The ratio of intensities of two waves is 2. the ratio of intensities of maxima and minima when these waves interfere is approximately

Options

A. 9
B. 8
C. 34
D. 36

Answer

To solve the problem, we need to find the ratio of intensities of maxima and minima when two waves interfere, given that the ratio of their intensities is 2.

### Step-by-Step Solution:

1. **Identify the Given Information:**
- The ratio of intensities of two waves is given as:
\[ \frac{I_1}{I_2} = 2 \]
- This means we can express \(I_1\) in terms of \(I_2\):
\[ I_1 = 2I_2 \]

2. **Use the Formula for Maxima and Minima:**
- The intensity of the resultant wave when two waves interfere can be calculated using the formula for maximum and minimum intensities:
- Maximum intensity:
\[ I_{\text{max}} = ( \sqrt{I_1} + \sqrt{I_2} )^2 \]
- Minimum intensity:
\[ I_{\text{min}} = ( \sqrt{I_1} - \sqrt{I_2} )^2 \]

3. **Calculate the Square Roots:**
- First, we find \(\sqrt{I_1}\) and \(\sqrt{I_2}\):
\[ \sqrt{I_1} = \sqrt{2I_2} = \sqrt{2} \cdot \sqrt{I_2} \]
\[ \sqrt{I_2} = \sqrt{I_2} \]

4. **Substitute into the Maxima and Minima Formulas:**
- Substitute \(\sqrt{I_1}\) and \(\sqrt{I_2}\) into the formulas for maximum and minimum intensities:
- Maximum intensity:
\[ I_{\text{max}} = ( \sqrt{2} \cdot \sqrt{I_2} + \sqrt{I_2} )^2 = ( \sqrt{2} + 1 )^2 \cdot I_2 \]
- Minimum intensity:
\[ I_{\text{min}} = ( \sqrt{2} \cdot \sqrt{I_2} - \sqrt{I_2} )^2 = ( \sqrt{2} - 1 )^2 \cdot I_2 \]

5. **Calculate the Ratios:**
- Now, we can find the ratio of maximum to minimum intensities:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{( \sqrt{2} + 1 )^2}{( \sqrt{2} - 1 )^2} \]

6. **Simplify the Expression:**
- Expanding both squares:
\[ ( \sqrt{2} + 1 )^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} \]
\[ ( \sqrt{2} - 1 )^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]
- Thus, the ratio becomes:
\[ \frac{3 + 2\sqrt{2}}{3 - 2\sqrt{2}} \]

7. **Rationalize the Denominator:**
- Multiply numerator and denominator by the conjugate of the denominator:
\[ \frac{(3 + 2\sqrt{2})(3 + 2\sqrt{2})}{(3 - 2\sqrt{2})(3 + 2\sqrt{2})} \]
- The denominator simplifies to:
\[ 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]
- The numerator becomes:
\[ (3 + 2\sqrt{2})^2 = 9 + 12\sqrt{2} + 8 = 17 + 12\sqrt{2} \]

8. **Approximate the Result:**
- The final ratio of intensities of maxima to minima is approximately:
\[ I_{\text{max}} : I_{\text{min}} \approx 34 \]

### Final Answer:
The ratio of intensities of maxima to minima when these waves interfere is approximately **34**.