In two separate set-ups of the Young's double slit experiment, fringes of equal width are observed when lights of wavelength in the ratio of `1:2` are used. If the ratio of the slit separation in the two cases is `2:1`, the ratio of the distance between the plane of the slits and the screen in the two set-ups are

Options

A. `1:1`
B. `1:4`
C. `2:1`
D. `4:1`

Answer

To solve the problem, we need to analyze the Young's double slit experiment and the relationship between the fringe width, wavelength, slit separation, and distance to the screen.

### Step-by-Step Solution:

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

2. **Set Up the Ratios**:
We are given:
- The ratio of wavelengths: \( \frac{\lambda_1}{\lambda_2} = \frac{1}{2} \) (thus, \( \frac{\lambda_2}{\lambda_1} = 2 \))
- The ratio of slit separations: \( \frac{d_1}{d_2} = \frac{2}{1} \)

3. **Equate the Fringe Widths**:
Since the fringes are of equal width in both setups, we can set the fringe widths equal to each other:
\[ \beta_1 = \beta_2 \]
This gives us:
\[ \frac{D_1 \cdot \lambda_1}{d_1} = \frac{D_2 \cdot \lambda_2}{d_2} \]

4. **Substituting the Ratios**:
Rearranging the equation gives:
\[ \frac{D_1}{D_2} = \frac{\lambda_2}{\lambda_1} \cdot \frac{d_1}{d_2} \]
Now substituting the known ratios:
\[ \frac{D_1}{D_2} = \frac{2}{1} \cdot \frac{2}{1} \]

5. **Calculate the Final Ratio**:
Therefore:
\[ \frac{D_1}{D_2} = 2 \cdot 2 = 4 \]
Thus, the ratio of the distances between the plane of the slits and the screen in the two setups is:
\[ D_1 : D_2 = 4 : 1 \]

### Final Answer:
The ratio of the distances \( D_1 : D_2 \) is \( 4 : 1 \).