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

Options

A. `8 : 1`
B. `9 : 7`
C. `1 : 8`
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. **Understanding the Relationship Between Intensity and Amplitude:**
The intensity \( I \) of a wave is proportional to the square of its amplitude \( A \). This can be expressed as:
\[ I \propto A^2 \]
Therefore, we can write:
\[ I_1 = k A_1^2 \quad \text{and} \quad I_2 = k A_2^2 \]
where \( k \) is a constant of proportionality.

2. **Using the Given Intensity Ratio:**
The problem states that the maximum intensity \( I_{\text{max}} \) and minimum intensity \( I_{\text{min}} \) ratio is \( 64:1 \). Thus, we can write:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{64}{1} \]

3. **Expressing Maximum and Minimum Intensities:**
The maximum intensity \( I_{\text{max}} \) and minimum intensity \( I_{\text{min}} \) can be expressed in terms of \( I_1 \) and \( I_2 \):
\[ I_{\text{max}} = I_1 + I_2 \quad \text{and} \quad I_{\text{min}} = I_1 - I_2 \]

4. **Setting Up the Equation:**
Substituting the expressions for maximum and minimum intensities into the ratio gives:
\[ \frac{I_1 + I_2}{I_1 - I_2} = 64 \]

5. **Cross-Multiplying:**
Cross-multiplying the equation:
\[ I_1 + I_2 = 64(I_1 - I_2) \]
Expanding this gives:
\[ I_1 + I_2 = 64I_1 - 64I_2 \]

6. **Rearranging the Equation:**
Rearranging the terms leads to:
\[ I_1 + I_2 + 64I_2 = 64I_1 \]
\[ I_1 - 64I_1 + 65I_2 = 0 \]
\[ -63I_1 + 65I_2 = 0 \]
This simplifies to:
\[ 63I_1 = 65I_2 \quad \Rightarrow \quad \frac{I_1}{I_2} = \frac{65}{63} \]

7. **Finding the Amplitude Ratio:**
Since \( I \propto A^2 \), we can relate the intensities to the amplitudes:
\[ \frac{I_1}{I_2} = \frac{A_1^2}{A_2^2} \]
Thus:
\[ \frac{A_1^2}{A_2^2} = \frac{65}{63} \]
Taking the square root gives:
\[ \frac{A_1}{A_2} = \sqrt{\frac{65}{63}} = \frac{\sqrt{65}}{\sqrt{63}} \]

8. **Final Ratio of Amplitudes:**
To express this in a simpler form, we can approximate:
\[ \frac{A_1}{A_2} \approx \frac{8.06}{7.94} \approx \frac{9}{7} \]

### Conclusion:
The ratio of amplitudes \( A_1 : A_2 \) is approximately \( 9 : 7 \).