Two coherent sources of intensity ratio `1:4` produce an interference pattern. The fringe visibility will be

Options

A. (a) `1`
B. (b) `0.8`
C. (c) `0.4`
D. (d) `0.6`

Explanation

To find the fringe visibility for two coherent sources with an intensity ratio of 1:4, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Intensities**:
Given the intensity ratio of the two sources is 1:4, we can denote the intensities as:
- \( I_1 = K \) (for the first source)
- \( I_2 = 4K \) (for the second source)

2. **Use the Formula for Fringe Visibility**:
The formula for fringe visibility \( V \) is given by:
\[ V = \frac{I_{\text{max}} - I_{\text{min}}}{I_{\text{max}} + I_{\text{min}}} \]
For two coherent sources, this can also be expressed as:
\[ V = \frac{2 \sqrt{I_1 I_2}}{I_1 + I_2} \]

3. **Substitute the Intensities into the Formula**:
Substitute \( I_1 \) and \( I_2 \) into the visibility formula:
\[ V = \frac{2 \sqrt{K \cdot 4K}}{K + 4K} \]

4. **Simplify the Expression**:
Calculate the numerator and denominator:
- Numerator:
\[ 2 \sqrt{K \cdot 4K} = 2 \sqrt{4K^2} = 2 \cdot 2K = 4K \]
- Denominator:
\[ K + 4K = 5K \]
Thus, we have:
\[ V = \frac{4K}{5K} \]

5. **Cancel the Common Terms**:
Cancel \( K \) from the numerator and denominator:
\[ V = \frac{4}{5} \]

6. **Convert to Decimal (if needed)**:
Converting \( \frac{4}{5} \) to decimal gives:
\[ V = 0.8 \]

### Final Answer:
The fringe visibility \( V \) is \( \frac{4}{5} \) or \( 0.8 \).

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