The light beams of intensities in the ratio of `9:1` are allowed to interfere. What will be the ratio of the intensities of maxima and minima ?

Options

A. `3:1`
B. `4:1`
C. `25:9`
D. `81:1`

Explanation

To solve the problem, we need to find the ratio of the intensities of maxima (Imax) and minima (Imin) when two light beams with intensities in the ratio of 9:1 interfere.

### Step-by-Step Solution:

1. **Identify the Intensities**:
Let the intensities of the two beams be \( I_1 \) and \( I_2 \). According to the problem, we have:
\[ \frac{I_1}{I_2} = \frac{9}{1} \]
This implies:
\[ I_1 = 9I_2 \]

2. **Formula for Maximum Intensity (Imax)**:
The formula for the maximum intensity when two waves interfere is given by:
\[ I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \]

3. **Calculate Imax**:
Substitute \( I_1 = 9I_2 \) into the formula:
\[ I_{\text{max}} = (\sqrt{9I_2} + \sqrt{I_2})^2 \]
Simplifying this:
\[ I_{\text{max}} = (3\sqrt{I_2} + \sqrt{I_2})^2 = (4\sqrt{I_2})^2 = 16I_2 \]

4. **Formula for Minimum Intensity (Imin)**:
The formula for the minimum intensity when two waves interfere is given by:
\[ I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2 \]

5. **Calculate Imin**:
Substitute \( I_1 = 9I_2 \) into the formula:
\[ I_{\text{min}} = (\sqrt{9I_2} - \sqrt{I_2})^2 \]
Simplifying this:
\[ I_{\text{min}} = (3\sqrt{I_2} - \sqrt{I_2})^2 = (2\sqrt{I_2})^2 = 4I_2 \]

6. **Calculate the Ratio of Imax to Imin**:
Now, we can find the ratio of the maximum intensity to the minimum intensity:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{16I_2}{4I_2} = 4 \]

7. **Final Result**:
Thus, the ratio of the intensities of maxima and minima is:
\[ \text{Ratio of } I_{\text{max}} \text{ to } I_{\text{min}} = 4:1 \]