Coulomb's Law — F = kq₁q₂/r²

Coulomb's law F = kq₁q₂/r² explained: the electrostatic force between charges, the constant k, attraction vs repulsion, and a worked example.

What it means

Coulomb's law is the inverse-square law of electricity: the force between two charges grows with the product of the charges and weakens with the square of the distance. Like charges repel, opposite charges attract, and the force acts along the line joining them. Its form mirrors Newton's law of gravitation — both are inverse-square — but the electric force between elementary particles is vastly stronger than their gravitational attraction, which is why chemistry and materials are electrical phenomena first and gravitational ones almost never.

Variables

Assumptions

• Charges are point charges (or spherically symmetric)
• Charges are stationary (electrostatic case)
• Surrounded by vacuum or air — otherwise replace k with 1/(4πε)

Derivation

Coulomb measured the force between charged spheres with a torsion balance and found it proportional to each charge and inversely proportional to the square of their separation. In SI form the proportionality constant is $k = 1/(4\pi\varepsilon_0) = 8.99 \times 10^9\ \mathrm{N\cdot m^2/C^2}$.

Worked example

Given:

• q₁ = 2 µC = 2 × 10⁻⁶ C
• q₂ = 3 µC = 3 × 10⁻⁶ C
• r = 0.1 m
• k = 8.99 × 10⁹ N·m²/C²

1. Convert to coulombs: $q_1 = 2 \times 10^{-6}\ \mathrm{C}$, $q_2 = 3 \times 10^{-6}\ \mathrm{C}$

2. Substitute: $F = (8.99 \times 10^9)(2 \times 10^{-6})(3 \times 10^{-6}) / 0.1^2$

3. Numerator: $8.99 \times 10^9 \times 6 \times 10^{-12} = 5.394 \times 10^{-2}$

4. Divide by $0.01$: $F = 5.39\ \mathrm{N}$