An interference pattern has maximum and minimum intensities in the ratio of 36:1, what is the ratio of theire amplitudes?

Options

A. `3:7`
B. `7:4`
C. `4:7`
D. `7:5`

Answer

To find the ratio of the amplitudes given the ratio of maximum and minimum intensities in an interference pattern, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between intensity and amplitude:**
The intensity of light is proportional to the square of the amplitude. Therefore, if \( I_{\text{max}} \) and \( I_{\text{min}} \) are the maximum and minimum intensities, we can express them in terms of the amplitudes \( A_1 \) and \( A_2 \):
\[ I_{\text{max}} = (A_1 + A_2)^2 \]
\[ I_{\text{min}} = (A_1 - A_2)^2 \]

2. **Set up the ratio of intensities:**
From the problem, we know that:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{36}{1} \]
Substituting the expressions for \( I_{\text{max}} \) and \( I_{\text{min}} \):
\[ \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = 36 \]

3. **Take the square root of both sides:**
Taking the square root gives us:
\[ \frac{A_1 + A_2}{A_1 - A_2} = 6 \]

4. **Cross-multiply to eliminate the fraction:**
This leads to:
\[ A_1 + A_2 = 6(A_1 - A_2) \]
Expanding the right side:
\[ A_1 + A_2 = 6A_1 - 6A_2 \]

5. **Rearranging the equation:**
Bringing all terms involving \( A_1 \) and \( A_2 \) to one side gives:
\[ A_1 + A_2 + 6A_2 = 6A_1 \]
Simplifying this:
\[ 7A_2 = 5A_1 \]

6. **Finding the ratio of amplitudes:**
Rearranging gives us:
\[ \frac{A_1}{A_2} = \frac{7}{5} \]
Thus, the ratio of the amplitudes \( A_1 : A_2 \) is:
\[ A_1 : A_2 = 7 : 5 \]

### Final Answer:
The ratio of the amplitudes \( A_1 : A_2 \) is \( 7 : 5 \).