In a Young's double slit experiment, let `S_(1)` and `S_(2)` be the two slits, and C be the centre of the screen. If `angle S_(1)CS_(2)=theta` and `lambda` is the wavelength, the fringe width will be

Options

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

Explanation

To solve the problem regarding the fringe width in a Young's double slit experiment, we can follow these steps:

### Step 1: Understand the Setup
In a Young's double slit experiment, we have two slits \( S_1 \) and \( S_2 \) that are separated by a distance \( d \). A screen is placed at a distance \( D \) from the slits, and we denote the center of the screen as point \( C \). The angle \( \theta \) is formed between the lines connecting the slits to the center of the screen.

### Step 2: Recall the Formula for Fringe Width
The formula for fringe width \( \beta \) in a double slit experiment is given by:
\[ \beta = \frac{\lambda D}{d} \]
where \( \lambda \) is the wavelength of the light used, \( D \) is the distance from the slits to the screen, and \( d \) is the distance between the two slits.

### Step 3: Relate the Angle to Fringe Width
For small angles, we can relate the angle \( \theta \) to the fringe width. The angle \( \theta \) can be expressed in terms of the distances:
\[ \tan(\theta) \approx \theta \quad \text{(for small angles)} \]
Thus, we can write:
\[ \theta = \frac{d}{D} \]

### Step 4: Substitute \( \theta \) into the Fringe Width Formula
From the relationship established, we can substitute \( \theta \) into the fringe width formula. Since \( \theta = \frac{d}{D} \), we can rearrange the fringe width formula:
\[ \beta = \frac{\lambda D}{d} = \frac{\lambda}{\theta} \]

### Step 5: Final Expression for Fringe Width
Thus, we arrive at the final expression for the fringe width in terms of the angle \( \theta \):
\[ \beta = \frac{\lambda}{\theta} \]

### Conclusion
The fringe width \( \beta \) in the Young's double slit experiment, when considering the angle \( \theta \), is given by:
\[ \beta = \frac{\lambda}{\theta} \]