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. (a) `4:1`
B. (b) `1:1`
C. (c) `1:4`
D. (d) `2:1`

Explanation

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 from the slits to the screen.

### Step-by-Step Solution:

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

2. **Setting Up the Ratios**:
Given that the wavelengths are in the ratio \( \lambda_1 : \lambda_2 = 1 : 2 \) and the slit separations are in the ratio \( d_1 : d_2 = 2 : 1 \), we can express these relationships as:
\[ \lambda_1 = x, \quad \lambda_2 = 2x \]
\[ d_1 = 2d, \quad d_2 = d \]

3. **Equating Fringe Widths**:
Since the fringe widths are equal in both setups, we have:
\[ \beta_1 = \beta_2 \]
Therefore:
\[ \frac{D_1 \lambda_1}{d_1} = \frac{D_2 \lambda_2}{d_2} \]

4. **Substituting the Known Values**:
Substituting the values of \(\lambda\) and \(d\):
\[ \frac{D_1 \cdot x}{2d} = \frac{D_2 \cdot 2x}{d} \]

5. **Simplifying the Equation**:
Cancelling \(x\) and \(d\) from both sides:
\[ \frac{D_1}{2} = 2D_2 \]

6. **Finding the Ratio of Distances**:
Rearranging gives:
\[ D_1 = 4D_2 \]
Thus, the ratio of the distances is:
\[ \frac{D_1}{D_2} = 4 : 1 \]

### Final Answer:
The ratio of the distance between the plane of the slits and the screen in the two setups is \(4:1\).