If the amplitude ratio of two sources producing interference is 3 : 5, the ratio of intensities at maxima and minima is

Options

A. `25:16`
B. `5:3`
C. `16:1`
D. `25:9`

Explanation

To solve the problem of finding the ratio of intensities at maxima and minima given the amplitude ratio of two sources producing interference as 3:5, we can follow these steps:

### Step 1: Understand the relationship between amplitude and intensity
The intensity of a wave is proportional to the square of its amplitude. If we denote the amplitudes of the two sources as \( A_1 \) and \( A_2 \), then the intensities \( I_1 \) and \( I_2 \) can be expressed as:
\[ I_1 \propto A_1^2 \quad \text{and} \quad I_2 \propto A_2^2 \]

### Step 2: Assign values to the amplitudes
Given the amplitude ratio \( A_1 : A_2 = 3 : 5 \), we can assign:
\[ A_1 = 3k \quad \text{and} \quad A_2 = 5k \]
where \( k \) is a constant.

### Step 3: Calculate the intensities
Using the relationship between intensity and amplitude:
\[ I_1 = (3k)^2 = 9k^2 \quad \text{and} \quad I_2 = (5k)^2 = 25k^2 \]

### Step 4: Use the formulas for maximum and minimum intensity
The intensity at maxima \( I_{\text{max}} \) and minima \( I_{\text{min}} \) in interference can be calculated using the following formulas:
\[ I_{\text{max}} = (A_1 + A_2)^2 = (A_1 + A_2)^2 \]
\[ I_{\text{min}} = (A_1 - A_2)^2 = (A_1 - A_2)^2 \]

### Step 5: Substitute the amplitudes into the formulas
Calculating \( I_{\text{max}} \):
\[ I_{\text{max}} = (3k + 5k)^2 = (8k)^2 = 64k^2 \]

Calculating \( I_{\text{min}} \):
\[ I_{\text{min}} = (3k - 5k)^2 = (-2k)^2 = 4k^2 \]

### Step 6: Find the ratio of intensities at maxima and minima
Now, we can find the ratio of \( I_{\text{max}} \) to \( I_{\text{min}} \):
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{64k^2}{4k^2} = \frac{64}{4} = 16 \]

### Step 7: Write the final answer
Thus, the ratio of intensities at maxima and minima is:
\[ \text{Ratio of intensities} = 16 : 1 \]