Two incoherent sources of intensities l and 4l superpose then the resultant intensity is

Options

A. 5l
B. 9l
C. 3l
D. l

Explanation

To solve the problem of finding the resultant intensity when two incoherent sources of intensities \( I_1 = I \) and \( I_2 = 4I \) superpose, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Incoherent Sources**:
Incoherent sources do not have a fixed phase relationship. This means that the phase difference \( \phi \) between the two sources can vary randomly.

2. **Use the Formula for Resultant Intensity**:
The resultant intensity \( I_R \) for two sources is given by the formula:
\[ I_R = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos \phi \]
where \( I_1 \) and \( I_2 \) are the intensities of the two sources.

3. **Substitute the Intensities**:
In this case, we have:
\[ I_1 = I \quad \text{and} \quad I_2 = 4I \]
Substituting these values into the formula gives:
\[ I_R = I + 4I + 2\sqrt{I \cdot 4I} \cos \phi \]

4. **Consider the Phase Difference for Incoherent Sources**:
For incoherent sources, the average value of \( \cos \phi \) is zero. Therefore, we can simplify the equation:
\[ I_R = I + 4I + 2\sqrt{I \cdot 4I} \cdot 0 \]
This simplifies to:
\[ I_R = I + 4I = 5I \]

5. **Final Result**:
Thus, the resultant intensity when two incoherent sources of intensities \( I \) and \( 4I \) superpose is:
\[ I_R = 5I \]