The intensity variation in the interference pattern obtained with the help of two coherent sources is 5 % of the average intensity.find out the ratio of intensities of two sources.

Answer

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

### Step 1: Define average intensity
Let's assume the average intensity \( I_{\text{avg}} \) is 100 units.

### Step 2: Calculate maximum and minimum intensity
The intensity variation is given as 5% of the average intensity. Therefore, we can calculate:

- Maximum intensity \( I_{\text{max}} \):
\[ I_{\text{max}} = I_{\text{avg}} + 5\% \text{ of } I_{\text{avg}} = 100 + 5 = 105 \]

- Minimum intensity \( I_{\text{min}} \):
\[ I_{\text{min}} = I_{\text{avg}} - 5\% \text{ of } I_{\text{avg}} = 100 - 5 = 95 \]

### Step 3: Set up the ratio of intensities
The ratio of maximum to minimum intensity is given by:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{105}{95} \]

### Step 4: Simplify the ratio
We can simplify this ratio:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{21}{19} \]

### Step 5: Relate intensity to amplitude
We know that intensity is proportional to the square of the amplitude. Therefore, we can express the intensities in terms of amplitudes \( a_1 \) and \( a_2 \):
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{(a_1 + a_2)^2}{(a_1 - a_2)^2} \]

### Step 6: Set up the equation
From the ratio we derived:
\[ \frac{(a_1 + a_2)^2}{(a_1 - a_2)^2} = \frac{21}{19} \]

Taking the square root of both sides:
\[ \frac{a_1 + a_2}{a_1 - a_2} = \sqrt{\frac{21}{19}} \approx 1.05 \]

### Step 7: Cross-multiply to find a relationship
Cross-multiplying gives us:
\[ a_1 + a_2 = 1.05(a_1 - a_2) \]

### Step 8: Rearranging the equation
Rearranging the equation:
\[ a_1 + a_2 = 1.05a_1 - 1.05a_2 \]
\[ a_1 - 1.05a_1 + a_2 + 1.05a_2 = 0 \]
\[ -0.05a_1 + 2.05a_2 = 0 \]

### Step 9: Solve for the ratio of amplitudes
From the equation above, we can express the ratio of \( a_1 \) to \( a_2 \):
\[ 0.05a_1 = 2.05a_2 \implies \frac{a_1}{a_2} = \frac{2.05}{0.05} = 41 \]

### Step 10: Find the ratio of intensities
Since intensity is proportional to the square of the amplitude, we have:
\[ \frac{I_1}{I_2} = \left(\frac{a_1}{a_2}\right)^2 = 41^2 = 1681 \]

### Final Answer
Thus, the ratio of the intensities of the two sources is:
\[ \frac{I_1}{I_2} = 1681 : 1 \]