The intensity ratio of the maxima and minima in an interference pattern produced by two coherent sources of light is `9:1.` The intensities of the used light sources are in ratio

Options

A. `3:1`
B. `4:1`
C. `9:1`
D. `10:1`

Explanation

To solve the problem, we need to find the ratio of the intensities of two coherent light sources based on the given intensity ratio of maxima and minima in an interference pattern.

### Step-by-Step Solution:

1. **Understand the Given Ratio**: The intensity ratio of the maxima to the minima is given as \( \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{9}{1} \).

2. **Relate Intensity to Amplitude**: We know that intensity is proportional to the square of the amplitude. If we denote the amplitudes of the two sources as \( A \) and \( B \), we can express the intensities as:
\[ I_1 \propto A^2 \quad \text{and} \quad I_2 \propto B^2 \]

3. **Express Maxima and Minima Intensities**:
- The maximum intensity \( I_{\text{max}} \) occurs when the amplitudes add up:
\[ I_{\text{max}} = (A + B)^2 \]
- The minimum intensity \( I_{\text{min}} \) occurs when the amplitudes subtract:
\[ I_{\text{min}} = (A - B)^2 \]

4. **Set Up the Ratio**:
From the given ratio of intensities, we can write:
\[ \frac{(A + B)^2}{(A - B)^2} = \frac{9}{1} \]
This simplifies to:
\[ \frac{A + B}{A - B} = 3 \]

5. **Cross Multiply**:
Rearranging gives:
\[ A + B = 3(A - B) \]
Expanding the right side:
\[ A + B = 3A - 3B \]

6. **Rearranging the Equation**:
Bringing like terms together:
\[ A + B + 3B = 3A \]
This simplifies to:
\[ 4B = 2A \quad \Rightarrow \quad 2A = 4B \quad \Rightarrow \quad A = 2B \]

7. **Substituting Back**:
Now we can express \( A \) in terms of \( B \):
\[ A = 2B \]

8. **Finding the Ratio of Intensities**:
The intensities can now be expressed as:
\[ I_1 \propto A^2 = (2B)^2 = 4B^2 \]
\[ I_2 \propto B^2 \]
Thus, the ratio of the intensities is:
\[ \frac{I_1}{I_2} = \frac{4B^2}{B^2} = 4 \]

9. **Final Result**:
Therefore, the ratio of the intensities \( I_1 : I_2 \) is:
\[ 4 : 1 \]

### Conclusion:
The intensities of the two light sources are in the ratio \( 4:1 \).

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