Coulomb’s law was published by the French physicist, Charles Augustin de Coulomb. The Coulomb’s law was critical in the development of the theory of electromagnetism. Moreover, our world is in existence only because of the forces of attraction and repulsion. The particles in the universe, as well as our environment, remain in a balanced form only because of the forces of attraction; this renders one of the practical applications of the Coulomb’s law.

Let us discuss Coulomb’s law in more detail.

**Coulomb’s Law**

Coulomb’s law states that the electrostatic force between any two points is directly proportional to the product of the magnitude of these charges and inversely proportional to the square of the distance between them.

- Coulomb’s law gives us an idea about the amount of force between any two charged points separated by some distance.
- When we say point charge, we are actually referring to the size of linearly charged bodies; which is very small as compared to the distance between them. For easier calculations of the force of attraction and repulsion, we consider them to be point charges.

We can also prove Coulomb’s law. Let us consider two charges, ‘q_{1}‘ and ‘q_{2}‘; separated by a distance ‘r.’ The force of attraction or repulsion is ‘F’;

**F ∝ q _{1}q_{2 }or F ∝ 1/r^{2}**

Adding the constant of proportionality, we get;

In the above equation, k= 1/4 π ε_{0}. ε_{0 }describes the permittivity of a vacuum. The value of k is nearly 9 × 10^{9} Nm^{2}/ C^^{2}; if we consider the value of ε_{0 }in SI units as 8.854 × 10^^{-12} C^^{2} N^^{-1} m^^{-2}.

Moreover, this theory also describes that like charges repel each other and opposite charges attract each other.

**Vector Form of Coulomb’s Law**

The physical quantities fall into two categories;

- Scalar: with magnitude only
- Vector: with a magnitude as well as direction

Force is a vector quantity. It has a magnitude and direction. Therefore, Coulomb’s law can also be written in the vector form. Again, considering the two charges q_{1} and q_{2}; these charges have position vectors r_{1} and r_{2} respectively.

When the two charges q_{1} and q_{2 }carry the same sign, a repulsive force prevails between them. The force F_{12} is because of the force of q2 on q_{1}; the force F_{21} is because of the force of q_{1 }on the charge q_{2}. The vector from q_{1} to q_{2} will be r_{21}; r_{21 }= r_{2} – r_{1}.

Now, we can easily denote the direction of the vector from r_{1} to r_{2} and from r_{2} to r_{1};

The force on q_{2} due to q_{1} in the vector form can, now, be written as:

The above equation represents the Coulomb’s law in the vector form. However, while following the vector form, certain points should be taken into consideration:

- The vector form of the Coulomb’s law is independent of the nature of the sign carried by the charges because of the fact that both the forces are opposite in nature. F
_{12}is the repulsive force because of q_{2}on q_{1}; F_{21}is the repulsive force on q_{2 }because of q_{1}.

- r
_{12}is the position vector for force F_{12}; r_{21}is the position vector for F_{21}.

- The signs of the vectors r
_{21}and r_{12}are opposite in nature; therefore, they render forces with opposite signs. Now, we get to see that the Newton’s Third Law of Motion also validates the Coulomb’s law. Newton’s Third law states that to every action, there is an equal and opposite reaction. - Moreover, Coulomb’s law describes the force between the two charges only when they are present in the vacuum; because the charges in the vacuum are free from interference from other particles.

**Limitations of Coulomb’s Law**

- The charges must have symmetric distribution; preferably spherical. The charges can either be point charges or spherical metal.
- The point charges must be static and distinct.

why we take square of area in coulombs law? what happen if would take single area?