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Coulomb's Law (Scalar)

Force between two point charges q₁ and q₂ separated by distance r in vacuum. k = 9×10⁹ N m² C⁻².
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Derivation

Experimental basis

Coulomb (1785) used a torsion balance to measure forces between charged spheres. Three observations emerged:

  1. Force is proportional to each charge: F∝q1F \propto q_1, F∝q2F \propto q_2
  2. Force is inversely proportional to the square of separation: F∝1/r2F \propto 1/r^2
  3. Force acts along the line joining the charges

Combining these:

F∝q1q2r2F \propto \frac{q_1 q_2}{r^2}

Introducing the proportionality constant

In SI units, the constant of proportionality is written as k=14πε0k = \dfrac{1}{4\pi\varepsilon_0}, where ε0\varepsilon_0 is the permittivity of free space:

ε0=8.854×10−12 C2 N−1 m−2\varepsilon_0 = 8.854 \times 10^{-12} \text{ C}^2 \text{ N}^{-1} \text{ m}^{-2}

This gives k≈9×109 N m2 C−2k \approx 9 \times 10^9 \text{ N m}^2 \text{ C}^{-2}.

The factor 4π4\pi is introduced deliberately — it cancels in Gauss's law, making that equation cleaner for symmetric geometries.

F=14πε0q1q2r2\boxed{F = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}}

Sign convention

The sign of FF encodes the nature of the force:

  • q1q2>0q_1 q_2 > 0 (same sign): F>0F > 0 → repulsive
  • q1q2<0q_1 q_2 < 0 (opposite sign): F<0F < 0 → attractive
Remember
Coulomb's law applies to point charges in vacuum. For charges in a medium of relative permittivity $\varepsilon_r$, replace $\varepsilon_0$ with $\varepsilon_r \varepsilon_0$.