Electrostatic force - Coulomb’s law

Coulomb’s law is an experimental law published in 1786 by the French physicist Charles-Augustin de Coulomb that describes the interaction between two stationary point charges.

A point charge is an electric charge with irrelevant (very small) dimensions for the problem we try to solve.

Let’s consider two stationary point charges q1 and q2 like the two illustrated in the following figure:

The electrostatic force F12 experienced by a charge q2 in the vicinity of another charge q1 is:

Where r is the distance between the two charges, ur a unit vector in the radial direction and k is the Coulomb constant (or electrostatic constant). The value of this constant is dependent upon the medium that the charged objects are immersed in, and its value in vacuum is:

The Coulomb constant can be expressed as the following product of the vacuum permittivity (permittivity of free space or electric constant) ε0:

ε0 is one of the fundamental constants of Physics and in SI units its value is:

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The previous expression of the force between two stationary point charges is called the Coulomb’s law. The law is analogous to Newton’s law of universal gravitation but with some fundamental differences:

  • The gravitational force is always attractive, while the electrostatic force can be attractive or repulsive. It is attractive if the two charges have opposite signs (see left side of the upper figure) and it is repulsive if both charges have the same sign (right side of the upper figure).
  • The gravitational constant G is universal; its value is independent of the medium in which the masses are placed while the Coulomb constant k is dependent upon the medium that the charged objects are immersed in.
  • Both forces have long range as they obey the inverse square law; but the electrostatic force is far stronger than the gravitational force because, among other reasons, the Coulomb constant k is very large compared to the gravitational constant G.

To determine how strong is the electrostatic force relative to the gravitational force we will compare both forces between two protons located at one meter from each other (we will see later that the distance is irrelevant).

The magnitudes of the electrostatic and gravitational forces between the two protons are given respectively by:

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To compare both values we will simply divide them:

It means that the electrostatic force between the two protons is 36 orders of magnitude greater than the gravitational force. To get an ideal, the difference between the two forces is:

This is why when two charged particles interact, although they have a mass, we only take into account the effect of the electrostatic force between them.

When a charge (we will call it a test charge) is subjected to the effect of a set of charges, the net force it experiences is the sum of the individual forces exerted by each individual charge on it. Each one of these charges is called a source charge. The concepts of test charge and source charge will make sense when we will introduce the concept of electric field.

The electrostatic force experienced by a test charge q2 when it is subjected to an electric field E due to a source charge q1 is:

This expression is equivalent to the Coulomb’s law.

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