Coulomb’s Law: Definition, Equation & Derivation

Coulombs law

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. Charles_de_coulomb

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, ‘q1‘ and ‘q2‘; separated by a distance ‘r.’ The force of attraction or repulsion is ‘F’;

F  ∝ q1qor F  ∝  1/r2

Adding the constant of proportionality, we get;

coulombs law

In the above equation, k= 1/4 π ε0. εdescribes the permittivity of a vacuum. The value of k is nearly 9 × 109 Nm2/ C^2; if we consider the value of εin 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.

Force in Coulombs law

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 vector form. Again, considering the two charges q1 and q2; these charges have position vectors r1 and r2 respectively.

When the two charges q1 and qcarry the same sign, a repulsive force prevails between them. The force F12 is because of the force of q2 on q1; the force F21 is because of the force of qon the charge q2. The vector from q1 to q2 will be r21; r21 = r2 – r1.

Now, we can easily denote the direction of the vector from r1 to  r2 and from r2 to r1;

vectors in coulombs law

The force on q2 due to q1 in the vector form can, now, be written as:

Force vector in Coulombs law

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. F12 is the repulsive force because of q2 on q1; F21 is the repulsive force on qbecause of q1.

Force in Coulombs law

  • r12 is the position vector for force F12; r21 is the position vector for F21.

Force in Coulombs law

  • The signs of the vectors r21 and r12 are opposite in nature; therefore, they render forces with opposite signs. Now, we get to see that Newton’s Third Law of Motion also validates 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.


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