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Decibels

The power of ...

Question

The power of a loud speaker is increased from 20 W to 400 W. What is the intensity increase as compared to the original value?

A

13 dB

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B

7 dB

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C

4 dB

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D

2 dB

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Solution

The correct option is A 13 dB
As intensity of wave is directly proportional to power, i.e $I \propto P$
Loudness of sound, $L = 10 l o g_{10} \frac{I}{I_{o}}$
$⟹ L_{2} - L_{1} = 10 l o g_{10} \frac{I_{2}}{I_{1}}$
Thus, $L_{2} - L_{1} = 10 l o g_{10} \frac{P_{2}}{P_{1}}$
$L_{2} - L_{1} = 10 l o g_{10} \frac{400}{20}$

$L_{2} - L_{1} = 10 l o g_{10} 20$ $(l o g_{10} 20 = 1.3)$

$⟹ L_{2} - L_{1} = 13 d B$

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