Coulomb’s law

Coulomb’s law gives the size of the electric force between two point charges. It depends on the product of the charges and gets weaker with the square of the distance – just like gravity, but it can attract or repel and is enormously stronger. It explains how atoms hold together and how charged particles push each other around.

Key facts

  • F = kQ₁Q₂ ÷ r², with k = 9.0 × 10⁹ N m²/C².
  • Like charges repel; unlike charges attract. The forces on the two charges are equal and opposite.
  • Inverse square: doubling r makes F a quarter; tripling r makes it a ninth.
  • Always convert: µC → × 10⁻⁶ C, nC → × 10⁻⁹ C, cm → ÷ 100 m.
  • For several charges in a line, find each force separately (with its direction) and add them as vectors.

Worked examples

Example 1

Two charges of +3 µC and −2 µC are 30 cm apart. Find the force between them.

  1. Convert Q₁ = 3 × 10⁻⁶ C, Q₂ = 2 × 10⁻⁶ C, r = 0.30 m
  2. F = kQ₁Q₂ ÷ r² F = 9.0 × 10⁹ × 3 × 10⁻⁶ × 2 × 10⁻⁶ ÷ 0.30²
  3. Work it out F = 0.054 ÷ 0.09 = 0.6 N
  4. Opposite signs, so the force is attractive.

Answer: 0.6 N, attractive

Example 2

The force between two charges is 8 N. What is it if the distance is halved and one charge is doubled?

  1. Halving r: F × 2² = × 4.
  2. Doubling one charge: × 2.
  3. Together 8 × 4 × 2 = 64 N

Answer: 64 N

Example 3

Charges A (+4 µC), B (+2 µC) and C (−3 µC) are in a line with AB = 20 cm and BC = 30 cm. Find the resultant force on B.

  1. From A (repels B, pushes it towards C) F = 9.0 × 10⁹ × 4 × 10⁻⁶ × 2 × 10⁻⁶ ÷ 0.20² = 1.8 N
  2. From C (attracts B, pulls it towards C) F = 9.0 × 10⁹ × 3 × 10⁻⁶ × 2 × 10⁻⁶ ÷ 0.30² = 0.6 N
  3. Both forces point towards C, so they add.
  4. Resultant 1.8 + 0.6 = 2.4 N towards C

Answer: 2.4 N towards C