Coulomb’s law: Force between two point charges

French scientist Charles Coulomb first gave the empirical law regarding the force between two point charges and is known as Coulomb’s law. The law states that-
The force between two point charges at rest is directly proportional to the product of the magnitude of the charges and inversely proportional to the square of the distance between them. The force acts along the line joining the two charges and its value depends on the nature of the intervening medium.

Let r be the distance between two point charges q₁ and q₂ then following Coulomb’s law the force of interaction (F) between them is given by,

F\propto q_1q_2 and F\propto\frac1{r^2}

i.e, F\propto \frac{q_1q_2}{r^2}

or, F=K\frac{q_1q_2}{r^2}

where K is a constant of proportionality, known as electrostatic force constant or Coulomb’s constant and depends upon the system of units used and the medium intervening the charges.

The charges q₁ and q₂ are scalars, but electrostatic force is vector, so Coulomb’s force acting on charge q₁ due to charge q₂ is given by,

\vec {F_12}K\frac{q_1q_2}{r^2} \widehat r_{21}

where \widehat r_{12} is the unit vector directed from q₁ to q₂

Similarly, force acting on q₁ due to q₂ is given by \vec {F_21}K\frac{q_1q_2}{r^2} \widehat r_{12}

where \widehat r_{12} is the unit vector in the direction from q₁ to q₂ .

i.e., the forces exerted by the two charges on each other are equal and opposite. So, Newton’s third
law of motion is obeyed.
As already been stated value of K depends on the system of units and nature of the intervening
medium, so in vacuum or free space, K=\frac1{4\pi\epsilon_0}=9\times{10}^9 Nm^2C^{-2} (in SI)

The value of K in CGS system is 1.