Coulomb’s Law: Definition, Formula, & Constant

Contents

💡

Key learnings:

Coulomb’s Law Definition: Coulomb’s law defines the force between two stationary, electrically charged particles, known as the electrostatic force.
Electrostatic Force: The electrostatic force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
Coulomb’s Law Formula: The formula for Coulomb’s law is F = k * (Q1 * Q2) / d², where F is the force, Q1 and Q2 are the charges, d is the distance, and k is Coulomb’s constant.
Coulomb’s Constant: Coulomb’s constant (k) in a vacuum is approximately 8.99 x 10⁹ N m²/C², and it varies with the medium.
Historical Background: Charles-Augustin de Coulomb formalized Coulomb’s law in 1785, building on earlier observations by Thales of Miletus.

What is Coulomb’s Law?

Coulomb’s law (also known as Coulomb’s inverse-square law) is a law of physics that defines the amount of force between two stationary, electrically charged particles (known as the electrostatic force). Coulomb’s law was discovered by Charles-Augustin de Coulomb in 1785. Hence the law and the associated formula was named after him.

Coulomb’s Law Definition

Coulomb’s law states that the magnitude of the electrostatic force between two stationary point charges is proportional to the product of their charge magnitudes and inversely proportional to the square of their separation. The force acts along the line joining the charges. Like signs repel and unlike signs attract.

Doubling the distance reduces the force magnitude to one quarter, which is why this relation is called Coulomb’s inverse-square law.

The scalar formula below gives the force magnitude. Direction must be handled separately with the charge signs and a unit vector. For more than two charges, calculate each pairwise force and add the force vectors by superposition.

Coulomb’s Law Formula

Let Q₁ and Q₂ be two point charges.
Let d be the distance between their positions.

For a homogeneous, linear and isotropic medium, let its permittivity be ε_oε_r.

The force magnitude F can then be written as:

This form calculates the magnitude of the equal and opposite forces that the two point charges exert on each other. A full vector form also specifies the direction along their line of separation.

Statement of Coulomb’s Law

The video below gives another explanation of the charge and distance relationships:

Coulomb’s First Law

In a common teaching split, Coulomb’s first law states the direction rule: like point charges repel, while unlike point charges attract.

Coulomb’s Second Law

In the same teaching convention, Coulomb’s second law gives the magnitude: force is proportional to the product of the two charge magnitudes and inversely proportional to the square of their separation.

Where,

‘F’ is the magnitude of the attraction or repulsion force between the two point charges.
‘Q₁’ and ‘Q₂’ are the signed electric charges, measured in coulombs.
‘d’ is the distance between the point-charge positions.
‘k’ is 1/(4πε) for the stated medium. In the SI system, ε = ε_oε_r for a simple homogeneous, linear and isotropic dielectric, giving the form below.

The 2022 CODATA value is ε_o = 8.854 187 8188(14) × 10^-12 C²/(N·m²), which is equivalent to F/m.

For the simple dielectric model, Coulomb’s law becomes:

In vacuum, ε_r = 1. Air at ordinary conditions is often approximated as vacuum for basic calculations. The vacuum form of Coulomb’s law is:

Relative permittivity ε_r is the ratio of a material’s permittivity to vacuum permittivity. Its applicable value can depend on material, temperature and frequency; use an ε_r value measured for the relevant electrostatic conditions.

Principle of Coulomb’s Law

Two stationary point charges with opposite signs attract. At fixed separation, increasing the magnitude of either charge in the model increases the force magnitude in direct proportion.

For example, doubling either charge magnitude doubles the force. Doubling both charge magnitudes makes the force four times as large, provided the separation and medium remain unchanged.

With charges fixed at Q₁ and Q₂, reducing their separation increases the force magnitude, while increasing their separation reduces it.

If their separation is d, the force magnitude is inversely proportional to d². Multiplying d by any factor divides the force by the square of that factor.

A dielectric can polarize in response to the charges, changing the macroscopic force from its vacuum value. The simple ε_r factor applies to a uniform, linear and isotropic material that fills the region. Interfaces, nonlinear media and microscopic separations require a more complete field model.

Limitations of Coulomb’s Law

Coulomb’s law applies directly to stationary point charges. A spherically symmetric charge distribution can be treated as a point charge from outside the distribution.
Moving charges also produce magnetic fields, and time-varying interactions require the wider framework of electromagnetism.
For an extended or irregular charge distribution, divide it into charge elements and integrate their vector contributions. In material media, use boundary conditions and an appropriate constitutive model.

Who Invented Coulomb’s Law?

Around 600 BC, Thales of Miletus recorded that amber rubbed with fur attracted small objects. This was an early observation of static electricity, not an observation or formulation of Coulomb’s inverse-square law.

In 1785, French physicist Charles-Augustin de Coulomb used a calibrated torsion balance to measure forces between small charged spheres. His experiments established the inverse-square relation for electrostatic attraction and repulsion now known as Coulomb’s law.