If two sources have a randomly varying phase difference f(t), the resultant intensity will be given by:

Options

A. `I//2I_(0)`
B. `I_(0)//2`
C. `2I_(0)`
D. `I_(0)//sqrt(2)`

Answer

To solve the problem of finding the resultant intensity when two sources have a randomly varying phase difference \( f(t) \), we can follow these steps:

### Step 1: Understand the Concept of Intensity
The intensity of a wave is proportional to the square of its amplitude. For two sources, if each source has an intensity \( I_0 \), we denote the intensities of the two sources as \( I_1 \) and \( I_2 \).

### Step 2: Determine the Nature of the Sources
Since the phase difference \( f(t) \) is randomly varying, the two sources are considered incoherent. Incoherent sources do not maintain a constant phase relationship, which means their interference effects average out over time.

### Step 3: Calculate the Resultant Intensity
For incoherent sources, the resultant intensity \( I_{\text{net}} \) is simply the sum of the individual intensities. Therefore, we can express this mathematically as:

\[ I_{\text{net}} = I_1 + I_2 \]

### Step 4: Substitute the Intensities
Given that both sources have the same intensity \( I_0 \):

\[ I_1 = I_0 \quad \text{and} \quad I_2 = I_0 \]

Substituting these values into the equation for resultant intensity gives:

\[ I_{\text{net}} = I_0 + I_0 = 2I_0 \]

### Step 5: Conclusion
Thus, the resultant intensity when two incoherent sources with randomly varying phase differences are combined is:

\[ I_{\text{net}} = 2I_0 \]

### Final Answer
The resultant intensity will be \( 2I_0 \).

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