The intensity of sound of 50 dB (Take reference of intensity `10^-12` W/`m^2`)

Options

A. `10^-10W/m^2`
B. `210^-5 W/m^2`
C. `10^-12 W/m^2`
D. `10^-7 W/m^2`

Answer

To find the intensity of sound corresponding to a loudness of 50 dB, we can use the formula for sound intensity level in decibels:

\[ L = 10 \log_{10} \left( \frac{I}{I_0} \right) \]

Where:
- \( L \) is the sound level in decibels (dB)
- \( I \) is the intensity of the sound in watts per meter squared (W/m²)
- \( I_0 \) is the reference intensity, which is given as \( 10^{-12} \, \text{W/m}^2 \)

### Step-by-Step Solution:

1. **Identify the given values**:
- Loudness \( L = 50 \, \text{dB} \)
- Reference intensity \( I_0 = 10^{-12} \, \text{W/m}^2 \)

2. **Substitute the values into the formula**:
\[ 50 = 10 \log_{10} \left( \frac{I}{10^{-12}} \right) \]

3. **Divide both sides by 10**:
\[ 5 = \log_{10} \left( \frac{I}{10^{-12}} \right) \]

4. **Convert the logarithmic equation to its exponential form**:
\[ 10^5 = \frac{I}{10^{-12}} \]

5. **Multiply both sides by \( 10^{-12} \)** to solve for \( I \):
\[ I = 10^5 \times 10^{-12} \]

6. **Combine the powers of 10**:
\[ I = 10^{-7} \, \text{W/m}^2 \]

### Final Answer:
The intensity of sound corresponding to 50 dB is \( I = 10^{-7} \, \text{W/m}^2 \).

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