Question

A 55 dB sound wave strikes an eardrum whose area is \(5.0 \times {10^{ - 5}}\;{{\rm{m}}^2}\). (a) How much energy is received by the eardrum per second? (b) At this rate, how long would it take your eardrum to receive a total energy of 1.0 J?

Short answer

(a) The energy received by the eardrum per second is \(1.58 \times {10^{ - 11}}\;{\rm{J/s}}\). (b) The time taken by the eardrum to take complete energy is \(2005.25\;{\rm{yr}}\).

Step-by-step solution

Understanding the concept for power of sound wave

In this problem, the power of the sound wave is calculated by using the product of sound intensity and the area covered by sound.

Given data

The level of sound wave is \(\beta = 55\;{\rm{dB}}\).

The surface area of eardrum is \(A = 5.0 \times {10^{ - 5}}\;{{\rm{m}}^2}\).

The total energy received by eardrum is \(E = 1.0\;{\rm{J}}\).

The standard value of threshold intensity is \({I_0} = 1.0 \times {10^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}\).

Evaluating the intensity of sound, and the energy received by eardrum per second

The intensity of sound is calculated below:

\(\begin{aligned}{c}\beta = 10\log \left( {\frac{I}{{{I_0}}}} \right)\\I = {10^{\left( {\frac{\beta }{{10}}} \right)}} \times {I_0}\end{aligned}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}I = {10^{\left( {\frac{{55\;{\rm{dB}}}}{{10}}} \right)}} \times \left( {1.0 \times {{10}^{ - 12}}\;{\rm{W/}}{{\rm{m}}^2}} \right)\\I = 3.16 \times {10^{ - 7}}\;{\rm{W/}}{{\rm{m}}^2}\end{aligned}\)

The energy received by the eardrum per second is calculated below:

\(P = IA\)

Substitute the values in the above equation.

\(\begin{aligned}{c}P = \left( {3.16 \times {{10}^{ - 7}}\;{\rm{W/}}{{\rm{m}}^2}} \right)\left( {5.0 \times {{10}^{ - 5}}\;{{\rm{m}}^2}} \right)\\P = 1.58 \times {10^{ - 11}}\;{\rm{J/s}}\end{aligned}\)

Hence, the energy received by the eardrum per second is \(1.58 \times {10^{ - 11}}\;{\rm{J/s}}\).

Evaluating the time taken by eardrum to take complete energy

The time taken by the eardrum to take complete energy is calculated below:

\(t = \frac{E}{P}\)

Substitute the values in the above equation.

\(\begin{aligned}{c}t = \frac{{1.0\;{\rm{J}}}}{{1.58 \times {{10}^{ - 11}}\;{\rm{J/s}}}}\\t = 6.32 \times {10^{10}}\;{\rm{s}} \times \left( {\frac{{1\;{\rm{yr}}}}{{3.154 \times {{10}^7}\;{\rm{s}}}}} \right)\\t = 2005.25\;{\rm{yr}}\end{aligned}\)

Hence, the time taken by the eardrum to take complete energy is \(2005.25\;{\rm{yr}}\).