Question

If the intensity is increased by a factor of $20,$ then how many decibels in the sound level increased?
(A) $18$
(B) $13$
(C) $9$
(D) $7$

Answer 1

Hint The power carried by a sound wave per unit area is called the sound intensity or acoustic intensity. It is perpendicular to the direction of the area. Sound pressure and sound intensity are not the same though they are related. Human hearing is affected by sound intensity.

Formula used:
$dB = 10{\log _{10}}\dfrac{I}{{{I_0}}}$(Where $dB$ stands for the decibels in sound, ${I_0}$ stands for the initial intensity and $I$ stands for the final intensity of the sound)

Complete step by step solution:
Decibel is the unit used to measure the loudness of sound. The decibel can be obtained by taking the logarithm of the ratio of the intensities of the sound.
Let us consider the initial intensity here be, ${I_0}$ and the final intensity be $I$
Now, we know that the change in decibel is,
$dB = 10{\log _{10}}\dfrac{I}{{{I_0}}}$
It is given that the intensity is increased by a factor of $20$ now we can write the final intensity to be,
$I = 20{I_0}$
Now substituting these values in the expression for decibel, we get
$dB = 10{\log _{10}}\dfrac{{20{I_0}}}{{{I_0}}}$
$\Rightarrow dB = 10 \times {\log _{10}}\left( {20} \right)$
$\Rightarrow dB = 10 \times 1.301$
$\Rightarrow dB = 13$

The correct answer is Option (B): $13$

Note
Decibel is used to measure the loudness of sound. It should be noted that the intensity of light and sound perceived by the human eyes and ears is calculated on a logarithmic scale, not on a linear scale. As the intensity of the sound increases the decibel will also increase.