Calculate the fringe width, when in YDSE the separation between the slits is halved and the distance between slit and the screen is doubled:

Options

A. Unchanged
B. Halved
C. Doubled
D. Four times

Explanation

To calculate the new fringe width in the Young's Double Slit Experiment (YDSE) when the separation between the slits is halved and the distance from the slits to the screen is doubled, we can follow these steps:

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

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

### Step 3: Substitute the new values into the fringe width formula
The new fringe width \( \beta' \) can be expressed as:
\[ \beta' = \frac{\lambda D'}{d'} \]

Substituting the new values:
\[ \beta' = \frac{\lambda (2D)}{\frac{d}{2}} \]

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

### Step 5: Relate the new fringe width to the old fringe width
From the original formula, we know:
\[ \beta = \frac{\lambda D}{d} \]
Thus, we can relate the new fringe width to the old fringe width:
\[ \beta' = 4 \beta \]

### Conclusion
The new fringe width is four times the old fringe width:
\[ \beta' = 4 \beta \]

### Final Answer
The new fringe width is 4 times the old fringe width.

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