Question

A source emits sound waves isotropically. The intensity of the waves$\mathbf{2}\mathbf{.}\mathbf{50}\mathit{m}$from the source is${\mathbf{\text{1.91×10}}}^{\mathbf{\text{-4}}}{\mathbf{\text{W/m}}}^{\mathbf{\text{2}}}$. Assuming that the energy of the waves is conserved, find the power of the source.

Short answer

The power of the source is $1.50\times {10}^{-2}\text{W}$.

Step-by-step solution

Given data

• Intensity of the wave is$I=1.91\times {10}^{-4}{\text{W/m}}^{\text{2}}$
• Radius$r=2.5\text{0m}$

Determining the concept

Intensity of the source is given as the rate of energy (power) per unit area. Let, that source be centered at the sphere, having area$\text{4π}{\mathrm{r}}^2$. From this, calculate the power of the source. P is the power of the source, r is the radius of the sphere, and I is the intensity of the source.

The expression for the power is given by,

$P=I\times A$

Here is the intensity and A is the area.

Determining the power of the source 

To calculate the power of the source,

$P=I\times A$

Substitute $1.91\times {10}^{-4}{\text{W/m}}^{\text{2}}$for l and$4\pi r^2$ for A into the above equation,

$P=(1.91\times {10}^{-4}{\text{W/m}}^2)\times \left(4\pi r^2\right)$

Substitute for into the above equation,

role="math" localid="1661538919438" $\begin{array}{l}P=(1.91\times {10}^{-4})\times (4\times 3.14\times {\left(2.5\right)}^2)\\ =1.50\times {10}^{-2}\text{W}\end{array}$

Hence, the power of the source is $1.50\times {10}^{-2}\text{W}$.