Question

What are the dimensions of ${\text{K = }}\dfrac{1}{{4\pi {\varepsilon _o}}}$ ? (Given, N is the unit of force, with dimensions ${\text{ML}}{{\text{T}}^{ - 2}}$)
A. ${{\text{C}}^2}{{\text{N}}^{ - 1}}{{\text{M}}^{ - 2}}$
B. ${\text{N}}{{\text{M}}^2}{{\text{C}}^{ - 2}}$
C. ${\text{N}}{{\text{M}}^2}{{\text{C}}^2}$
D. Unitless

Answer 1

To find the dimensions of a physical quantity we need to first write down the formulas and relation between the given quantities in terms of distance, time and mass and then find the dimensions of each one of them.

Complete step by step answer:
We know that Coulomb’s law gives the electric force between the two point charges.
It is defined by
$F = K\dfrac{{{q_1}{q_2}}}{{{r^2}}}$
Where $K$ is Coulomb’s constant $= 9.0 \times {10^9}{\text{N}}{{\text{m}}^2}{\text{/}}{{\text{C}}^2}$, ${q_1}{\text{and }}{q_2}$ are the point charges and $r$ is the distance between the two charges.
So we can already get the S.I. units for $K$ above, substituting the dimensions in the above formula will give us the dimensions of $K$.Therefore,
${\text{dim }}[K] = \dfrac{{{\text{N}}{{\text{M}}^2}}}{{{{\text{C}}^2}}}$
Where we know the dimensions of ${\text{N}}$ as it is mentioned in the question which is the unit of force.

Hence option B is correct.

Additional information: Some basic rules for dimensional analysis are: Two physical quantities can only be equated if they have the same dimensions. Two physical quantities can only be added if they have the same dimensions. The dimensions of the multiplication of two quantities are given by the multiplication of the dimensions of the two quantities.

Note: Dimensions are denoted with square brackets. In mechanics, mass, length and time are the basic quantities and the units used for the measurement of these quantities are called fundamental units. A dimensional equation is an equation obtained by equating the physical quantities with its dimensional formula. The dimensional formula of various physical quantities is expressed in terms of fundamental units. The dimensional formula is an expression that shows which fundamental units are required to represent the unit of the given physical quantity.