Two coherent light beams of intensities I and 4I produce interference pattern. The intensity at a point where the phase difference is zero, will b:

Options

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

Answer

To solve the problem of finding the intensity at a point where the phase difference between two coherent light beams is zero, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Intensities**:
- Let the intensity of the first beam be \( I_1 = I \).
- Let the intensity of the second beam be \( I_2 = 4I \).

2. **Use the Formula for Resultant Intensity**:
The formula for the resultant intensity \( I \) when two coherent beams interfere is given by:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi) \]
where \( \phi \) is the phase difference between the two beams.

3. **Substitute the Values**:
Since the phase difference \( \phi = 0 \) (as given in the problem), we have:
\[ \cos(0) = 1 \]
Therefore, the formula simplifies to:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \]

4. **Plug in the Intensities**:
Substituting \( I_1 \) and \( I_2 \) into the equation:
\[ I = I + 4I + 2\sqrt{I \cdot 4I} \]

5. **Calculate the Square Root**:
Calculate \( \sqrt{I \cdot 4I} \):
\[ \sqrt{I \cdot 4I} = \sqrt{4I^2} = 2I \]

6. **Combine the Terms**:
Now substitute back into the intensity equation:
\[ I = I + 4I + 2(2I) \]
\[ I = I + 4I + 4I \]
\[ I = 9I \]

7. **Final Result**:
Therefore, the intensity at the point where the phase difference is zero is:
\[ I = 9I \]

### Final Answer:
The intensity at the point where the phase difference is zero is \( 9I \).

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