Two coherent sources of equal intensities produce a maximum of 100 units. If the amplitude of one of the sources is reduced by 20%, then the maximum intensity produced will be :

Options

A. 100
B. 81
C. 89
D. 60

Answer

To solve the problem step by step, let's follow the reasoning outlined in the video transcript.

### Step 1: Understand the Initial Conditions
We have two coherent sources of equal intensity, denoted as \( I_0 \). The maximum intensity produced by these two sources is given as 100 units.

### Step 2: Determine the Initial Intensity
Since the maximum intensity \( I_{\text{max}} \) for two coherent sources of equal intensity is given by the formula:
\[ I_{\text{max}} = 4I_0 \]
We can set up the equation:
\[ 4I_0 = 100 \]
From this, we can solve for \( I_0 \):
\[ I_0 = \frac{100}{4} = 25 \text{ units} \]

### Step 3: Adjust the Amplitude of One Source
Now, we reduce the amplitude of one of the sources by 20%. If the original amplitude of the source is \( A \), the new amplitude becomes:
\[ A' = 0.8A \]

### Step 4: Relate Amplitude to Intensity
Intensity is proportional to the square of the amplitude. Therefore, if the intensity of the first source is \( I_0 \) (which corresponds to amplitude \( A \)), the new intensity \( I' \) of the modified source becomes:
\[ I' = k(0.8A)^2 = k(0.64A^2) = 0.64I_0 \]
where \( k \) is a constant of proportionality.

### Step 5: Calculate the New Intensity
Substituting \( I_0 = 25 \) units into the equation for \( I' \):
\[ I' = 0.64 \times 25 = 16 \text{ units} \]

### Step 6: Find the New Maximum Intensity
Now we have two intensities: one is \( I_0 = 25 \) units and the other is \( I' = 16 \) units. The new maximum intensity \( I_{\text{max}}' \) can be calculated using the formula:
\[ I_{\text{max}}' = \left( \sqrt{I_0} + \sqrt{I'} \right)^2 \]
Calculating the square roots:
\[ \sqrt{I_0} = \sqrt{25} = 5 \]
\[ \sqrt{I'} = \sqrt{16} = 4 \]
Now substituting back into the formula:
\[ I_{\text{max}}' = (5 + 4)^2 = 9^2 = 81 \text{ units} \]

### Final Answer
The maximum intensity produced after reducing the amplitude of one source by 20% is:
\[ \boxed{81 \text{ units}} \]