Ratio of amplitude for two wave is 1:4 .Find the ratio of intensity?

Answer

To solve the problem of finding the ratio of intensity given the ratio of amplitude, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information**:
- The ratio of amplitudes \( A_1 : A_2 = 1 : 4 \).
- We need to find the ratio of intensities \( I_1 : I_2 \).

2. **Understand the Relationship Between Intensity and Amplitude**:
- The intensity \( I \) of a wave is proportional to the square of its amplitude. This can be expressed mathematically as:
\[ I \propto A^2 \]
- Therefore, we can write:
\[ \frac{I_1}{I_2} = \left(\frac{A_1}{A_2}\right)^2 \]

3. **Substitute the Given Ratio of Amplitudes**:
- From the problem, we know:
\[ \frac{A_1}{A_2} = \frac{1}{4} \]
- Now, substitute this value into the intensity ratio formula:
\[ \frac{I_1}{I_2} = \left(\frac{1}{4}\right)^2 \]

4. **Calculate the Ratio of Intensities**:
- Calculating the square:
\[ \frac{I_1}{I_2} = \frac{1}{16} \]

5. **Express the Ratio in Standard Form**:
- Thus, the ratio of intensities can be expressed as:
\[ I_1 : I_2 = 1 : 16 \]

### Final Answer:
The ratio of intensity \( I_1 : I_2 \) is \( 1 : 16 \).

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