Question

A point source of power \[5\,{\text{W}}\] is emitting sound waves in a medium. If the medium does not absorb any energy, then the intensity of the sound wave at a distance of \[5\,{\text{m}}\] from the source is
A. \[5\,{\text{W}} \cdot {{\text{m}}^{ - 2}}\]
B. \[\dfrac{1}{5}\,{\text{W}} \cdot {{\text{m}}^{ - 2}}\]
C. \[\dfrac{1}{{25}}\,{\text{W}} \cdot {{\text{m}}^{ - 2}}\]
D. \[\dfrac{1}{{20\pi }}\,{\text{W}} \cdot {{\text{m}}^{ - 2}}\]

Answer 1

Hint:Use the formula for the intensity of the sound wave at a distance r from the source of the sound. This equation gives the relation between the power output of the sound source and the distance of the wave from the sound source.

Formula used:
The intensity of the sound wave at a distance from the source of sound is given by
\[I = \dfrac{P}{{4\pi {r^2}}}\] …… (1)
Here, \[I\] is the intensity of the source at a distance \[r\] from the source of sound and \[P\] is the power output of the sound source.

Complete step by step answer:
The power output \[P\] of the source of sound is \[5\,{\text{W}}\].
\[P = 5\,{\text{W}}\]
Determine the intensity of the sound wave at a distance of \[5\,{\text{m}}\] from the source of sound.
Rewrite equation (1) for the intensity of sound.
\[I = \dfrac{P}{{4\pi {r^2}}}\]

Substitute \[5\,{\text{W}}\] for \[P\] and \[5\,{\text{m}}\] for \[r\] in the above equation.
\[I = \dfrac{{5\,{\text{W}}}}{{4\pi {{\left( {5\,{\text{m}}} \right)}^2}}}\]
\[ \Rightarrow I = \dfrac{5}{{4\pi \left( {25} \right)}}\]
\[ \Rightarrow I = \dfrac{5}{{100\pi }}\]
\[ \therefore I = \dfrac{1}{{20\pi }}\,{\text{W}} \cdot {{\text{m}}^{{\text{ - 2}}}}\]

Hence, the correct option is D.

Additional information:
The intensity of the sound wave is the ratio of the power of the sound wave to the area in which the sound wave travels.The larger the intensity of sound, larger is the amplitude of the sound and the listener will hear a larger sound. Any normal person with a normal hearing ability can hear the sound with minimum intensity \[{10^{ - 12}}\,{\text{W}} \cdot {{\text{m}}^{{\text{ - 2}}}}\]. The equation (1) for the intensity of sound waves at a distance r from the source is also used for the intensity of the light wave at a distance r from the source of light.

Note:The students may substitute the value of \[\pi \] while calculating the intensity of the sound wave using equation (1). But if the value of \[\pi \] is substituted in the respective equation, the final intensity of the sound after solving the equation will be different and the students may get confused that the correct answer is not given in the options. So avoid directly substituting the value of\[\pi \] in the equation.