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 find the ratio of intensities of maxima and minima when two waves interfere, we can follow these steps:

### Step 1: Define the Intensities
Let the intensities of the two waves be \( I_1 \) and \( I_2 \). According to the problem, the ratio of their intensities is given as:
\[ \frac{I_1}{I_2} = 2 \]
This implies that we can express \( I_1 \) in terms of \( I_2 \):
\[ I_1 = 2I_2 \]

### Step 2: Calculate Maximum Intensity
The maximum intensity \( I_{max} \) when two waves interfere is given by the formula:
\[ I_{max} = I_1 + I_2 + 2\sqrt{I_1 I_2} \]
Substituting \( I_1 = 2I_2 \) into the equation:
\[ I_{max} = 2I_2 + I_2 + 2\sqrt{(2I_2)(I_2)} \]
\[ I_{max} = 3I_2 + 2\sqrt{2I_2^2} \]
\[ I_{max} = 3I_2 + 2I_2\sqrt{2} \]
\[ I_{max} = I_2(3 + 2\sqrt{2}) \]

### Step 3: Calculate Minimum Intensity
The minimum intensity \( I_{min} \) when two waves interfere is given by:
\[ I_{min} = I_1 + I_2 - 2\sqrt{I_1 I_2} \]
Substituting \( I_1 = 2I_2 \):
\[ I_{min} = 2I_2 + I_2 - 2\sqrt{(2I_2)(I_2)} \]
\[ I_{min} = 3I_2 - 2\sqrt{2I_2^2} \]
\[ I_{min} = 3I_2 - 2I_2\sqrt{2} \]
\[ I_{min} = I_2(3 - 2\sqrt{2}) \]

### Step 4: Calculate the Ratio of Intensities
Now, we can find the ratio of maximum intensity to minimum intensity:
\[ \text{Ratio} = \frac{I_{max}}{I_{min}} = \frac{I_2(3 + 2\sqrt{2})}{I_2(3 - 2\sqrt{2})} \]
The \( I_2 \) cancels out:
\[ \text{Ratio} = \frac{3 + 2\sqrt{2}}{3 - 2\sqrt{2}} \]

### Step 5: Simplify the Ratio
To simplify this expression, we can multiply the numerator and denominator by the conjugate of the denominator:
\[ \text{Ratio} = \frac{(3 + 2\sqrt{2})(3 + 2\sqrt{2})}{(3 - 2\sqrt{2})(3 + 2\sqrt{2})} \]
Calculating the denominator:
\[ (3 - 2\sqrt{2})(3 + 2\sqrt{2}) = 9 - 8 = 1 \]
Calculating the numerator:
\[ (3 + 2\sqrt{2})^2 = 9 + 12\sqrt{2} + 8 = 17 + 12\sqrt{2} \]
Thus, the ratio simplifies to:
\[ \text{Ratio} = 17 + 12\sqrt{2} \]

### Final Answer
The ratio of intensities of maxima and minima when these waves interfere is approximately:
\[ \text{Ratio} \approx 34 \quad (\text{after numerical evaluation}) \]