The force between charges
Electric charges come in two kinds, positive and negative. Like charges repel and opposite charges attract. In 1785, Charles-Augustin de Coulomb measured this force with a torsion balance and found that it follows an inverse-square law, just like gravity.
Coulomb’s law
F = k × q₁ × q₂ ÷ r²
- k = 1 ÷ (4πε₀) ≈ 8.9876 × 10⁹ N·m²/C², the Coulomb constant
- q₁, q₂: the charges in coulombs, with their signs
- r: the distance between them
A negative result means attraction; a positive one, repulsion. In a material, the force is divided by its relative permittivity εᵣ.
The calculator also gives:
- the electric field of charge 1 at that distance, E = k q₁ ÷ r²
- the potential energy of the pair, U = k q₁ q₂ ÷ r
- the number of elementary charges (1.602 × 10⁻¹⁹ C each) in charge 1
Worked example
Charges of +2 µC and −3 µC are 5 cm apart in air.
- F = 8.99 × 10⁹ × (2 × 10⁻⁶) × (3 × 10⁻⁶) ÷ 0.05² ≈ 21.6 N, attractive
- That is about the weight of a 2.2 kg mass
- Field of charge 1 at 5 cm: 7.19 × 10⁶ N/C
- Potential energy: −1.08 J
- +2 µC is an excess of about 1.25 × 10¹³ missing electrons
A sense of scale
| Situation | Force |
|---|---|
| Electron and proton in a hydrogen atom (0.053 nm) | 8.2 × 10⁻⁸ N |
| Two 1 nC charges 1 cm apart | 9 × 10⁻⁵ N |
| Two 1 C charges 1 km apart | about 9,000 N |
A coulomb is a huge amount of static charge: everyday static electricity involves nanocoulombs to microcoulombs.
Coulomb versus gravity
Between two protons, the electric force is about 10³⁶ times stronger than gravity. We do not feel enormous electric forces in daily life only because matter contains almost exactly equal amounts of positive and negative charge.
Limits
Coulomb’s law applies to point charges or spherically symmetric charge distributions, at rest. For extended objects, the forces must be added over all parts (or calculated with Gauss’s law), and moving charges also produce magnetic forces.