Question

Decibel (dB) is a unit of loudness of sound. It is defined in a manner such that when the amplitude of the sound is multiplied by a factor of $\sqrt{10}$, the decibel level increases by 10 units. Loud music of 70 dB is being played at a function. To reduce the loudness to a level of 30 dB, the amplitude of the instrument playing music has to be reduced by a factor of:

• A. 10
• B. $10\sqrt{10}$
• C. 100
• D. $100\sqrt{100}$

Answer 1

The correct option is C

100

Loudness of sound,

$\beta =10log\left(\frac{I}{I_0}\right)dBwhere{I}_0={10}^{-12}W{m}^{-2}$ is the reference unit. Intensity $I\propto a^2$ where a = amplitude.
For a sound of 70 dB and 30 dB let Intensity be $I_1$ and $I_2$ respectively.
For $\beta =70dB:$
$70=10log\left(\frac{I_1}{{10}^{-12}}\right)$
${10}^7=\left(\frac{I_1}{{10}^{-12}}\right)$
∴ $I_1={10}^{-5}$
For $\beta =30dB:$
$30=10log\left(\frac{I_2}{{10}^{-12}}\right)$
${10}^3=\left(\frac{I_2}{{10}^{-12}}\right)$
∴ $I_2={10}^{-9}$
The intensity reduces by a factor of ${10}^4$
$I\propto a^2$
∴ Amplitude reduces by a factor of ${10}^2$