In Young’s experiment, the ratio of maximum to minimum intensities of the fringe system is `4 : 1`. The amplitudes of the coherent sources are in the ratio

Options

A. `1:1`
B. `3:1`
C. `1:4`
D. `5:1`

Answer

To solve the problem, we need to find the ratio of the amplitudes of the coherent sources based on the given ratio of maximum to minimum intensities in Young's double-slit experiment.

### Step-by-Step Solution:

1. **Understanding the Relationship Between Intensity and Amplitude**:
The intensity (I) of a wave is proportional to the square of its amplitude (A). This can be expressed as:
\[ I \propto A^2 \]
Therefore, we can write:
\[ I_1 = k A_1^2 \quad \text{and} \quad I_2 = k A_2^2 \]
where \(k\) is a constant.

2. **Expressing Maximum and Minimum Intensities**:
In Young's experiment, the maximum intensity \(I_{max}\) and minimum intensity \(I_{min}\) can be expressed as:
\[ I_{max} = (A_1 + A_2)^2 \quad \text{and} \quad I_{min} = (A_1 - A_2)^2 \]

3. **Setting Up the Ratio**:
Given that the ratio of maximum to minimum intensity is \(4:1\), we can write:
\[ \frac{I_{max}}{I_{min}} = \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = 4 \]

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

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

6. **Rearranging the Equation**:
Rearranging gives:
\[ A_1^2 + 2A_1A_2 + A_2^2 = 4A_1^2 - 8A_1A_2 + 4A_2^2 \]
Bringing all terms to one side:
\[ 0 = 3A_1^2 - 10A_1A_2 + 3A_2^2 \]

7. **Using the Quadratic Formula**:
This can be treated as a quadratic equation in terms of \(A_1/A_2\). Let \(x = \frac{A_1}{A_2}\), then:
\[ 3x^2 - 10x + 3 = 0 \]
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\[ x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 3 \cdot 3}}{2 \cdot 3} \]
\[ x = \frac{10 \pm \sqrt{100 - 36}}{6} \]
\[ x = \frac{10 \pm \sqrt{64}}{6} \]
\[ x = \frac{10 \pm 8}{6} \]
This gives two possible values:
\[ x = \frac{18}{6} = 3 \quad \text{or} \quad x = \frac{2}{6} = \frac{1}{3} \]

8. **Conclusion**:
The ratio of the amplitudes \(A_1 : A_2\) is \(3 : 1\).

### Final Answer:
The ratio of the amplitudes of the coherent sources is \(3 : 1\).