When two coherent monochromatic light beams of intensities I and 4I are superimposed, the ratio between maximum and minimum intensities in the resultant beam is

Options

A. A. `9 : 1`
B. B. `1 : 9`
C. C. `4 : 1`
D. D. `1 : 9`

Answer

To solve the problem of finding the ratio between maximum and minimum intensities when two coherent monochromatic light beams of intensities \( I \) and \( 4I \) are superimposed, we can follow these steps:

### Step 1: Understand the formulas for maximum and minimum intensity
The maximum intensity \( I_{\text{max}} \) and minimum intensity \( I_{\text{min}} \) for two coherent light sources can be calculated using the following formulas:
- Maximum intensity:
\[ I_{\text{max}} = \left( \sqrt{I_1} + \sqrt{I_2} \right)^2 \]
- Minimum intensity:
\[ I_{\text{min}} = \left( \sqrt{I_1} - \sqrt{I_2} \right)^2 \]

### Step 2: Assign the values of \( I_1 \) and \( I_2 \)
In this case, we have:
- \( I_1 = I \)
- \( I_2 = 4I \)

### Step 3: Calculate \( I_{\text{max}} \)
Substituting the values into the formula for maximum intensity:
\[ I_{\text{max}} = \left( \sqrt{I} + \sqrt{4I} \right)^2 \]
This simplifies to:
\[ I_{\text{max}} = \left( \sqrt{I} + 2\sqrt{I} \right)^2 = \left( 3\sqrt{I} \right)^2 = 9I \]

### Step 4: Calculate \( I_{\text{min}} \)
Now, substituting the values into the formula for minimum intensity:
\[ I_{\text{min}} = \left( \sqrt{I} - \sqrt{4I} \right)^2 \]
This simplifies to:
\[ I_{\text{min}} = \left( \sqrt{I} - 2\sqrt{I} \right)^2 = \left( -\sqrt{I} \right)^2 = I \]

### Step 5: Find the ratio \( \frac{I_{\text{max}}}{I_{\text{min}}} \)
Now, we can find the ratio of maximum intensity to minimum intensity:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{9I}{I} = 9 \]

### Conclusion
Thus, the ratio between maximum and minimum intensities in the resultant beam is:
\[ \text{Ratio} = 9:1 \]