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

Answer

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: Understanding the Intensities
We have two coherent light beams with the following intensities:
- \( I_1 = I \)
- \( I_2 = 4I \)

### Step 2: Finding Maximum Intensity
The formula for the maximum intensity \( I_{\text{max}} \) when two coherent beams interfere is given by:
\[ I_{\text{max}} = \left( \sqrt{I_1} + \sqrt{I_2} \right)^2 \]
Substituting the values of \( I_1 \) and \( I_2 \):
\[ I_{\text{max}} = \left( \sqrt{I} + \sqrt{4I} \right)^2 \]
Calculating \( \sqrt{4I} \):
\[ \sqrt{4I} = 2\sqrt{I} \]
Now substituting back:
\[ I_{\text{max}} = \left( \sqrt{I} + 2\sqrt{I} \right)^2 = \left( 3\sqrt{I} \right)^2 = 9I \]

### Step 3: Finding Minimum Intensity
The formula for the minimum intensity \( I_{\text{min}} \) when two coherent beams interfere is given by:
\[ I_{\text{min}} = \left( \sqrt{I_1} - \sqrt{I_2} \right)^2 \]
Substituting the values of \( I_1 \) and \( I_2 \):
\[ I_{\text{min}} = \left( \sqrt{I} - \sqrt{4I} \right)^2 \]
Calculating \( \sqrt{4I} \) again:
\[ I_{\text{min}} = \left( \sqrt{I} - 2\sqrt{I} \right)^2 = \left( -\sqrt{I} \right)^2 = I \]

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

### Final Answer
The maximum and minimum possible intensities are \( 9I \) and \( I \), respectively.

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