Two coherent monochromatic light beams of intensities I and 4I are superposed. The maximum and minimum possible intensities in the resulting beam are

Options

A. 5I and 3I
B. 9I and 3I
C. 4I and I
D. 9I and I

Explanation

To solve the problem of finding the maximum and minimum possible intensities when two coherent monochromatic light beams of intensities \( I_1 = I \) and \( I_2 = 4I \) are superposed, we can follow these steps:

### Step 1: Identify the Intensities
Let the intensities of the two beams be:
- \( I_1 = I \)
- \( I_2 = 4I \)

### Step 2: Calculate the Maximum Intensity
The maximum intensity \( I_{\text{max}} \) when two coherent beams interfere constructively is given by the formula:
\[ I_{\text{max}} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
Substituting the values:
\[ I_{\text{max}} = (\sqrt{I} + \sqrt{4I})^2 \]
Calculating the square root:
\[ \sqrt{4I} = 2\sqrt{I} \]
Thus, we have:
\[ I_{\text{max}} = (\sqrt{I} + 2\sqrt{I})^2 = (3\sqrt{I})^2 = 9I \]

### Step 3: Calculate the Minimum Intensity
The minimum intensity \( I_{\text{min}} \) when the beams interfere destructively is given by the formula:
\[ I_{\text{min}} = (\sqrt{I_1} - \sqrt{I_2})^2 \]
Substituting the values:
\[ I_{\text{min}} = (\sqrt{I} - \sqrt{4I})^2 \]
Again, calculating the square root:
\[ I_{\text{min}} = (\sqrt{I} - 2\sqrt{I})^2 = (-\sqrt{I})^2 = I \]

### Step 4: Final Results
Thus, the maximum and minimum possible intensities in the resulting beam are:
- Maximum Intensity \( I_{\text{max}} = 9I \)
- Minimum Intensity \( I_{\text{min}} = I \)

### Conclusion
The maximum and minimum intensities when the two coherent light beams are superposed are \( 9I \) and \( I \), respectively.

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