In Young's double slit experiment, the ratio of maximum and minimum intensities in the fringe system is `9:1` the ratio of amplitudes of coherent sources is

Options

A. `9:1`
B. `3:1`
C. `2:1`
D. `1:1`

Explanation

To solve the problem, we need to find the ratio of amplitudes of two coherent sources in Young's double slit experiment, given that the ratio of maximum intensity (Imax) to minimum intensity (Imin) is 9:1.

### Step-by-Step Solution:

1. **Understanding Intensity in Terms of Amplitude**:
The intensity of light is proportional to the square of the amplitude. Therefore, we can express the maximum and minimum intensities in terms of the amplitudes \( A_1 \) and \( A_2 \) of the two coherent sources:
\[ I_{\text{max}} = (A_1 + A_2)^2 \]
\[ I_{\text{min}} = (A_1 - A_2)^2 \]

2. **Setting Up the Ratio**:
According to the problem, the ratio of maximum intensity to minimum intensity is given as:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{9}{1} \]
Substituting the expressions for \( I_{\text{max}} \) and \( I_{\text{min}} \):
\[ \frac{(A_1 + A_2)^2}{(A_1 - A_2)^2} = 9 \]

3. **Cross-Multiplying**:
Cross-multiplying gives us:
\[ (A_1 + A_2)^2 = 9(A_1 - A_2)^2 \]

4. **Expanding Both Sides**:
Expanding both sides leads to:
\[ A_1^2 + 2A_1A_2 + A_2^2 = 9(A_1^2 - 2A_1A_2 + A_2^2) \]

5. **Rearranging the Equation**:
Rearranging gives:
\[ A_1^2 + 2A_1A_2 + A_2^2 = 9A_1^2 - 18A_1A_2 + 9A_2^2 \]
Bringing all terms to one side:
\[ 0 = 8A_1^2 - 20A_1A_2 + 8A_2^2 \]

6. **Dividing by 4**:
Simplifying the equation by dividing everything by 4:
\[ 0 = 2A_1^2 - 5A_1A_2 + 2A_2^2 \]

7. **Using the Quadratic Formula**:
This is a quadratic equation in terms of \( A_1 \):
\[ 2A_1^2 - 5A_1A_2 + 2A_2^2 = 0 \]
Using the quadratic formula \( A_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 2 \), \( b = -5A_2 \), and \( c = 2A_2^2 \):
\[ A_1 = \frac{5A_2 \pm \sqrt{(-5A_2)^2 - 4 \cdot 2 \cdot 2A_2^2}}{2 \cdot 2} \]
\[ A_1 = \frac{5A_2 \pm \sqrt{25A_2^2 - 16A_2^2}}{4} \]
\[ A_1 = \frac{5A_2 \pm \sqrt{9A_2^2}}{4} \]
\[ A_1 = \frac{5A_2 \pm 3A_2}{4} \]

8. **Finding Possible Values**:
This gives us two possible values for \( A_1 \):
- \( A_1 = \frac{8A_2}{4} = 2A_2 \)
- \( A_1 = \frac{2A_2}{4} = \frac{1}{2}A_2 \)

9. **Calculating the Ratio**:
The ratio of amplitudes \( \frac{A_1}{A_2} \) can be calculated:
- For \( A_1 = 2A_2 \), the ratio \( \frac{A_1}{A_2} = 2:1 \)
- For \( A_1 = \frac{1}{2}A_2 \), the ratio \( \frac{A_1}{A_2} = \frac{1}{2}:1 \)

10. **Conclusion**:
Since we are interested in the ratio of amplitudes, the relevant ratio is:
\[ \frac{A_1}{A_2} = 2:1 \]

### Final Answer:
The ratio of the amplitudes of the coherent sources is \( 2:1 \).