Question

The dimensional formula for electric flux density is given by:
A. $[ML{{T}^{-3}}A-1]$
B. $[ML{{T}^{3}}A-1]$
C. $[ML{{T}^{-3}}A1]$
D. $[ML{{T}^{3}}A1]$

Answer 1

The unit of electric flux density is the same as that of the electric field. So, the dimensional formula for electric flux density is the same as the dimensional formula for electric field.

Complete step by step solution:
In an electric field, the ratio of electric flux through a surface to the area of the surface is called the ‘electric flux density’ at the location of the surface.
Electric flux density $=\dfrac{\text{Electric flux}}{\text{Area}}=\dfrac{EA\cos \theta }{A}$
Where E is the magnitude of electric field, A is the area of the surface and $\theta$ is the angle between electric field vector $\overrightarrow{E}$ and the area vector $d\overrightarrow{A}$.
For a plane surface normal to the electric field, $\theta =0$, so
Electric flux density $=\dfrac{EA}{A}=E$

Therefore, the unit of electric flux density is the same as that of the electric field.
Now, the S.I. the unit of electric field is ‘newton/coulomb’, $\text{N/C}$.
So,
$\dfrac{\text{newton}}{\text{coulomb}}=\dfrac{\text{kg}\times \text{metre second}{{\text{d}}^{-2}}}{\text{ampere}\times \text{second}}=\text{kg metre second}{{\text{d}}^{-3}}\text{amper}{{\text{e}}^{-1}}$
The dimension of length is denoted by $[L]$, mass by $[M]$, time by $[T]$ and current by $[A]$.
Therefore, the dimensional formula of electric field is $[ML{{T}^{-3}}{{A}^{-1}}]$, which is also the dimensional formula of electric flux density.

So, option A. is the correct answer.

Note: $\theta$ is a dimensionless constant.
Another S.I. the unit of electric flux density (or electric field) is volt/metre ($\text{V/m}$).
$\dfrac{\text{volt}}{\text{metre}}=\dfrac{\text{kg}\times \text{metr}{{\text{e}}^{2}}\times \text{secon}{{\text{d}}^{-3}}\times \text{amper}{{\text{e}}^{-1}}}{\text{metre}}=\text{kg metre second}{{\text{d}}^{-3}}\text{amper}{{\text{e}}^{-1}}$
The dimensional formula of electric flux density is $[ML{{T}^{-3}}{{A}^{-1}}]$.
The dimensions of a physical quantity are the powers to which the fundamental units are raised in order to obtain the derived unit of that quantity. In other words, the dimensional formula of a physical quantity remains the same, irrespective of the unit of measurement.