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 the Problem**:
We are given that the ratio of maximum intensity (I_max) to minimum intensity (I_min) is 25. We need to find the ratio of the intensities of the two sources, I1 and I2.

2. **Using the Formula**:
The relationship between maximum and minimum intensity in terms of the intensities of the two sources is given by:
\[ \frac{I_{max}}{I_{min}} = \frac{(\sqrt{I_1} + \sqrt{I_2})^2}{(\sqrt{I_1} - \sqrt{I_2})^2} \]
Given that \(\frac{I_{max}}{I_{min}} = 25\), we can write:
\[ 25 = \frac{(\sqrt{I_1} + \sqrt{I_2})^2}{(\sqrt{I_1} - \sqrt{I_2})^2} \]

3. **Setting Up the Equation**:
Let \(\sqrt{I_1} = a\) and \(\sqrt{I_2} = b\). Then the equation becomes:
\[ 25 = \frac{(a + b)^2}{(a - b)^2} \]

4. **Cross Multiplying**:
Cross-multiplying gives us:
\[ 25(a - b)^2 = (a + b)^2 \]

5. **Expanding Both Sides**:
Expanding both sides, we have:
\[ 25(a^2 - 2ab + b^2) = a^2 + 2ab + b^2 \]
This simplifies to:
\[ 25a^2 - 50ab + 25b^2 = a^2 + 2ab + b^2 \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ 24a^2 - 52ab + 24b^2 = 0 \]

7. **Dividing by 4**:
Dividing the entire equation by 4 simplifies it to:
\[ 6a^2 - 13ab + 6b^2 = 0 \]

8. **Using the Quadratic Formula**:
This is a quadratic equation in terms of \(a\). We can use the quadratic formula \(a = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A}\) where \(A = 6\), \(B = -13b\), and \(C = 6b^2\):
\[ a = \frac{13b \pm \sqrt{(-13b)^2 - 4 \cdot 6 \cdot 6b^2}}{2 \cdot 6} \]
Simplifying gives:
\[ a = \frac{13b \pm \sqrt{169b^2 - 144b^2}}{12} = \frac{13b \pm 5b}{12} \]

9. **Finding Values of a**:
This results in two possible values for \(a\):
\[ a = \frac{18b}{12} = \frac{3b}{2} \quad \text{or} \quad a = \frac{8b}{12} = \frac{2b}{3} \]

10. **Finding the Ratio of Intensities**:
The ratio of intensities \(I_1\) to \(I_2\) can be found as follows:
- For \(a = \frac{3b}{2}\):
\[ I_1 = a^2 = \left(\frac{3b}{2}\right)^2 = \frac{9b^2}{4}, \quad I_2 = b^2 \]
Thus,
\[ \frac{I_1}{I_2} = \frac{\frac{9b^2}{4}}{b^2} = \frac{9}{4} \]

### Final Result:
The ratio of the intensities of the two sources \(I_1 : I_2 = 9 : 4\).