Question

An increase in intensity level of 1dB implies an increase in the intensity by(given $anti{{\log }_{10}}0.1=1.2589$
a) 1%
b) 3.01%
c) 26%
d) 0.1%

Answer 1

Hint: Decibel (dB) is a relative unit of measurement corresponding to one tenth of bel(B). it is given in the question that the intensity level on the decibel scale is by 1dB. Hence we can use the decibel scale i.e. $L=10\log \left( \dfrac{I}{{{I}_{0}}} \right)$ where ${{I}_{0}}$ is the intensity for threshold hearing and I is the intensity corresponding to a particular value on the decibel scale to calculate the increase in intensity when the value on the scale increases by 1dB.

Complete answer:
Let the initial reading on the decibel scale be M. Hence the corresponding intensity i.e. (${{I}_{1}}$) for that is given by,
$M=10\log \left( \dfrac{{{I}_{1}}}{{{I}_{0}}} \right)...(1)$ Now in the question it is given that the value of the scale increases by 1dB, hence the intensity(${{I}_{2}}$) corresponding to that is given by,
$M+1=10\log \left( \dfrac{{{I}_{2}}}{{{I}_{0}}} \right)...(2)$ . Now let us subtract equation 1 from 2

$\begin{align}
  & \Rightarrow M+1-M=10\log \left( \dfrac{{{I}_{2}}}{{{I}_{0}}} \right)-10\log \left( \dfrac{{{I}_{1}}}{{{I}_{0}}} \right) \\
 & \Rightarrow 1=10\log \left( \dfrac{\dfrac{{{I}_{2}}}{{{I}_{0}}}}{\dfrac{{{I}_{1}}}{{{I}_{0}}}} \right)=10\log \left( \dfrac{{{\text{I}}_{\text{2}}}}{{{\text{I}}_{\text{1}}}} \right) \\
 & \Rightarrow \dfrac{1}{10}=\log \left( \dfrac{{{\text{I}}_{\text{2}}}}{{{\text{I}}_{\text{1}}}} \right) \\
 & \Rightarrow 0.1=\log \left( \dfrac{{{\text{I}}_{\text{2}}}}{{{\text{I}}_{\text{1}}}} \right) \\
 & \Rightarrow \dfrac{{{\text{I}}_{\text{2}}}}{{{\text{I}}_{\text{1}}}}={{10}^{0.1}}=1.26 \\
 & \Rightarrow {{\text{I}}_{\text{2}}}=1.26{{\text{I}}_{\text{1}}} \\
\end{align}$

Hence the increase in the intensity is 26%.

Note:
The Decibel scale is used to the ratio of one power to another on a logarithmic scale, The logarithmic quantity being called the power level or field level, respectively. The definition of decibel is based on the measurement of power in telephony of the early 20th century in the bell system in the United states. This was basically named in honor of Alexander Graham Bell.. Today the decibel is used for a wide variety of measurements in science and engineering most prominently in acoustics, electronics etc. In electronics, the gains of amplifiers, attenuation of signals and signal to noise ratios are often expressed in decibels.