Find the maximum intensity in case of interference of infinite identical coherent sources having intensities `I_(0),(I_(0))/(4),(I_(0))/(16),(I_(0))/(64),(I_(0))/(256)……..` (upto infinite).

Options

A. `I_(0)`
B. `2I_(0)`
C. `4I_(0)`
D. `8I_(0)`

Answer

To solve the problem of finding the maximum intensity in the case of interference of infinite identical coherent sources with given intensities, we can follow these steps:

### Step 1: Understand the intensity of the sources
The intensities of the sources are given as:
- \( I_0 \)
- \( \frac{I_0}{4} \)
- \( \frac{I_0}{16} \)
- \( \frac{I_0}{64} \)
- \( \frac{I_0}{256} \)
- and so on...

This sequence can be expressed as:
\[ I_n = \frac{I_0}{4^{n-1}} \]
for \( n = 1, 2, 3, \ldots \)

### Step 2: Calculate the total intensity
The total intensity \( I \) when combining these sources can be calculated using the formula for the intensity of coherent sources:
\[ I = I_1 + I_2 + I_3 + \ldots + I_n + 2 \sum_{i < j} \sqrt{I_i I_j} \cos(\phi_{ij}) \]

For maximum intensity, we consider the case where the phase difference \( \phi_{ij} = 0 \) (constructive interference), thus \( \cos(\phi_{ij}) = 1 \).

### Step 3: Sum the intensities
The total intensity becomes:
\[ I_{\text{max}} = I_1 + I_2 + I_3 + \ldots \]
Substituting the values of \( I_n \):
\[ I_{\text{max}} = I_0 + \frac{I_0}{4} + \frac{I_0}{16} + \frac{I_0}{64} + \frac{I_0}{256} + \ldots \]

### Step 4: Recognize the series
This is an infinite geometric series where:
- First term \( a = I_0 \)
- Common ratio \( r = \frac{1}{4} \)

The sum of an infinite geometric series is given by:
\[ S = \frac{a}{1 - r} \]
Thus,
\[ I_{\text{max}} = \frac{I_0}{1 - \frac{1}{4}} = \frac{I_0}{\frac{3}{4}} = \frac{4I_0}{3} \]

### Step 5: Finalize the maximum intensity
The maximum intensity \( I_{\text{max}} \) is:
\[ I_{\text{max}} = \frac{4I_0}{3} \]

### Conclusion
Thus, the maximum intensity in the case of interference of infinite identical coherent sources is:
\[ \boxed{\frac{4I_0}{3}} \]

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