In the case of interference, the maximum and minimum intensities are in the 16 : 9. Then

Options

A. the maximum and minimum amplitudes will be in the ratio 9 : 5
B. The intensities of the individual waves will be in the ratio 4 : 3
C. The amplitudes of the individual waves will be in the ratio 7 : 1
D. The amplitudes of the individual waves will be in the ratio 4 : 1

Answer

To solve the problem of finding the ratio of the amplitudes of two waves given the ratio of their maximum and minimum intensities in an interference pattern, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
We are given that the ratio of maximum intensity (I_max) to minimum intensity (I_min) is 16:9.
\[ \frac{I_{max}}{I_{min}} = \frac{16}{9} \]

2. **Use the Relationship Between Intensity and Amplitude**:
The intensity of a wave is proportional to the square of its amplitude. Therefore, we can write:
\[ I \propto A^2 \]
This means:
\[ \frac{I_{max}}{I_{min}} = \frac{A_{max}^2}{A_{min}^2} \]

3. **Set Up the Equation**:
Substituting the given ratio into the equation:
\[ \frac{A_{max}^2}{A_{min}^2} = \frac{16}{9} \]

4. **Take the Square Root**:
To find the ratio of the amplitudes, we take the square root of both sides:
\[ \frac{A_{max}}{A_{min}} = \sqrt{\frac{16}{9}} = \frac{4}{3} \]

5. **Relate Individual Intensities**:
We also know that:
\[ \frac{I_{max}}{I_{min}} = \left(\frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}}\right)^2 \]
Setting this equal to the given ratio:
\[ \frac{16}{9} = \left(\frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}}\right)^2 \]

6. **Solve for the Intensities**:
Taking the square root:
\[ \frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}} = \frac{4}{3} \]
Cross-multiplying gives:
\[ 3(\sqrt{I_1} + \sqrt{I_2}) = 4(\sqrt{I_1} - \sqrt{I_2}) \]
Expanding and rearranging:
\[ 3\sqrt{I_1} + 3\sqrt{I_2} = 4\sqrt{I_1} - 4\sqrt{I_2} \]
\[ 7\sqrt{I_2} = \sqrt{I_1} \]
Thus:
\[ \frac{\sqrt{I_1}}{\sqrt{I_2}} = 7 \]

7. **Find the Ratio of Intensities**:
Since \(I \propto A^2\), we have:
\[ \frac{I_1}{I_2} = \left(\frac{\sqrt{I_1}}{\sqrt{I_2}}\right)^2 = 7^2 = 49 \]

8. **Find the Ratio of Amplitudes**:
Using the relationship \(I \propto A^2\):
\[ \frac{A_1^2}{A_2^2} = 49 \implies \frac{A_1}{A_2} = 7 \]

### Final Result:
The ratio of the amplitudes of the two waves is:
\[ \frac{A_1}{A_2} = 7:1 \]