Question

The dimension of intensity of wave is:
A). $\left[ M{{L}^{2}}{{T}^{-3}} \right]$
B). $\left[ M{{L}^{0}}{{T}^{-3}} \right]$
C). $\left[ M{{L}^{-2}}{{T}^{-3}} \right]$
D). $\left[ M{{L}^{2}}{{T}^{3}} \right]$

Answer 1

Hint: To find the dimension of any physical quantity first express it in terms of the fundamental quantities. Here, intensity can be expressed as power per unit area. Again, power can be expressed in terms of force and displacement. This way, express the intensity in terms of the fundamental quantities and then we can find the dimension of the quantity.

Complete step by step answer:
Intensity of waves can be defined as the energy carried by the wave per unit area per unit seconds or directly power of the wave per unit area.
$\text{Intensity = }\dfrac{\text{power}}{\text{area}}$
Now power can be defined as energy used per unit time or work done per unit time.
$\text{Power = }\dfrac{\text{energy}}{\text{time}}$
Again, energy can be defined as the force per unit displacement.
So, $\text{energy = force }\\!\\!\times\\!\\!\text{ displacement}$
Force can be found out as a product of mass and acceleration.
So,
$\text{power = }\dfrac{\text{energy}}{\text{time}}$
Again, energy is given as the product of force and displacement.
$\text{power = }\dfrac{\text{force }\\!\\!\times\\!\\!\text{ displacement}}{\text{time}}$
Force is defined as the product of mass and acceleration.
$\text{power = }\dfrac{\text{mass }\\!\\!\times\\!\\!\text{ acceleration }\\!\\!\times\\!\\!\text{ displacement}}{\text{time}}$
Now, the dimension of mass is $\left[ M \right]$
Dimension of acceleration is $\left[ {{L}^{1}}{{T}^{-2}} \right]$
Dimension of displacement is $\left[ L \right]$
Dimension of time is $\left[ T \right]$
So, the dimension of power will be
$\text{power }=\dfrac{\left[ M \right]\left[ L{{T}^{-2}} \right]\left[ L \right]}{\left[ T \right]}=\left[ M{{L}^{2}}{{T}^{-3}} \right]$
Now dimension of area is $\left[ {{L}^{2}} \right]$
So, dimension of intensity will be
$\text{intensity = }\dfrac{\text{power}}{\text{area}}=\dfrac{\left[ M{{L}^{2}}{{T}^{-3}} \right]}{\left[ {{L}^{2}} \right]}=\left[ M{{L}^{0}}{{T}^{-3}} \right]$
The correct option is (B).

Note: The SI unit of power is watt. The SI unit of intensity is watts per square-meter.
All the derived physical quantities can be expressed in terms of the fundamental quantities. The derived units are dependent on the 7 fundamental quantities. Fundamental units are mutually independent of each other.
Dimension of a physical quantity is the power to which the fundamental quantities are raised to express that physical quantity.