{(for small } \theta\text{)} \]
However, for simplicity in this context, we can directly relate the angle to the distance between the slits and the distance to the screen.

### Step 3: Use the Formula for Fringe Width
The formula for fringe width \( \beta \) in Young's double-slit experiment is given by:
\[ \beta = \frac{\lambda D}{d} \]
where:
- \( \lambda \) is the wavelength of the light,
- \( D \) is the distance from the slits to the screen,
- \( d \) is the distance between the two slits.

### Step 4: Substitute the Angle
From the geometry of the setup, we can relate the angle \( \theta \) to the distances:
\[ \theta = \frac{d}{D} \]
This means we can express \( d \) in terms of \( \theta \) and \( D \):
\[ d = D \cdot \theta \]

### Step 5: Substitute \( d \) in the Fringe Width Formula
Substituting \( d \) into the fringe width formula gives:
\[ \beta = \frac{\lambda D}{D \cdot \theta} = \frac{\lambda}{\theta} \]

### Conclusion
Thus, the fringe width \( \beta \) is given by:
\[ \beta = \frac{\lambda}{\theta} \]

### Final Answer
The fringe width for the light of wavelength \( \lambda \) is:
\[ \beta = \frac{\lambda}{\theta} \]