Two waves having the intensities in the ratio of `9 : 1` produce interference. The ratio of maximum to minimum intensity is equal to

Options

A. `10:8`
B. `9:1`
C. `4:1`
D. `2:1`

Answer

To solve the problem of finding the ratio of maximum to minimum intensity produced by two waves with intensities in the ratio of 9:1, we can follow these steps:

### Step 1: Define the Intensities
Let the intensities of the two waves be:
- \( I_1 = 9 I_0 \)
- \( I_2 = 1 I_0 \)

### Step 2: Write the Formula for Resultant Intensity
The resultant intensity \( I \) when two waves interfere can be expressed as:
\[ I = I_1 + I_2 + 2 \sqrt{I_1 I_2} \cos \phi \]
where \( \phi \) is the phase difference between the two waves.

### Step 3: Calculate Maximum Intensity
The maximum intensity occurs when \( \cos \phi = 1 \):
\[ I_{\text{max}} = I_1 + I_2 + 2 \sqrt{I_1 I_2} \]
Substituting the values:
\[ I_{\text{max}} = 9 I_0 + 1 I_0 + 2 \sqrt{9 I_0 \cdot 1 I_0} \]
\[ I_{\text{max}} = 10 I_0 + 2 \sqrt{9 I_0^2} \]
\[ I_{\text{max}} = 10 I_0 + 6 I_0 = 16 I_0 \]

### Step 4: Calculate Minimum Intensity
The minimum intensity occurs when \( \cos \phi = -1 \):
\[ I_{\text{min}} = I_1 + I_2 - 2 \sqrt{I_1 I_2} \]
Substituting the values:
\[ I_{\text{min}} = 9 I_0 + 1 I_0 - 2 \sqrt{9 I_0 \cdot 1 I_0} \]
\[ I_{\text{min}} = 10 I_0 - 6 I_0 = 4 I_0 \]

### Step 5: Calculate the Ratio of Maximum to Minimum Intensity
Now, we find the ratio of maximum intensity to minimum intensity:
\[ \text{Ratio} = \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{16 I_0}{4 I_0} = 4 \]

### Final Answer
Thus, the ratio of maximum to minimum intensity is:
\[ \boxed{4} \]