A point source is emitting sound in all directions. The ratio of distance of two points from the point source where the difference in loudness levels is 3 dB is: ( $\log_{10} 2 = 0.3$ )

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$\frac{1}{\sqrt{2}}$

$\frac{2}{3}$

$\frac{1}{2}$

$\frac{1}{4}$

answer is B.

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Detailed Solution

dB = $10 \log (\frac{I}{I_{0}}) = 10 \log (\frac{\frac{K}{r^{2}}}{I_{0}})$

= 10[log $(K^{'}) - 2 \log r]$

${dB}_{1} = 10 (\log K^{'} - 2 \log r_{1})$

${dB}_{2} = 10 (\log K^{'} - 2 \log r_{2})$

3 = ${dB}_{1} - {dB}_{2} = 20 \log (\frac{r_{2}}{r_{1}})$

(0.3) = log ${(\frac{r_{2}}{r_{1}})}^{2} \Rightarrow (\frac{r_{1}}{r_{2}}) = \frac{1}{\sqrt{2}}$

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