Coulomb's Law (F = k · Q₁ · Q₂ / r²)

In 2026, a high-school physics teacher running the rubbed-balloons demo wants their students to feel — not just compute — that the same law governs the hydrogen atom and the cling between balloon and wall. The Charge-Pair Simulator on this page exists to make that single law visible across 30 orders of magnitude of distance: from one femtometre between two protons in a nucleus, to a metre between two balloons, to a kilometre between a thundercloud and the ground. The same F = k · Q₁ · Q₂ / r² governs every step.

The law itself is named for Charles-Augustin de Coulomb (1736-1806), a French military engineer who, between 1785 and 1791, published seven memoirs to the Académie Royale des Sciences describing his torsion balance experiments. Coulomb suspended a horizontal rod from a fine silver wire inside a glass case to eliminate air currents, charged a small pith ball at one end of the rod, brought a second charged ball nearby, and measured the angular twist of the wire. From the elastic constant of the wire (which he calibrated independently) he extracted the force magnitude. His seventh memoir established the 1/r² scaling to within 4% accuracy — extraordinary for 1785 instrumentation. The original apparatus survives in the Musée des Arts et Métiers in Paris.

Henry Cavendish, working in private at his estate at Clapham Common, had independently established the 1/r² law a decade earlier (1772-1773) using a different experimental method — observing that a hollow conductor placed inside a charged shell experienced no net force, which mathematically requires the 1/r² distance law (Gauss's flux theorem in disguise). Cavendish's results were not published in his lifetime; James Clerk Maxwell rediscovered the manuscripts in 1879 and published them as The Electrical Researches of the Honourable Henry Cavendish, fully crediting Cavendish's 1772 priority.

Michael Faraday at the Royal Institution in London brought the field-theoretic interpretation in the 1830s and 1840s. Where Coulomb saw a force acting between two distant charges, Faraday saw each charge sourcing a field that permeated space — the force on a test charge was always local, exerted by the field at its location. Faraday's "lines of force" were the first field-theoretic picture in physics and laid the conceptual ground for what Maxwell would later formalize mathematically.

James Clerk Maxwell's 1865 paper "A Dynamical Theory of the Electromagnetic Field" and the 1873 Treatise on Electricity and Magnetism recast Coulomb's law as the static limit of a fully dynamical field theory. Maxwell's equation div(E) = ρ/ε₀ — Gauss's law — implies Coulomb's law for a point charge. The constant k = 1/(4πε₀) appears naturally when you integrate Gauss's law over a Gaussian sphere around a point charge. By 2026 every textbook derives Coulomb's law as a corollary of Maxwell, not as a primary axiom.

The SI redefinition of 20 May 2019 reshaped the foundations: the ampere is now defined by fixing the elementary charge e ≡ 1.602176634 × 10⁻¹⁹ C exactly, which means the coulomb is literally a count of electrons. The constants ε₀ and k_Coulomb are now measured quantities with small experimental uncertainties rather than defined values — although the uncertainty is so small (parts in 10¹¹) that no engineering or laboratory work is affected. The 11th CGPM in 1960 had formally adopted the coulomb as the SI base derived unit of charge, naming it for Coulomb in recognition of his 1785 torsion-balance memoirs.

On the standards side, IEC 62305 (Protection against lightning) uses Coulomb's law inside its risk-assessment chapters to estimate the field strength and induced voltages on tall structures during cloud-to-ground discharges with typical charge transfer of 5-50 C across 2-5 km altitude. The largest Coulomb force anyone might routinely calculate on this page comes from those lightning numbers — still many orders of magnitude smaller than the proton-proton force at 1 fm inside a nucleus. The simulator's auto-prefix (fN to MN) is built to make that 30-decade range visually legible.