Question

A small speaker delivers $2\text{W}$ of audio output. At what distance from the speaker will one detect $120\text{dB}$ intensity sound? [Given reference intensity of sound as ${10}^{-12}{\text{W/m}}^2]$

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

Answer 1

The correct option is A $40\text{cm}$

Given:
$\beta =120\text{dB};I_0={10}^{-12}{\text{W/m}}^2$

Loudness of sound in decibel is given by,

$\beta =10{\log }_{10}(\frac{I}{I_0})$,

$\Rightarrow 120=10{\log }_{10}(\frac{I}{{10}^{-12}})$

$\Rightarrow 12={\log }_{10}(\frac{I}{{10}^{-12}})$

taking antilog

$\Rightarrow {10}^{12}=\frac{I}{{10}^{-12}}$

$\Rightarrow I=1{\text{W/m}}^2$

This is the intensity of sound reaching the observer.

Now, intensity, $I=\frac{P}{A}=\frac{P}{4\pi r^2}$

$\Rightarrow 1=\frac{2}{4\pi r^2}$

$\Rightarrow r=\sqrt{\frac{1}{2\pi }}=0.40\text{m}=40\text{cm}$

Hence, option $\left(a\right)$ is the correct answer.

Why this question? Note:- The lower limit of intensity for human hearing range is ${10}^{-12}{\text{W/m}}^2$ (or $0\text{dB})$. Thus , it is chosen as the reference level.