Two coherent sources of different intensities send waves which interfere. The ratio of maximum intensity to the minimum intensity is 25. The intensities of the sources are in the ratio

Options

A. `25 : 1`
B. `5 : 1`
C. `9 : 4`
D. `625 : 1`

Answer

To solve the problem, we need to find the ratio of the intensities of two coherent sources based on the given ratio of maximum intensity to minimum intensity.

### Step-by-Step Solution:

1. **Understanding Maximum and Minimum Intensity**:
- The maximum intensity \( I_{max} \) when two coherent sources interfere is given by:
\[ I_{max} = (\sqrt{I_1} + \sqrt{I_2})^2 \]
- The minimum intensity \( I_{min} \) is given by:
\[ I_{min} = (\sqrt{I_1} - \sqrt{I_2})^2 \]

2. **Setting Up the Ratio**:
- We are given that the ratio of maximum intensity to minimum intensity is:
\[ \frac{I_{max}}{I_{min}} = 25 \]
- Substituting the expressions for \( I_{max} \) and \( I_{min} \):
\[ \frac{(\sqrt{I_1} + \sqrt{I_2})^2}{(\sqrt{I_1} - \sqrt{I_2})^2} = 25 \]

3. **Taking the Square Root**:
- Taking the square root of both sides:
\[ \frac{\sqrt{I_1} + \sqrt{I_2}}{\sqrt{I_1} - \sqrt{I_2}} = 5 \]

4. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ \sqrt{I_1} + \sqrt{I_2} = 5(\sqrt{I_1} - \sqrt{I_2}) \]

5. **Rearranging the Equation**:
- Rearranging the equation:
\[ \sqrt{I_1} + \sqrt{I_2} = 5\sqrt{I_1} - 5\sqrt{I_2} \]
- This simplifies to:
\[ 6\sqrt{I_2} = 4\sqrt{I_1} \]

6. **Finding the Ratio**:
- Dividing both sides by \( \sqrt{I_2} \):
\[ 6 = 4\frac{\sqrt{I_1}}{\sqrt{I_2}} \]
- Therefore:
\[ \frac{\sqrt{I_1}}{\sqrt{I_2}} = \frac{6}{4} = \frac{3}{2} \]
- Squaring both sides gives:
\[ \frac{I_1}{I_2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]

7. **Conclusion**:
- The ratio of the intensities of the two sources \( I_1 : I_2 \) is \( 9 : 4 \).

### Final Answer:
The intensities of the sources are in the ratio \( 9 : 4 \).