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

Hint: Intensity of a wave is the energy transferred per unit area in one unit of time. i.e. $I={\displaystyle \frac{E}{At}}$. 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:
$I={\displaystyle \frac{E}{At}}$
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. $I={\displaystyle \frac{E}{At}}$
Therefore, the dimensional formula of intensity will be $\left[I\right]=\left[{\displaystyle \frac{E}{At}}\right]={\displaystyle \frac{\left[E\right]}{\left[A\right]\left[t\right]}}$ …. (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).
$\Rightarrow \left[I\right]={\displaystyle \frac{\left[E\right]}{\left[A\right]\left[t\right]}}={\displaystyle \frac{\left[M{L}^2{T}^{-2}\right]}{\left[L^2\right]\left[T\right]}}=\left[M{L}^0{T}^{-3}\right]$.
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 ${\displaystyle \frac{kg{m}^2{s}^{-2}}{m^{2}s}}=kg{m}^0{s}^{-3}$.
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 $\left[M{L}^0{T}^{-3}\right]$.