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A small speaker delivers 2w of audio output. At a distance x cm from the speaker will one detect 120dB intensity of sound? [Given reference intensity of sound as $10^{- 12} w / m^{2}$ ]. Find value of x.

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answer is 40.

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Detailed Solution

Loudness of sound is given by $d B = 10 \log \frac{I}{I_{0}}$ [I is intensity of sound, $I_{0}$ is reference intensity of sound]
$∴ 120 = 10 \log (\frac{I}{I_{0}}) \Rightarrow I = 1 w / m^{2}$

Also $I = \frac{P}{4 π r^{2}} = \frac{2}{4 π r^{2}}$

$∴ r = \sqrt{\frac{2}{4 π}} = \sqrt{\frac{1}{2 π} m} = 0.399 m \approx 40 c m$

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