Question

The intensity level $1\text{m}$ away from a point source is $40\text{dB}$. Threshold intensity of hearing is ${10}^{-12}{\text{W/m}}^2$. If there is no loss of power in air, then intensity level at $10\text{m}$ from the source is

• A. $30\text{dB}$
• B. $10\text{dB}$
• C. $20\text{dB}$
• D. $40\text{dB}$

Answer 1

The correct option is C $20\text{dB}$

Given that,
$I_0={10}^{-12}{\text{W/m}}^2,r_1=1\text{m},r_2=10\text{m},{\beta }_1=40\text{dB}$

We know that, for a point source,

Intensity $\left(I\right)\propto \frac{1}{r^2}$

$\Rightarrow \frac{I_1}{I_2}={\left(\frac{r_2}{r_1}\right)}^2={\left(\frac{10}{1}\right)}^2=100$
So, from the relation between the loudness and intensity of sound, we can write that,

${\beta }_1=10{\log }_{10}\frac{I_1}{I_0}$ and ${\beta }_2=10{\log }_{10}\frac{I_2}{I_0}$
$\Rightarrow {\beta }_2-{\beta }_1=10{\log }_{10}\frac{I_2}{I_1}$

$\Rightarrow {\beta }_2=10{\log }_{10}\left(\frac{1}{100}\right)+40\Rightarrow {\beta }_2=-20+40=20\text{dB}$

Thus, option (c) is the correct answer.