Monochromatic light from a narrow slit illuminates two parallel slits producing an interference pattern on a screen. The separation between the two slits is now doubled and the distance between the screen and the slits is reduced to half. The fringe width

Options

A. is doubled
B. becomes four times
C. becomes one - fourth
D. remains the same

Answer

To solve the problem of finding the new fringe width when the separation between the two slits is doubled and the distance between the screen and the slits is halved, we can follow these steps:

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

### Step 2: Define the initial conditions
Let:
- The initial separation between the slits be \( d \).
- The initial distance from the slits to the screen be \( D \).

### Step 3: Define the new conditions
According to the problem:
- The new separation between the slits is \( d' = 2d \) (doubled).
- The new distance from the slits to the screen is \( D' = \frac{D}{2} \) (halved).

### Step 4: Write the new fringe width formula
Using the new values in the fringe width formula, we have:
\[ \beta' = \frac{\lambda D'}{d'} \]

### Step 5: Substitute the new values into the formula
Substituting \( D' \) and \( d' \):
\[ \beta' = \frac{\lambda \left(\frac{D}{2}\right)}{2d} \]

### Step 6: Simplify the expression
Now simplify the expression:
\[ \beta' = \frac{\lambda D}{2 \cdot 2d} = \frac{\lambda D}{4d} \]

### Step 7: Relate the new fringe width to the old fringe width
From the original fringe width \( \beta = \frac{\lambda D}{d} \), we can express the new fringe width as:
\[ \beta' = \frac{1}{4} \cdot \frac{\lambda D}{d} = \frac{1}{4} \beta \]

### Conclusion
Thus, the new fringe width \( \beta' \) is one-fourth of the original fringe width \( \beta \):
\[ \beta' = \frac{1}{4} \beta \]