The formula

Magnitude of the force
F = k·|q₁q₂| / r²
Coulomb constant from the permittivity of free space
k = 1 / (4πε₀)

What the symbols mean

SymbolMeaningUnit
FElectrostatic force on each chargeN
kCoulomb constant8.99 × 10⁹ N·m²/C²
q₁, q₂The two charges, taken as magnitudes hereC
rCenter-to-center separation of the chargesm
ε₀Vacuum electric permittivity8.854 × 10⁻¹² F/m

When it applies

  • The charges are point charges, or uniformly charged spheres viewed from outside, where the separation is measured center to center.
  • The charges are at rest. The law is the electrostatic case, with the source charges held fixed.
  • In air or vacuum. A different medium replaces ε₀ with the permittivity of that material.
  • For three or more charges, apply the law to each pair separately and add the forces as vectors.

Worked example

Problem. A +2.0 μC charge and a -3.0 μC charge sit 0.30 m apart in air. What force does each one feel?

  1. Convert to SI units and take magnitudes: q₁ = 2.0 × 10⁻⁶ C, q₂ = 3.0 × 10⁻⁶ C, r = 0.30 m.
  2. Multiply the charges: |q₁q₂| = 6.0 × 10⁻¹² C².
  3. Square the separation: r² = 0.090 m².
  4. Substitute: F = (8.99 × 10⁹)(6.0 × 10⁻¹²) / 0.090 = 0.05394 / 0.090 = 0.60 N.
  5. The charges have opposite signs, so the force on each is attractive and directed along the line joining them.

Answer. 0.60 N on each charge, attractive, along the line between them.

Common mistakes

  • Leaving the charges in microcoulombs. A factor of 10⁻⁶ on each charge is 10⁻¹² in the product, which is the single most common source of a wildly wrong answer here.
  • Putting the signs of the charges into the magnitude formula and then also assigning a direction, which counts the attraction twice.
  • Using the gap between two spheres as r. The distance in the formula runs center to center.
  • Expecting the larger charge to feel the bigger force. Both charges feel the same magnitude, in opposite directions.

Related formulas

Sources