If two interfering waves have intensities in the ratio 9: 1 ,then the ratio of maximum to minimum amplitude is

Options

A. ` 10: 8`
B. ` 4: 2`
C. ` 100 : 64`
D. ` 16 : 4`

Answer

To solve the problem of finding the ratio of maximum to minimum amplitude when two interfering waves have intensities in the ratio of 9:1, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between intensity and amplitude**:
The intensity \( I \) of a wave is directly proportional to the square of its amplitude \( A \). This can be expressed as:
\[ I \propto A^2 \]
Therefore, if we have two waves with intensities \( I_1 \) and \( I_2 \), we can write:
\[ I_1 = k A_1^2 \quad \text{and} \quad I_2 = k A_2^2 \]
where \( k \) is a constant.

2. **Set up the ratio of intensities**:
Given that the ratio of intensities is \( \frac{I_1}{I_2} = \frac{9}{1} \), we can express this as:
\[ \frac{A_1^2}{A_2^2} = \frac{9}{1} \]

3. **Find the ratio of amplitudes**:
Taking the square root of both sides gives us the ratio of amplitudes:
\[ \frac{A_1}{A_2} = \sqrt{\frac{9}{1}} = \frac{3}{1} \]
Thus, \( A_1 = 3A_2 \).

4. **Calculate maximum amplitude**:
In the case of constructive interference (when the phase difference is 0), the maximum amplitude \( A_{max} \) is given by:
\[ A_{max} = A_1 + A_2 = 3A_2 + A_2 = 4A_2 \]

5. **Calculate minimum amplitude**:
In the case of destructive interference (when the phase difference is 180 degrees), the minimum amplitude \( A_{min} \) is given by:
\[ A_{min} = |A_1 - A_2| = |3A_2 - A_2| = 2A_2 \]

6. **Find the ratio of maximum to minimum amplitude**:
Now, we can find the ratio of maximum amplitude to minimum amplitude:
\[ \frac{A_{max}}{A_{min}} = \frac{4A_2}{2A_2} = 2 \]

### Final Answer:
The ratio of maximum to minimum amplitude is \( 2:1 \).

---