Question

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 $3dB$ is: $({\log }_{10}2$ $=0.3$ )

• A. $\frac{1}{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{4}$
• D. $\frac{2}{3}$

Answer 1

Answer: (B)

$dB=10\log \left(\frac{I}{I_0}\right)$
$=10\log \left(\frac{K/r^2}{I_0}\right)$
$=10\left[\log \left(K^{\prime }\right)-2\log r\right]$
$d{B}_1=10\left(\log K^{\prime }-2\log r_1\right)$
$d{B}_2=10\left(\log K^{\prime }-2\log r_2\right)$
$3=d{B}_1-d{B}_2=20\log \left(\frac{r_2}{r_1}\right)$
$(0.3)=\log {\left(\frac{r_2}{r_1}\right)}^2$
$\Rightarrow \left(\frac{r_1}{r_2}\right)=\frac{1}{\sqrt{2}}$