The two interfering waves have intensities in the ratio `9 : 4`. The ratio of intensities of maxima and minima in the interference pattern will be

Options

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

Answer

To find the ratio of intensities of maxima and minima in the interference pattern of two waves with intensities in the ratio of 9:4, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Intensities**:
Let the intensities of the two waves be \( I_1 \) and \( I_2 \). Given the ratio of intensities is \( I_1 : I_2 = 9 : 4 \).

2. **Relate Intensities to Amplitudes**:
The intensity \( I \) of a wave is proportional to the square of its amplitude \( A \). Thus, we can express the amplitudes in terms of the intensities:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]
From the given ratio \( \frac{9}{4} \), we can write:
\[ \frac{A_1^2}{A_2^2} = \frac{9}{4} \]

3. **Calculate the Amplitude Ratio**:
Taking the square root of both sides gives us the ratio of amplitudes:
\[ \frac{A_1}{A_2} = \frac{3}{2} \]

4. **Determine Maximum and Minimum Amplitudes**:
The maximum amplitude \( A_{max} \) when the waves interfere constructively is given by:
\[ A_{max} = A_1 + A_2 \]
The minimum amplitude \( A_{min} \) when the waves interfere destructively is given by:
\[ A_{min} = A_1 - A_2 \]

5. **Express \( A_1 \) and \( A_2 \)**:
Let \( A_2 = 2k \) (for some constant \( k \)). Then:
\[ A_1 = 3k \]

6. **Calculate \( A_{max} \) and \( A_{min} \)**:
Substitute \( A_1 \) and \( A_2 \):
\[ A_{max} = 3k + 2k = 5k \]
\[ A_{min} = 3k - 2k = k \]

7. **Calculate Intensities**:
The intensity is proportional to the square of the amplitude:
\[ I_{max} \propto (A_{max})^2 = (5k)^2 = 25k^2 \]
\[ I_{min} \propto (A_{min})^2 = (k)^2 = k^2 \]

8. **Find the Ratio of Intensities**:
The ratio of maximum intensity to minimum intensity is:
\[ \frac{I_{max}}{I_{min}} = \frac{25k^2}{k^2} = 25 \]

### Final Result:
The ratio of intensities of maxima and minima in the interference pattern is:
\[ \frac{I_{max}}{I_{min}} = 25 : 1 \]