If the separation between slits in Young’s double slit experiment is reduced to `(1)/(3)` rd, the fringe width becomes n times. The value of n

Options

A. 3
B. `1/3`
C. 9
D. `1/9`

Answer

To solve the problem, we need to analyze how the fringe width in Young's double slit experiment changes when the separation between the slits is altered.

### Step-by-Step Solution:

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

2. **Initial Conditions**:
Let the initial separation between the slits be \( d \). Therefore, the initial fringe width is:
\[ \beta = \frac{\lambda D}{d} \]

3. **Change in Slit Separation**:
According to the problem, the separation between the slits is reduced to \( \frac{1}{3} \) of the initial value. Thus, the new separation \( d' \) can be expressed as:
\[ d' = \frac{d}{3} \]

4. **Calculate the New Fringe Width**:
The new fringe width \( \beta' \) when the slit separation is \( d' \) is given by:
\[ \beta' = \frac{\lambda D}{d'} \]
Substituting \( d' \):
\[ \beta' = \frac{\lambda D}{\frac{d}{3}} = \frac{\lambda D \cdot 3}{d} = 3 \cdot \frac{\lambda D}{d} = 3\beta \]

5. **Determine the Value of n**:
From the above calculation, we find that the new fringe width \( \beta' \) is three times the original fringe width \( \beta \). Therefore, we can conclude:
\[ n = 3 \]

### Final Answer:
The value of \( n \) is \( 3 \).

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