Question

The dimensions of intensity of wave are
$\text{A}\text{. }\left[ M{{L}^{2}}{{T}^{-3}} \right]$
$\text{B}\text{. }\left[ M{{L}^{0}}{{T}^{-3}} \right]$
$\text{C}\text{. }\left[ M{{L}^{-2}}{{T}^{-3}} \right]$
$\text{D}\text{. }\left[ M{{L}^{2}}{{T}^{-3}} \right]$

Answer 1

Intensity of a wave is the energy transferred per unit area in one unit of time. i.e. $I=\dfrac{E}{At}$. Find the dimensional formula of energy by using W=Fd. The dimensional formulas of area and time are $\left[ {{L}^{2}} \right]$ and [T] respectively. With this, find the dimensional formula of intensity.

Formula Used :
$I=\dfrac{E}{At}$
W=Fd

** Complete step-by-step answer :**
Intensity of a wave is defined as the energy transferred per unit area in one unit of time. Take an example of radiations emitted by a black body. The intensity of the radiations emitted by a black body is the amount of radiation energy emitted by the body through a unit area in one unit of time.
We can simply say that intensity is energy (E) divided by the product of area (A) and time (t).
i.e. $I=\dfrac{E}{At}$
Therefore, the dimensional formula of intensity will be $\left[ I \right]=\left[ \dfrac{E}{At} \right]=\dfrac{\left[ E \right]}{\left[ A \right]\left[ t \right]}$ …. (i).
Let us calculate the dimensional formula for energy [E]. We know that change in energy is equal to work done. And work (W) is equal to the product of force (F) and displacement (d).
Therefore, we get $\left[ E \right]=\left[ F \right]\left[ d \right]$.
Dimensional formula of force is $\left[ F \right]=\left[ ML{{T}^{-2}} \right]$.
Dimensional formula of displacement is [d]=[L].
Hence, $\left[ E \right]=\left[ F \right]\left[ d \right]=\left[ ML{{T}^{-2}} \right]\left[ L \right]=\left[ M{{L}^{2}}{{T}^{-2}} \right]$.
The dimensional formula of area is $\left[ A \right]=\left[ {{L}^{2}} \right]$.
The dimensional formula of time is [t]=[T].
Substitute the dimensional formulas of energy, area and time in equation (i).
$\Rightarrow \left[ I \right]=\dfrac{\left[ E \right]}{\left[ A \right]\left[ t \right]}=\dfrac{\left[ M{{L}^{2}}{{T}^{-2}} \right]}{\left[ {{L}^{2}} \right]\left[ T \right]}=\left[ M{{L}^{0}}{{T}^{-3}} \right]$.
This means that the dimensional formula of intensity is $\left[ M{{L}^{0}}{{T}^{-3}} \right]$.
Hence, the correct option is B.

Note :You can also find the dimensional formula with the help of the SI units of the quantity.
The unit of energy is $kg{{m}^{2}}{{s}^{-2}}$.
The unit of area is ${{m}^{2}}$.
The unit of time is s.
Hence, the unit of intensity will be $\dfrac{kg{{m}^{2}}{{s}^{-2}}}{{{m}^{2}}s}=kg{{m}^{0}}{{s}^{-3}}$.
Mass has the unit of kg, length has the unit of m and time has the unit of s.
Therefore, the dimensional formula of intensity is $\left[ M{{L}^{0}}{{T}^{-3}} \right]$.