According to Coulomb's law, the magnitude of force of attraction or repulsion between any two point charges at rest is directly proportional to the product of the magnitude of charges and inversely proportional to the square of the distance between them.
Let $r$ be the distance between two static point charges $q_1$ and $q_2$. Then according to Coulomb's law, the force between charges is given by
$F \propto q_1 q_2$
$F \propto \frac{1}{r^2}$ or $F \propto \frac{q_1 q_2}{r^2}$
$F = \frac{k q_1 q_2}{r^2} \quad \dots (1)$
where $k$ is the constant of proportionality, known as electrostatic force constant} or Coulomb's constant}.
The value of electrostatic force constant ($k$) depends upon (i) the system of unit used and (ii) the nature of medium in which charges are placed.
Coulomb's Law in SI :
In SI, $k = \frac{1}{4\pi \epsilon} = \frac{1}{4\pi \epsilon_0 \epsilon_r}$, where, $\epsilon_0 = 8.854 \times 10^{-12} \text{C}^2 \text{N}^{-1} \text{m}^{-2}$ and is called absolute permittivity of free space (i.e., vacuum or air). $\epsilon_r$ is the relative permittivity of the medium. It has no unit. It is a pure number. The permittivity of a medium is given by $\epsilon = \epsilon_0 \epsilon_r$.
Hence, in SI, Coulomb's law for a medium other than air can be written as
$F_m = \frac{1}{4\pi \epsilon_0 \epsilon_r} \frac{q_1 q_2}{r^2} \quad \dots (2)$
In free space or vacuum, $\epsilon_r = 1$. Hence eqn. (2) becomes
$F_a = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2} \quad \dots (3)$
where
$\frac{1}{4\pi \epsilon_0} = 9 \times 10^9 \text{ N m}^2 \text{C}^{-2}$
Using eqn. (3), the magnitude of electrostatic force between two point charges $q_1$ and $q_2$ separated by a distance $r$ in free space can be calculated.
In CGS system, $k = \frac{1}{K}$, where K is dielectric constant of a medium.
Hence, Coulomb's law in CGS system can be written as
$F_m = \frac{1}{K} \frac{q_1 q_2}{r^2} \quad \dots (4)$
For air or vacuum (i.e., free space), $K = 1$. Hence, eqn. (4) becomes
$F_a = \frac{q_1 q_2}{r^2} \quad \dots (5)$
S.I. Unit of $\epsilon_0$
We know that,
$F = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2}$
$ \epsilon_0 = \frac{1}{4\pi F} \frac{q_1 q_2}{r^2}$
Now, in SI, unit of $q_1$ and $q_2$ (i.e., charge) is coulomb (C), the unit of force (F) is newton (N) and the unit of distance (r) is metre (m). Therefore, unit of $\epsilon_0$ in SI is given by
$ \epsilon_0 = \frac{\text{C}^2}{\text{N m}^2} = \text{C}^2 \text{N}^{-1} \text{m}^{-2}$
Dimension of $\epsilon_0$
We know,
$\epsilon_0 = \frac{q_1 q_2}{4\pi F r^2}$
Now
$[q_1]= [q_2] = [IT](\because I = q/t)$
$[F]= [MLT^{-2}]$
$[r] = [L]$
$\therefore [\epsilon_0] = \frac{[IT][IT]}{[MLT^{-2}][L]^2} = [M^{-1}L^{-3}T^4A^2]$
Limitations of Coulomb's Law :
1. It holds good for point charges at rest.
2. It is basically an experimental law.
3. For distances less than $10^{-15}$ m, it loses its validity.
4. It is applicable upto a few kilometres only.
5. It is a medium dependent law.
6. It is not a universal law.