Find intensity of sound in `dB` if its intensity in `W // m^(2)` is `10^(-10)`.

Explanation

To find the intensity of sound in decibels (dB) given its intensity in watts per meter squared (W/m²) is \(10^{-10}\), we can follow these steps:

### Step 1: Understand the formula for sound intensity in decibels
The sound level in decibels (\(\beta\)) is given by the formula:
\[ \beta = 10 \log_{10} \left( \frac{I}{I_0} \right) \]
where:
- \(I\) is the intensity of the sound in W/m²,
- \(I_0\) is the reference intensity, typically taken as \(10^{-12}\) W/m².

### Step 2: Identify the given values
From the problem statement:
- The given intensity \(I = 10^{-10}\) W/m²,
- The reference intensity \(I_0 = 10^{-12}\) W/m².

### Step 3: Substitute the values into the formula
Substituting the values into the formula:
\[ \beta = 10 \log_{10} \left( \frac{10^{-10}}{10^{-12}} \right) \]

### Step 4: Simplify the logarithmic expression
We can simplify the fraction inside the logarithm:
\[ \frac{10^{-10}}{10^{-12}} = 10^{-10 - (-12)} = 10^{2} \]
So now we have:
\[ \beta = 10 \log_{10} (10^{2}) \]

### Step 5: Calculate the logarithm
Using the property of logarithms:
\[ \log_{10} (10^{2}) = 2 \]
Thus, substituting this back into our equation:
\[ \beta = 10 \times 2 = 20 \text{ dB} \]

### Final Answer
The intensity of sound in decibels is \(20\) dB.

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