Coulomb’s Law – Vector Form, Limitations, Examples, Key Points


What is Coulomb’s Law?

According to Coulomb’s law, the force of attraction or repulsion between two charged bodies is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. It acts along the line joining the two charges considered to be point charges.

Coulomb’s Law
Coulomb’s Law

Coulomb’s Law Formula

Coulombs law 1In Short: F ∝ q1q2/d2


  • ε is absolute permittivity,
  • K or εr is the relative permittivity or specific inductive capacity 
  • ε0 is the permittivity of free space.
  • K or εr is also called a dielectric constant of the medium in which the two charges are placed.

Coulomb’s law is a quantitative statement about the force between two point charges.

When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and the charged bodies are treated as point charges.

Coulomb measured the force between two point charges and found that it varies inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of the two charges and acted along the line joining the two charges.

Thus, if two point charges q1, q2 are separated by a distance r in vacuum, the magnitude of the force (F) between them is given by

where k is a constant of proportionality, called electrostatic force constant. The value of k depends on the nature of the medium between the two charges and the system of units chosen to measure F, q,q2 and r.

where ε0 is called permittivity of free space.

Units of charges:

  • The SI unit of charge is coulomb. In the above equation, if q= q= 1C and r = 1m, then

  • In electrostatic cgs system, the unit of charge is known as electrostatic unit of charge (e.s.u. of charge) or statcoulomb (stat C)

  • In electromagnetic cgs system, the unit of charge is abcoulomb or electromagnetic unit of charge (e.m.u of charge).

Coulomb’s Law in Vector Form

  1. Let the position vectors of charges q1 and q2 be r1 and r2 respectively [see Fig 1].

Fig. 1 (a) Geometry and (b) Forces between charges

  • We denote force on q1 due to q2 by F12 and force on q2 due to q1 by F21.
  • The two-point charges q1 and q2 have been numbered 1 and 2 for convenience and the vector leading from 1 to 2 is denoted by r21:

In the same way, the vector leading from 2 to 1 is denoted by r12:

Coulomb’s force law between two point charges q1 and q2 located at r1 and r2 is then expressed as

The force F12 on charge q1 due to charge q2,

Thus, Coulomb’s law agrees with Newton’s third law.

Limitations of Coulomb’s Law

  • The law is applicable only for the point charges at rest.
  • Coulomb’s Law can only be applied in those cases where the inverse square law is obeyed.
  • It is difficult to implement Coulomb’s law where charges are in arbitrary shape because in such cases, we cannot determine the distance between the charges.
  • The law can’t be used directly to calculate the charge on the big planets.

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