The ratio of maximum and minimum intensities in the imterference pattern of two sources is `4:1`. The ratio of their amplitudes is

Options

A. `1:3`
B. `3:1`
C. `1:9`
D. `1:16`

Explanation

To solve the problem of finding the ratio of amplitudes given the ratio of maximum and minimum intensities in the interference pattern of two sources, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Intensity Ratio**:
Given that the ratio of maximum intensity (I_max) to minimum intensity (I_min) is 4:1, we can express this mathematically as:
\[ \frac{I_{max}}{I_{min}} = 4 \]

2. **Using Intensity Formulas**:
The maximum intensity (I_max) in terms of amplitudes (A1 and A2) is given by:
\[ I_{max} = (A_1 + A_2)^2 \]
The minimum intensity (I_min) is given by:
\[ I_{min} = (A_1 - A_2)^2 \]

3. **Setting Up the Equation**:
From the intensity ratio, we can set up the equation:
\[ \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = 4 \]

4. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ (A_1 + A_2)^2 = 4(A_1 - A_2)^2 \]

5. **Expanding Both Sides**:
Expanding both sides, we have:
\[ A_1^2 + 2A_1A_2 + A_2^2 = 4(A_1^2 - 2A_1A_2 + A_2^2) \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ A_1^2 + 2A_1A_2 + A_2^2 = 4A_1^2 - 8A_1A_2 + 4A_2^2 \]
Simplifying this leads to:
\[ 0 = 3A_1^2 - 10A_1A_2 + 3A_2^2 \]

7. **Using the Quadratic Formula**:
This is a quadratic equation in terms of A1 and A2. We can solve for the ratio \(\frac{A_1}{A_2}\) by substituting \(x = \frac{A_1}{A_2}\):
\[ 3x^2 - 10x + 3 = 0 \]

8. **Finding the Roots**:
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 3 \cdot 3}}{2 \cdot 3} \]
\[ x = \frac{10 \pm \sqrt{100 - 36}}{6} \]
\[ x = \frac{10 \pm \sqrt{64}}{6} \]
\[ x = \frac{10 \pm 8}{6} \]

9. **Calculating the Values**:
This gives us two possible values:
\[ x = \frac{18}{6} = 3 \quad \text{and} \quad x = \frac{2}{6} = \frac{1}{3} \]

10. **Final Ratio of Amplitudes**:
Since \(x = \frac{A_1}{A_2}\), we can conclude that the ratio of amplitudes is:
\[ \frac{A_1}{A_2} = 3:1 \]

### Conclusion:
The ratio of the amplitudes of the two sources is \(3:1\).