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 64:1, we can follow these steps:

### Step 1: Understand the relationship between intensity and amplitude
The intensity (I) of a wave is directly proportional to the square of its amplitude (A). Mathematically, this can be expressed as:
\[ I \propto A^2 \]
This means that if we have two waves with intensities \( I_1 \) and \( I_2 \), we can write:
\[ I_1 = k A_1^2 \]
\[ I_2 = k A_2^2 \]
where \( k \) is a proportionality constant.

### Step 2: Set up the intensity ratio
Given that the maximum intensity (I_max) and minimum intensity (I_min) ratio is 64:1, we can express this as:
\[ \frac{I_{max}}{I_{min}} = \frac{64}{1} \]

### Step 3: Express maximum and minimum intensity in terms of amplitudes
The maximum intensity for two interfering waves can be expressed as:
\[ I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
And the minimum intensity can be expressed as:
\[ I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2 \]

### Step 4: Substitute the intensity expressions
Substituting the expressions for \( I_1 \) and \( I_2 \):
\[ I_{max} = (A_1 + A_2)^2 \]
\[ I_{min} = (A_1 - A_2)^2 \]

### Step 5: Set up the equation using the intensity ratio
Now, substituting these into the ratio:
\[ \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = 64 \]

### Step 6: Take the square root of both sides
Taking the square root of both sides gives:
\[ \frac{A_1 + A_2}{A_1 - A_2} = 8 \]

### Step 7: Cross-multiply to solve for the ratio
Cross-multiplying gives:
\[ A_1 + A_2 = 8(A_1 - A_2) \]
Expanding this:
\[ A_1 + A_2 = 8A_1 - 8A_2 \]

### Step 8: Rearranging the equation
Rearranging the terms gives:
\[ A_1 + A_2 + 8A_2 = 8A_1 \]
\[ A_1 - 8A_1 + 9A_2 = 0 \]
\[ -7A_1 + 9A_2 = 0 \]
Thus:
\[ 7A_1 = 9A_2 \]

### Step 9: Find the ratio of amplitudes
Dividing both sides by \( A_2 \) gives:
\[ \frac{A_1}{A_2} = \frac{9}{7} \]

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