Question

The dimensions of permittivity (\[{{\varepsilon }_{0}}\]) are ______. Take Q as the dimension of charge.
A. \[{{M}^{1}}{{L}^{-2}}{{T}^{-2}}{{Q}^{-2}}\]
B. \[{{M}^{-1}}{{L}^{-3}}{{T}^{2}}{{Q}^{2}}\]
C. \[{{M}^{-1}}{{L}^{2}}{{T}^{-3}}{{Q}^{-1}}\]
D. \[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}{{Q}^{-2}}\]

Answer 1

Hint: In this question we have been asked to find the dimensions of permittivity. It is given that the dimension of charge is Q. We know that Coulomb's law gives us the relation between force, charge, distance between charge and the charge of particles and the permittivity. Therefore, using Coulomb's law we shall find the dimensions of permittivity. Permittivity is the ability of a material to store electrical potential energy under the influence of an electric field.

Formula used:
\[F=k\dfrac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}\]
Where,
\[k=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\]

Complete step by step answer:
From the coulomb's law, we know that electrostatic force is given by,
\[F=k\dfrac{{{q}_{1}}{{q}_{2}}}{{{r}^{2}}}\]
Where
\[k=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\]
\[{{\varepsilon }_{0}}\] is the permittivity
Now, to calculate dimensions of permittivity,
We know that
\[F=ma\]
Therefore, units of force can be given as \[kgm/{{s}^{2}}\]. Therefore, the dimensions of force are given as,
\[F={{M}^{1}}{{L}^{1}}{{T}^{-2}}\] …………….. (1)
Writing the Coulomb's law as dimensional formula,
Since, it is given that dimension of charge is Q and that of r is L
\[{{M}^{1}}{{L}^{1}}{{T}^{-2}}=k\dfrac{{{Q}^{2}}}{{{L}^{2}}}\]
On solving,
\[\left[ K \right]={{M}^{1}}{{L}^{3}}{{T}^{-2}}{{Q}^{-2}}\]
We know that,
\[k=\dfrac{1}{4\pi {{\varepsilon }_{0}}}\]
Therefore,
\[\left[ {{\varepsilon }_{0}} \right]={{M}^{-1}}{{L}^{-3}}{{T}^{2}}{{Q}^{2}}\]
Therefore, the correct answer is option B.

Note:
Coulomb’s law states that electrical force between two charged particles is directly proportional to product of charge on the objects and inversely proportional to the square of the distance between both particles. As the distance between the charge increases, the electric field and the force decreases. This force acts along the line joining the two particles. Permittivity is the measure of polarizability of a dielectric.