For an interference pattern, the maximum and minimum intensity ratio is 64 : 1 , then what will be the ratio of amplitudes ?

Options

A. (a)`8:1`
B. (b)`9:7`
C. (c)`1:8`
D. (d)`7:9`

Answer

To solve the problem of finding the ratio of amplitudes given the intensity ratio of an interference pattern, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Relationship Between Intensity and Amplitude**:
The intensity \( I \) of a wave is related to its amplitude \( A \) by the formula:
\[ I \propto A^2 \]
Therefore, if we have two waves with amplitudes \( A_1 \) and \( A_2 \), their intensities can be expressed as:
\[ I_1 \propto A_1^2 \quad \text{and} \quad I_2 \propto A_2^2 \]

2. **Express Maximum and Minimum Intensities**:
The maximum intensity \( I_{max} \) and minimum intensity \( I_{min} \) for two interfering waves can be expressed as:
\[ I_{max} = (A_1 + A_2)^2 \quad \text{and} \quad I_{min} = (A_1 - A_2)^2 \]

3. **Set Up the Ratio of Intensities**:
Given that the ratio of maximum to minimum intensity is \( \frac{I_{max}}{I_{min}} = \frac{64}{1} \), we can write:
\[ \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = 64 \]

4. **Take the Square Root**:
Taking the square root of both sides gives:
\[ \frac{A_1 + A_2}{A_1 - A_2} = 8 \]

5. **Cross Multiply**:
Cross multiplying leads to:
\[ A_1 + A_2 = 8(A_1 - A_2) \]

6. **Expand and Rearrange**:
Expanding the equation gives:
\[ A_1 + A_2 = 8A_1 - 8A_2 \]
Rearranging terms results in:
\[ A_1 + A_2 + 8A_2 = 8A_1 \]
\[ 9A_2 = 7A_1 \]

7. **Find the Ratio of Amplitudes**:
Dividing both sides by \( A_1A_2 \) gives:
\[ \frac{A_1}{A_2} = \frac{9}{7} \]

### Final Answer:
The ratio of the amplitudes \( A_1 : A_2 \) is \( 9 : 7 \).