The clocktower ("ghantaghar") of Dehradun is famous for the sound of its bell, which can be heard, albeit faintly, upto the outskirts of the city $8 \,km$ away. Let the intensity of this faint sound be $30 \,dB$. The clock is situated $80 \,m$ high. The intensity at the base of the tower is .............$\,dB$
KVPY 2020, Advanced
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(b)
Given, $r_{1}=8 \times 10^{3} \,m , L_{1}=30 \,dB$
$r_{2}=80 \,m , L_{2}=?$
As, $L=10 \log _{10}\left(\frac{I}{I_{0}}\right)$
Also, $\quad I=\frac{P}{A}=\frac{P}{4 \pi r^{2}} \Rightarrow I \propto \frac{1}{r^{2}}$
$\therefore L_{2}-L_{1} =10 \log _{10}\left(\frac{I_{2}}{I_{0}}\right)-10 \log _{10}\left(\frac{I_{1}}{I_{0}}\right)$ $=10 \log _{10}\left(\frac{I_{2}}{I_{1}}\right)$ $=10 \log _{10}\left(\frac{r_{1}}{r_{2}}\right)^{2}$
$L_{2}-30=10 \log _{10}\left(\frac{8000}{80}\right)^{2}$ $=20 \times 2=40$ $\Rightarrow \quad L_{2} =70 \,dB$
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