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

Options

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

Explanation

To solve the problem of finding 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
In interference, the maximum intensity \( I_{max} \) and minimum intensity \( I_{min} \) can be expressed in terms of the amplitudes \( A_1 \) and \( A_2 \) of the two waves:
- \( I_{max} = (A_1 + A_2)^2 \)
- \( I_{min} = (A_1 - A_2)^2 \)

### Step 2: Set up the ratio of intensities
Given that the ratio of maximum intensity to minimum intensity is \( \frac{I_{max}}{I_{min}} = \frac{36}{1} \), we can write:
\[ \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = 36 \]

### Step 3: 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 4: Cross-multiply to eliminate the fraction
Cross-multiplying leads to:
\[ A_1 + A_2 = 6(A_1 - A_2) \]

### Step 5: Expand and rearrange the equation
Expanding the right side gives:
\[ A_1 + A_2 = 6A_1 - 6A_2 \]
Rearranging the equation to group terms involving \( A_1 \) and \( A_2 \):
\[ A_1 + 6A_2 = 6A_1 - A_2 \]
\[ A_1 - 6A_1 + 6A_2 + A_2 = 0 \]
\[ -5A_1 + 7A_2 = 0 \]

### Step 6: Solve for the ratio of amplitudes
Rearranging gives:
\[ 5A_1 = 7A_2 \]
Thus, we can express the ratio of amplitudes as:
\[ \frac{A_1}{A_2} = \frac{7}{5} \]

### Conclusion
The ratio of the amplitudes \( A_1 \) to \( A_2 \) is \( \frac{7}{5} \).