Two interfering sources have an intensity ratio `16 : 1`. Deduce ratio and ratio of intensity between the maxima and minima in interference pattern.

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the intensity ratio
We are given that the intensity ratio of two interfering sources is \( I_1 : I_2 = 16 : 1 \).

### Step 2: Relate intensity to amplitude
The intensity \( I \) of a wave is proportional to the square of its amplitude \( A \). Therefore, we can express the intensities in terms of amplitudes:
\[ I_1 = A_1^2 \quad \text{and} \quad I_2 = A_2^2 \]
Given the ratio \( I_1 : I_2 = 16 : 1 \), we can write:
\[ \frac{A_1^2}{A_2^2} = \frac{16}{1} \]
Taking the square root of both sides gives us the ratio of amplitudes:
\[ \frac{A_1}{A_2} = \frac{4}{1} \]

### Step 3: Calculate the intensity at maxima and minima
The intensity at the maxima \( I_{max} \) and minima \( I_{min} \) in an interference pattern can be calculated using the formulas:
\[ I_{max} = (A_1 + A_2)^2 \quad \text{and} \quad I_{min} = (A_1 - A_2)^2 \]

### Step 4: Substitute the amplitude ratio
Using the amplitude ratio \( \frac{A_1}{A_2} = 4 \), we can express \( A_1 \) as \( 4A_2 \). Substituting this into the formulas for \( I_{max} \) and \( I_{min} \):
\[ I_{max} = (4A_2 + A_2)^2 = (5A_2)^2 = 25A_2^2 \]
\[ I_{min} = (4A_2 - A_2)^2 = (3A_2)^2 = 9A_2^2 \]

### Step 5: Find the ratio of intensities
Now we can find the ratio of the intensity at maxima to the intensity at minima:
\[ \frac{I_{max}}{I_{min}} = \frac{25A_2^2}{9A_2^2} = \frac{25}{9} \]

### Conclusion
Thus, the ratio of intensity between the maxima and minima in the interference pattern is:
\[ \text{Ratio of intensities} = 25 : 9 \]