Coulomb's Law and Electric Fields: From Point Charges to Field Lines

Coulomb's law

Charles-Augustin de Coulomb measured the force between charged spheres in 1785 and found it follows an inverse-square law: like charges repel, unlike charges attract, and the force falls off as 1/r² — at double the distance, the force is four times weaker.

F = k|q₁q₂| / r²
k = 1/(4πε₀) ≈ 8.988×10⁹ N·m²/C²
ε₀ = 8.854×10⁻¹² F/m  (permittivity of free space)

This is the same power law as Newtonian gravity, but the electromagnetic force between an electron and proton is about 10³⁶ times stronger than their gravitational attraction. Electric charge is also quantised — every free charge is an integer multiple of the elementary charge e = 1.602×10⁻¹⁹ C — and always conserved.

The electric field and superposition

The electric field E(r) at a point is the force per unit positive test charge that would be felt there. For multiple charges, the total field is simply the vector sum of each charge's individual contribution — the principle of superposition.

E(r) = k·q·r̂ / r²                [N/C = V/m]
E_total(r) = Σᵢ k·qᵢ·(r − rᵢ) / |r − rᵢ|³

A naïve sum over every pair of charges costs O(N²) — for large N, simulations instead use hierarchical tree algorithms such as Barnes-Hut, which approximate distant clusters of charge as a single macro-charge and reduce the cost to O(N log N).

Electric potential and Gauss's law

The electric potential V is the scalar quantity whose negative gradient is the field: E = −∇V, with V(r) = kq/r for a single point charge. Since the electrostatic field is conservative, the work done moving a charge between two points depends only on the potential difference, W = q(V_B − V_A). Gauss's law, one of Maxwell's four equations, gives a shortcut whenever symmetry is present: ∮E·dA = Q_enc/ε₀. It shows a spherical shell behaves like a point charge outside itself with E = 0 inside, an infinite line charge falls off as 1/r rather than 1/r², and an infinite charged plane produces a uniform field independent of distance.

Field lines and conductors

Field lines are integral curves of E(r): they start on positive charges, end on negative charges, and never cross, with their density indicating field strength. Equipotential surfaces are always perpendicular to field lines. For a dipole, the field forms a characteristic figure-eight pattern whose far field falls off as 1/r³. In a conductor at electrostatic equilibrium, free charges rearrange until the field inside is exactly zero — all excess charge sits on the surface, concentrated most where the surface curves sharply (which is why lightning rods work), and a fully enclosed conductor forms a Faraday cage that shields its interior from external fields.

Frequently asked questions

Try it live

Everything above runs in your browser — open Electric Field Lines, add positive and negative charges, drag them around, and watch the field lines and equipotential surfaces update in real time. Nothing is installed, nothing is uploaded.