The minimum intensity of audibility of sound is `10^(-12) W//m^(2) s` and density of air ` = 1.293 kg//m^(3)`. If the frequency of sound in `1000 Hz` , then the corresponding amplitude the vibration of the air particles is
[ Take velocity of sound `= 332 m//s`]

Options

A. `1.1 xx 10^(-7) m`
B. `1.1 xx 10^(-9) m`
C. `1.1 xx 10^(-11) m`
D. `1.1 xx 10^(-14) m`

Explanation

To find the amplitude of the vibration of air particles corresponding to the given intensity of sound, we can use the formula for intensity:

\[ I = 2 \pi^2 n^2 a^2 \rho v \]

where:
- \( I \) = intensity of sound (in \( W/m^2 \))
- \( n \) = frequency of sound (in Hz)
- \( a \) = amplitude of vibration (in m)
- \( \rho \) = density of air (in \( kg/m^3 \))
- \( v \) = velocity of sound (in m/s)

### Step 1: Write down the given values
- Intensity \( I = 10^{-12} \, W/m^2 \)
- Density of air \( \rho = 1.293 \, kg/m^3 \)
- Frequency \( n = 1000 \, Hz \)
- Velocity of sound \( v = 332 \, m/s \)

### Step 2: Rearrange the formula to solve for amplitude \( a \)
From the intensity formula, we can solve for \( a^2 \):

\[ a^2 = \frac{I}{2 \pi^2 n^2 \rho v} \]

### Step 3: Substitute the known values into the equation
Substituting the values into the rearranged formula:

\[ a^2 = \frac{10^{-12}}{2 \pi^2 (1000)^2 (1.293) (332)} \]

### Step 4: Calculate the denominator
Calculating \( 2 \pi^2 (1000)^2 (1.293) (332) \):

1. Calculate \( \pi^2 \):
\[ \pi^2 \approx 9.87 \]

2. Calculate \( 2 \pi^2 \):
\[ 2 \pi^2 \approx 19.74 \]

3. Calculate \( (1000)^2 \):
\[ (1000)^2 = 1000000 \]

4. Now calculate the entire denominator:
\[ 19.74 \times 1000000 \times 1.293 \times 332 \]

Calculating step by step:
- \( 19.74 \times 1000000 = 19740000 \)
- \( 19740000 \times 1.293 \approx 25500000 \)
- \( 25500000 \times 332 \approx 8466000000 \)

### Step 5: Substitute back into the equation for \( a^2 \)
Now we have:

\[ a^2 = \frac{10^{-12}}{8466000000} \]

### Step 6: Calculate \( a^2 \)
Calculating \( a^2 \):

\[ a^2 \approx 1.179 \times 10^{-21} \]

### Step 7: Take the square root to find \( a \)
Now, take the square root to find \( a \):

\[ a \approx \sqrt{1.179 \times 10^{-21}} \approx 1.077 \times 10^{-11} \, m \]

### Final Answer
Thus, the corresponding amplitude of the vibration of the air particles is approximately:

\[ a \approx 1.077 \times 10^{-11} \, m \]

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