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 is zero for two coherent light beams with intensities \( I \) and \( 4I \), we can follow these steps:

### Step 1: Understand the formula for intensity in interference
The intensity \( I \) at a point where two coherent beams interfere can be calculated using the formula:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi) \]
where \( I_1 \) and \( I_2 \) are the intensities of the two beams, and \( \phi \) is the phase difference between them.

### Step 2: Assign values to the intensities
Given:
- \( I_1 = I \)
- \( I_2 = 4I \)

### Step 3: Substitute the values into the formula
At a point where the phase difference \( \phi = 0 \), we have:
\[ \cos(0) = 1 \]
Thus, substituting the values into the intensity formula gives:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cdot 1 \]
Substituting \( I_1 \) and \( I_2 \):
\[ I = I + 4I + 2\sqrt{I \cdot 4I} \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ I = I + 4I + 2\sqrt{4I^2} \]
\[ I = I + 4I + 2 \cdot 2I \]
\[ I = I + 4I + 4I \]
\[ I = 9I \]

### Step 5: Write the final answer
Thus, the total intensity at a point where the phase difference is zero is:
\[ I = 9I \]

### Conclusion
The intensity at a point where the phase difference is zero is \( 5I \).