Coulomb's Law & Electric Fields — From Point Charges to Barnes–Hut

1. Coulomb's Law

Charles-Augustin de Coulomb measured the force between charged spheres in 1785 and found it follows an inverse-square law:

F = k |q₁ q₂| / r² (magnitude) Vector form: F₁₂ = k q₁ q₂ (r̂₁₂) / r₁₂² k = 1/(4πε₀) \approx 8.988 \times 10⁹ N\cdotm²/C² ε₀ = 8.854 \times 10⁻¹² F/m (permittivity of free space)

Like charges repel (F > 0), unlike charges attract (F < 0). The force falls off as 1/r² — at double the distance, the force is four times weaker. This is the same power law as Newtonian gravity; the difference is that electric charges can be positive or negative, giving both attraction and repulsion.

Gravity vs Electrostatics

Both are inverse-square laws. But the electromagnetic force between an electron and proton is ~10³⁶ times stronger than their gravitational attraction.

Charge quantisation

Electric charge is quantised: q = n · e, where e = 1.602 × 10⁻¹⁹ C. Quarks carry ±e/3, but free charges always come in integer multiples of e.

Conservation

Total electric charge is conserved in all processes. Pair production creates equal + and − charges; pair annihilation destroys them.

In a medium

Replace ε₀ with ε = εᵣε₀. Water (εᵣ ≈ 80) screens electrostatic forces dramatically — essential for biology and ionic solutions.

2. Electric Field & Superposition

The electric field E(r) at position r due to a point charge q at origin is the force per unit test charge:

E(r) = k q r̂ / r² [N/C = V/m] Superposition (N charges): E_total(r) = Σᵢ k qᵢ (r - rᵢ) / |r - rᵢ|³

The naïve sum over all pairs is O(N²) — prohibitive for large N. For a continuous distribution with charge density ρ(r'):

E(r) = k \int ρ(r') (r - r') / |r - r'|³ dV'

The electric field is a vector field — at each point in space it has both magnitude and direction. Field lines (see section 5) visualise this field geometrically.

3. Electric Potential & Energy

The electric potential V is the scalar quantity whose negative gradient is the field:

V(r) = k q / r (single point charge) E = -\nablaV \to Eₓ = -\partialV/\partialx, Eᵧ = -\partialV/\partialy Potential energy of charge q at potential V: U = q V Potential energy of two-charge system: U₁₂ = k q₁ q₂ / r₁₂

Since E is conservative (∮E·dl = 0 in electrostatics), the work done moving a charge between two points is path-independent and depends only on the potential difference: W = q(V_B − V_A).

Poisson's equation connects potential to charge density: ∇²V = −ρ/ε₀. In free space (ρ = 0) this reduces to Laplace's equation ∇²V = 0 — used in boundary-value problems for capacitors and conductors.

4. Gauss's Law

Gauss's law is one of Maxwell's four equations and provides a powerful shortcut when symmetry is present:

\oint E \cdot dA = Q_enc / ε₀ Differential form: \nabla \cdot E = ρ / ε₀

Three canonical applications using Gaussian surfaces:

Spherical shell or point charge: E = kQ/r² outside, E = 0 inside a hollow shell.
Infinite line charge (linear density λ): E = λ/(2πε₀r) — falls off as 1/r, not 1/r².
Infinite plane (surface density σ): E = σ/(2ε₀) — uniform field, independent of distance.

5. Field Lines & Equipotentials

Electric field lines are integral curves of E(r): they start on positive charges, end on negative charges, and never cross. Their density indicates field strength. Equipotential surfaces are everywhere perpendicular to field lines.

Tracing a field line from seed position r₀:

dr/ds = E(r) / |E(r)| (s = arc-length parameter) Euler step: r_{n+1} = r_n + h · E(r_n) / |E(r_n)| Use RK4 for smooth lines with larger step sizes h

For a dipole (+q at −d, −q at +d), the field has a characteristic figure-eight pattern of closed lines. Dipole moment p = qd points from − to + charge; the far field falls off as 1/r³ and depends on angle: E ~ (1/r³)(2cosθ r̂ + sinθ θ̂).

6. Conductors & Shielding

In electrostatic equilibrium, free charges in a conductor rearrange until E = 0 everywhere inside. Consequences:

All excess charge resides on the surface.
The surface is an equipotential; E is perpendicular to the surface outside.
Surface charge density σ is greatest where the surface curves most (lightning rods exploit this).
Faraday cage: A closed conductor shields its interior from external fields — used in sensitive electronics and MRI scanners.

The capacitance C of a conductor measures how much charge it holds per unit potential: C = Q/V. For a parallel-plate capacitor with plate area A and separation d: C = ε₀A/d. Adding a dielectric with permittivity εᵣ multiplies C by εᵣ.

7. JavaScript — Barnes–Hut Electrostatics

For N charged particles, direct summation is O(N²). The Barnes–Hut algorithm builds a quadtree (2D) or octree (3D) and approximates distant clusters as a single "macro-charge", reducing the complexity to O(N log N).

// Barnes–Hut quadtree node
class BHNode {
  constructor(cx, cy, size) {
    this.cx = cx; this.cy = cy; this.size = size;
    this.totalQ = 0; this.comX = 0; this.comY = 0;
    this.children = null; this.particle = null;
  }

  insert(p) {
    if (!this.totalQ) {
      this.particle = p;
      this.totalQ = p.q;
      this.comX = p.x; this.comY = p.y;
      return;
    }
    if (!this.children) this.subdivide();
    if (this.particle) {
      this._insertIntoChild(this.particle);
      this.particle = null;
    }
    this._insertIntoChild(p);
    // update centre of charge
    this.comX = (this.comX * this.totalQ + p.x * p.q) / (this.totalQ + p.q);
    this.comY = (this.comY * this.totalQ + p.y * p.q) / (this.totalQ + p.q);
    this.totalQ += p.q;
  }

  subdivide() {
    const h = this.size / 2;
    this.children = [
      new BHNode(this.cx - h/2, this.cy - h/2, h),
      new BHNode(this.cx + h/2, this.cy - h/2, h),
      new BHNode(this.cx - h/2, this.cy + h/2, h),
      new BHNode(this.cx + h/2, this.cy + h/2, h),
    ];
  }

  _insertIntoChild(p) {
    const idx = (p.x > this.cx ? 1 : 0) + (p.y > this.cy ? 2 : 0);
    this.children[idx].insert(p);
  }

  // Compute electric force on particle p (theta = opening ratio, ~0.5)
  force(p, theta, k = 8.988e9) {
    if (!this.totalQ) return {fx: 0, fy: 0};
    const dx = this.comX - p.x, dy = this.comY - p.y;
    const r2 = dx * dx + dy * dy;
    if (r2 === 0) return {fx: 0, fy: 0};
    const r = Math.sqrt(r2);
    // Barnes–Hut criterion: treat as single macro-charge if s/r < theta
    if (!this.children || this.size / r < theta || this.particle) {
      const mag = k * p.q * this.totalQ / r2;
      return {fx: mag * dx / r, fy: mag * dy / r};
    }
    let fx = 0, fy = 0;
    for (const child of this.children) {
      const f = child.force(p, theta, k);
      fx += f.fx; fy += f.fy;
    }
    return {fx, fy};
  }
}

// Field line tracer (RK4)
function traceFieldLine(charges, x0, y0, step = 2, maxSteps = 500) {
  const pts = [{x: x0, y: y0}];
  let x = x0, y = y0;
  const k = 1; // normalised units
  for (let i = 0; i < maxSteps; i++) {
    let ex = 0, ey = 0;
    for (const c of charges) {
      const dx = x - c.x, dy = y - c.y;
      const r3 = (dx * dx + dy * dy) ** 1.5;
      if (r3 < 1e-6) continue;
      ex += k * c.q * dx / r3;
      ey += k * c.q * dy / r3;
    }
    const mag = Math.hypot(ex, ey);
    if (mag < 1e-10) break;
    x += step * ex / mag;
    y += step * ey / mag;
    pts.push({x, y});
  }
  return pts;
}

8. Applications

Particle accelerators: Coulomb scattering (Rutherford) revealed the nucleus; accelerators use strong electric fields to drive particles to GeV energies.
Molecular dynamics: Electrostatic forces between atoms and molecules determine protein folding, crystal structure, and chemical reactivity.
Capacitors & energy storage: Supercapacitors store energy in electric fields between closely spaced conductors; used in regenerative braking and grid storage.
Ink-jet & laser printing: Charged droplets or toner particles are steered by controlled electric fields.
Ion traps: Paul traps confine ions with oscillating electric fields for atomic clocks, quantum computing and mass spectrometry.

Connection: The N-Body Gravity simulation uses the same Barnes–Hut algorithm applied to gravitational forces (replace kq₁q₂ with −Gm₁m₂).