Coulomb's law between two charged spheres.

• Coulomb's law: the force between two point charges at rest is proportional to the product of the charges, inversely proportional to the square of the distance between them, and acts along the line joining them: F = k|q₁q₂|/r².
• Coulomb measured the force with a torsion balance (arriving at the law in 1785). He did not know the charges; he halved a sphere's charge by touching it to an identical uncharged sphere and compared forces.
• In SI units k = 1/(4πε₀) ≈ 9 × 10⁹ N m² C⁻², where ε₀ = 8.854 × 10⁻¹² C² N⁻¹ m⁻² is the permittivity of free space. Two 1 C charges 1 m apart would repel with 9 × 10⁹ N, so the coulomb is a very large unit; μC and nC are used in practice.
• In vector form F₂₁ = kq₁q₂ r̂₂₁/r₂₁², with r̂₂₁ pointing from q₁ to q₂. The same formula covers like charges (repulsion) and unlike charges (attraction), through the sign of q₁q₂.
• F₁₂ = −F₂₁: the two charges pull or push each other equally and oppositely, as Newton's third law requires.
• The law has been checked from laboratory distances down to about 10⁻¹⁰ m. It is the force in vacuum; matter between the charges changes the situation.
• Electric forces dwarf gravity: between an electron and a proton the ratio of electric to gravitational force is about 2.4 × 10³⁹, and between two protons about 1.3 × 10³⁶.

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