In `YDSE` ratio of width of slit is `4:1`, then ratio of maximum to minimum intensity

Options

A. `25:9`
B. `9:1`
C. `4:1`
D. `(sqrt(3)-1)^(4):16`

Answer

To solve the problem of finding the ratio of maximum to minimum intensity in Young's double slit experiment (YDSE) when the ratio of the widths of the slits is 4:1, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between intensity and slit width**:
In YDSE, the intensity of light from each slit is proportional to the square of the amplitude of the light wave, which in turn is proportional to the width of the slit. Therefore, if we denote the widths of the two slits as \( W_1 \) and \( W_2 \), we can express the intensities \( I_1 \) and \( I_2 \) as:
\[ I_1 \propto W_1 \quad \text{and} \quad I_2 \propto W_2 \]

2. **Assign values based on the given ratio**:
Given the ratio of the widths of the slits is \( 4:1 \), we can assign:
\[ W_1 = 4k \quad \text{and} \quad W_2 = k \]
where \( k \) is a constant.

3. **Calculate the intensities**:
From the relationship between intensity and width, we have:
\[ I_1 = k_1 W_1 = k_1 (4k) = 4k_1 k \]
\[ I_2 = k_2 W_2 = k_2 (k) = k_2 k \]
For simplicity, we can assume \( k_1 = k_2 = 1 \) (the proportionality constants can be ignored for the ratio), thus:
\[ I_1 = 4k \quad \text{and} \quad I_2 = k \]

4. **Calculate the maximum and minimum intensities**:
The maximum intensity \( I_{\text{max}} \) and minimum intensity \( I_{\text{min}} \) can be calculated using the formulas:
\[ I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
\[ I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2 \]
Substituting the values:
\[ I_{\text{max}} = (\sqrt{4k} + \sqrt{k})^2 = (2\sqrt{k} + \sqrt{k})^2 = (3\sqrt{k})^2 = 9k \]
\[ I_{\text{min}} = (\sqrt{4k} - \sqrt{k})^2 = (2\sqrt{k} - \sqrt{k})^2 = (\sqrt{k})^2 = k \]

5. **Find the ratio of maximum to minimum intensity**:
Now, we can find the ratio:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{9k}{k} = 9 \]

### Final Answer:
The ratio of maximum to minimum intensity is \( 9:1 \).