Question

Spherical waves are emitted from a $1.0\;{\rm{W}}$ source in an isotropic non- absorbing medium. What is the wave intensity $1.0\;{\rm{m}}$from the source?
A. $\dfrac{1}{{3\pi }}\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}$
B. $\dfrac{1}{{4\pi }}\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}$
C. $\dfrac{1}{{2\pi }}\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}$
D. $\dfrac{\pi }{4}\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}$

Answer 1

The above problem can be resolved by using the fundamental expression for the intensity of the wave. This expression can be determined by taking the ratio of the magnitude of power and area for which the intensity is required to be calculated. In this problem, we are given the values of power and the distance upto which the intensity is required to be calculated.

Complete step by step answer:
Given:
The magnitude of power from the source is, $P = 1.0\;{\rm{W}}$.
The distance from the source to the target is, $r = 1.0\;{\rm{m}}$.
The mathematical expression for the wave intensity from the source is,
$\Rightarrow I = \dfrac{P}{A}$………………………………………. (1)
Here, A is the area of the spherical wave and its value is, $A = 4\pi {r^2}$.
We can find the value of intensity by substituting some basic formulas and values in equation 1 as,

$$\Rightarrow I = \dfrac{P}{A}\\\ \Rightarrow I = \dfrac{{1.0\;{\rm{W}}}}{{\left( {4\pi {r^2}} \right)}}\\\ \Rightarrow I = \dfrac{{1.0\;{\rm{W}}}}{{\left( {4\pi {{\left( {1.0\;{\rm{m}}} \right)}^2}} \right)}}\\\ \Rightarrow I = \dfrac{1}{{4\pi }}\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}$$

Therefore, the magnitude of wave intensity is $\dfrac{1}{{4\pi}}\;{\rm{W/}}{{\rm{m}}^{\rm{2}}}$ and hence from the given options, option (B) is correct.

Note: To solve the given problem, it is important to remember the concept and fundamentals of sound waves. Along with this, it is also necessary to keep in check with the mathematical formulation for the intensity of sound waves for any specific distance. Moreover, it is also necessary to remember the factor affecting the intensity of the sound wave. And with the general relation, it is much easier to predict the relation and dependency of such parameters. Moreover, the basic applications of this concept are also of great use to resolve such problems.