Coulomb's law

Lore & Background

Coulomb's law states that the magnitude of the attractive or repulsive electrostatic force between two point charges is directly proportional to the product of the magnitudes of their charges and inversely proportional to the square of the distance between them. Coulomb discovered that bodies with like electrical charges repel, and also showed that oppositely charged bodies attract according to an inverse-square law. The force is along the straight line joining the two bodies; if the charges have the same sign, they repel; if different signs, they attract.

Reader's Guide

Coulomb's law is a foundational principle in electromagnetism, providing the first quantitative description of electrostatic force. Its inverse-square form is similar to Newton's law of universal gravitation, but unlike gravity, electrostatic forces can be either attractive or repulsive. The law has been tested extensively and upheld on scales from 10⁻¹⁶ m to 10⁸ m. It can be used to derive Gauss's law, and vice versa; for a single point charge at rest, the two laws are equivalent. The time-dependent generalization of Coulomb's law is given by Jefimenko's equations, which describe electric and magnetic fields generated by time-dependent charge and current distributions.

Did You Know?

• The law states that electrostatic force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.
• Coulomb used a torsion balance consisting of a bar suspended from its middle by a thin fiber to measure the force between charged balls.
• The law has been tested and upheld on scales from 10⁻¹⁶ m to 10⁸ m.

The Mathematical Heart of the Law

At its core, Coulomb's law provides a precise mathematical relationship governing how two stationary electrically charged particles interact. The magnitude of the force they exert on one another scales directly with the product of their individual charge magnitudes while scaling inversely with the square of the separation distance between them. This force always acts along the straight line connecting the two bodies, and its character—attraction or repulsion—depends entirely on whether the charges share the same sign or carry opposite signs. The proportionality constant that completes the equation varies depending on which unit system is chosen for charge, and is sometimes referred to as the Coulomb constant. The law's significance extends far beyond a single formula: it served as a foundational pillar for the broader theory of electromagnetism and arguably marks its starting point, because it gave physicists a concrete, quantitative handle on the amount of electric charge residing in a particle for the first time.

A Two-Millennium Road to Discovery

The story of how humanity came to understand electrostatic force spans more than two millennia. Around 600 BC, Thales of Miletus observed that rubbing amber with fur allowed it to attract small objects like feathers, marking the first recorded description of static electricity. Centuries later, in 1600, William Gilbert carefully separated the lodestone effect from friction-generated electricity and coined the Neo-Latin term electricus, derived from the Greek word for amber, which eventually gave rise to the English words electric and electricity. By the mid-eighteenth century, several investigators suspected the electrical force diminished with distance in the same inverse-square fashion as gravity. Henry Cavendish, in the early 1770s, had already worked out the distance and charge dependence in private notes but never published.

The Torsion Balance and Coulomb's Method

The device consisted of an insulating rod with a small metal-coated sphere fixed to one end, the whole assembly hung from a silk thread that acted as an extraordinarily weak torsion spring. To test his hypothesis, Coulomb charged this suspended ball with a known quantity of static electricity, then brought a second ball carrying the same polarity close to it. The mutual repulsion between the two like-charged spheres twisted the delicate fiber through a measurable angle. By varying the separation distance and the amount of charge on each ball, Coulomb could read off how the force scaled with both quantities. His measurements confirmed that the mutual repulsion between two spheres carrying identical types of charge diminished in proportion to the square of their separation. He then extended the same methodology to oppositely charged bodies, demonstrating that attraction obeyed the identical inverse-square relationship. This careful experimental verification transformed what had been a conjecture into a published, quantitative law of nature.

Kinship with Gravity, Gauss, and Beyond

Coulomb's law occupies a unique position in physics because of its structural kinship with Newton's inverse-square law of universal gravitation. Both describe forces that diminish with the square of distance, yet they differ in crucial ways: gravitational interaction is exclusively attractive and enormously weaker, whereas electrostatic interaction can either pull charges together or push them apart depending on their signs. The law also stands in a remarkable duality with Gauss's law; each can be derived from the other, and for a single stationary point charge the two formulations express the identical physical content in different mathematical languages. Experimental scrutiny has been exhaustive: observations have confirmed the inverse-square behavior across an astonishing range, from distances as small as 10 to the minus 16th power of a meter up to 10 to the 8th power of a meter. When charges and currents are no longer static, the picture must be extended. Jefimenko's equations provide the time-dependent generalization, describing how moving distributions of charge and current generate both electric and magnetic fields, thereby linking Coulomb's static framework to the fuller theory of electromagnetism.