Ratio of amplitude for two wave is 1:5.Find the ratio of maximum Intensity to minimum intensity.

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Given Information
We are given the ratio of amplitudes of two waves, \( A_1 : A_2 = 1 : 5 \). This means we can express the amplitudes as:
- \( A_1 = k \) (some constant)
- \( A_2 = 5k \)

### Step 2: Relate Intensity to Amplitude
We know that intensity \( I \) is directly proportional to the square of the amplitude:
\[ I \propto A^2 \]
Thus, we can express the intensities in terms of the amplitudes:
- \( I_1 \propto A_1^2 \)
- \( I_2 \propto A_2^2 \)

### Step 3: Calculate the Intensities
Using the amplitudes we defined:
- \( I_1 = k^2 \)
- \( I_2 = (5k)^2 = 25k^2 \)

### Step 4: Find the Ratio of Intensities
Now, we can find the ratio of the two intensities:
\[ \frac{I_1}{I_2} = \frac{k^2}{25k^2} = \frac{1}{25} \]

### Step 5: Calculate Maximum Intensity
The maximum intensity \( I_{\text{max}} \) for two waves is given by:
\[ I_{\text{max}} = (A_1 + A_2)^2 \]
Substituting the values of \( A_1 \) and \( A_2 \):
\[ I_{\text{max}} = (k + 5k)^2 = (6k)^2 = 36k^2 \]

### Step 6: Calculate Minimum Intensity
The minimum intensity \( I_{\text{min}} \) is given by:
\[ I_{\text{min}} = (A_1 - A_2)^2 \]
Substituting the values of \( A_1 \) and \( A_2 \):
\[ I_{\text{min}} = (k - 5k)^2 = (-4k)^2 = 16k^2 \]

### Step 7: Find the Ratio of Maximum to Minimum Intensity
Now, we can find the ratio of maximum intensity to minimum intensity:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{36k^2}{16k^2} = \frac{36}{16} = \frac{9}{4} \]

### Final Answer
The ratio of maximum intensity to minimum intensity is:
\[ \frac{I_{\text{max}}}{I_{\text{min}}} = \frac{9}{4} \]

---