In a Young's double slit experiment, the fringe width will remain same, if ( D = distance between screen and plane of slits, d = separation between two slits and `lambda` = wavelength of light used)

Options

A. Both `lambda` and D are doubled
B. Both d and D are doubled
C. D is doubled but d is halved
D. `lamba` is doubled but d is halved

Explanation

To solve the problem regarding the fringe width in a Young's double slit experiment, we need to analyze how changes in the parameters affect the fringe width. The formula for fringe width (β) is given by:

\[ \beta = \frac{\lambda D}{d} \]

where:
- \( \beta \) = fringe width
- \( \lambda \) = wavelength of light used
- \( D \) = distance between the screen and the plane of the slits
- \( d \) = separation between the two slits

### Step-by-Step Solution:

1. **Understand the Formula**:
The fringe width is directly proportional to the wavelength (\( \lambda \)) and the distance to the screen (\( D \)), and inversely proportional to the slit separation (\( d \)).

2. **Analyze the Options**:
We will check each option to see if the fringe width remains the same (i.e., \( \beta' = \beta \)).

3. **Option 1: Both \( \lambda \) and \( D \) are doubled**:
- New fringe width \( \beta' = \frac{2\lambda \cdot 2D}{d} = \frac{4\lambda D}{d} \)
- This is not equal to \( \beta \). Thus, this option is incorrect.

4. **Option 2: Both \( d \) and \( D \) are doubled**:
- New fringe width \( \beta' = \frac{\lambda \cdot 2D}{2d} = \frac{\lambda D}{d} \)
- This is equal to \( \beta \). Thus, this option is correct.

5. **Option 3: \( d \) is doubled and \( D \) is halved**:
- New fringe width \( \beta' = \frac{\lambda \cdot \frac{D}{2}}{2d} = \frac{\lambda D}{4d} \)
- This is not equal to \( \beta \). Thus, this option is incorrect.

6. **Option 4: \( \lambda \) is doubled and \( d \) is halved**:
- New fringe width \( \beta' = \frac{2\lambda \cdot D}{\frac{d}{2}} = \frac{4\lambda D}{d} \)
- This is not equal to \( \beta \). Thus, this option is incorrect.

### Conclusion:
The only option that keeps the fringe width the same is **Option 2**, where both \( d \) and \( D \) are doubled.