An interference pattern is observed by Young's double slit experiment.If now the separation between coherent source is halved and the distance of screen from coheren sources is doubled, then now fringe width

Options

A. becomes double
B. becomes ne-fourth
C. remains same
D. becomes four times

Explanation

To solve the problem regarding the change in fringe width in Young's double slit experiment when the separation between coherent sources is halved and the distance of the screen from the coherent sources is doubled, we can follow these steps:

### Step 1: Understand the formula for fringe width
The fringe width (β) in Young's double slit experiment is given by the formula:
\[ \beta = \frac{\lambda D}{d} \]
where:
- \( \lambda \) = wavelength of light used
- \( D \) = distance from the slits to the screen
- \( d \) = separation between the slits

### Step 2: Identify the changes in the parameters
According to the problem:
- The separation between the coherent sources (slits) is halved, so the new separation \( d' = \frac{d}{2} \).
- The distance of the screen from the coherent sources is doubled, so the new distance \( D' = 2D \).

### Step 3: Substitute the new values into the fringe width formula
Now, we can calculate the new fringe width \( \beta' \) using the modified values:
\[ \beta' = \frac{\lambda D'}{d'} \]
Substituting the new values:
\[ \beta' = \frac{\lambda (2D)}{\frac{d}{2}} \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ \beta' = \frac{\lambda \cdot 2D}{\frac{d}{2}} = \frac{\lambda \cdot 2D \cdot 2}{d} = \frac{4\lambda D}{d} \]

### Step 5: Relate the new fringe width to the original fringe width
Recall that the original fringe width \( \beta \) is given by:
\[ \beta = \frac{\lambda D}{d} \]
Thus, we can express the new fringe width in terms of the original fringe width:
\[ \beta' = 4 \cdot \frac{\lambda D}{d} = 4\beta \]

### Conclusion
Therefore, the new fringe width \( \beta' \) is four times the original fringe width \( \beta \):
\[ \beta' = 4\beta \]

### Final Answer
The new fringe width will be 4 times the original fringe width.

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