If an interference pattern have maximum and minimum intensities in `36:1` ratio, then what will be the ratio of amplitudes?

Options

A. (a) `5:7`
B. (b) `7:4`
C. (c) `4:7`
D. (d) `7:5`

Explanation

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

### Step 1: Understand the relationship between intensity and amplitude
The intensity of light in an interference pattern is related to the square of the amplitude. Therefore, if we denote the amplitudes as \( A_1 \) and \( A_2 \), the intensities can be expressed as:
\[ I_1 \propto A_1^2 \quad \text{and} \quad I_2 \propto A_2^2 \]

### Step 2: Set up the equations for maximum and minimum intensities
The maximum intensity \( I_{\text{max}} \) and minimum intensity \( I_{\text{min}} \) in terms of amplitudes are given by:
\[ I_{\text{max}} = (A_1 + A_2)^2 \quad \text{and} \quad I_{\text{min}} = (A_1 - A_2)^2 \]

### Step 3: Use the given ratio of intensities
We are given that the ratio of maximum to minimum intensities is:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{36}{1} \]
This can be written as:
\[ \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = 36 \]

### Step 4: Take the square root of both sides
Taking the square root of both sides gives:
\[ \frac{A_1 + A_2}{A_1 - A_2} = 6 \]

### Step 5: Set up the equation
From the above equation, we can express it as:
\[ A_1 + A_2 = 6(A_1 - A_2) \]

### Step 6: Rearrange the equation
Expanding and rearranging gives:
\[ A_1 + A_2 = 6A_1 - 6A_2 \]
\[ A_2 + 6A_2 = 6A_1 - A_1 \]
\[ 7A_2 = 5A_1 \]

### Step 7: Find the ratio of amplitudes
From the equation \( 7A_2 = 5A_1 \), we can express the ratio of amplitudes as:
\[ \frac{A_1}{A_2} = \frac{7}{5} \]

### Conclusion
Thus, the ratio of the amplitudes \( A_1 : A_2 \) is:
\[ \boxed{\frac{7}{5}} \]