Coulomb's Law Explained

How electric charges attract and repel — gravity's far stronger electrical cousin, and how to calculate the force.

Electric charges exert forces on one another: like charges repel, opposites attract. Coulomb's law, named after the French physicist Charles-Augustin de Coulomb who measured it in the 1780s, tells us exactly how strong that force is. Strikingly, it has the very same mathematical shape as Newton's law of gravitation — an inverse-square law — even though the two forces are utterly different in origin and strength.

The equation

F = k_e q₁ q₂r² where F is the electric force, q₁ and q₂ are the charges, r is the distance between them, and k_e is Coulomb's constant

F — the electric force in newtons (N), attractive or repulsive along the line between the charges.
k_e — Coulomb's constant, about 8.99 × 10⁹ N·m²·C⁻².
q₁, q₂ — the two charges, in coulombs (C).
r — the distance between them, in metres.

Gravity's far stronger cousin

The resemblance to gravitation is no accident — both forces spread their influence through three-dimensional space, so both dilute as one over distance squared. But the resemblance ends at strength. The electric force between two protons is roughly 10³⁶ times stronger than the gravitational force between them. Gravity wins out over cosmic distances only because large bodies are electrically neutral, their positive and negative charges almost perfectly cancelling. Charge comes in two signs that balance; mass comes in only one, and so it always adds up.

Sign convention: if you keep track of the signs of q₁ and q₂, a positive result for F means repulsion and a negative result means attraction. Many people simply use the magnitudes and decide the direction by inspection — like repels like, opposites attract.

A worked example

Take two charges of one microcoulomb each (1 × 10⁻⁶ C) held one metre apart. The force is Coulomb's constant times the product of the charges divided by the distance squared, which works out to about 9 × 10⁻³ N — roughly nine millinewtons. That may sound tiny, but a microcoulomb is actually a substantial amount of charge; the forces between charges in atoms, separated by mere nanometres, are enormous on the atomic scale.

Try it yourselfOpen the calculator with F = ke*q1*q2 / r^2 ready to go.

Open calculator

The constant behind the constant

Coulomb's constant k_e is often written in terms of a more fundamental quantity, the permittivity of free space ε₀, through the relation that k_e equals one divided by four pi times ε₀. Permittivity describes how readily an electric field passes through empty space, and it appears throughout electromagnetism — including, remarkably, in the formula for the speed of light. The threads of electricity, magnetism and light are all woven from the same constants.

Key takeaways

Coulomb's law gives the force between electric charges and follows an inverse-square form.
Like charges repel, opposite charges attract.
The electric force is vastly stronger than gravity but is usually hidden because matter is electrically neutral.
Coulomb's constant is built from the permittivity of free space, ε₀.