Coulomb's Law

Coulomb's law

Force between two point charges q₁ and q₂ separated by distance r:

F = k · q₁ · q₂ / r²

where k = 8.99 × 10⁹ N·m²/C² is Coulomb's constant. Equivalently:

F = q₁ · q₂ / (4π · ε₀ · r²)

with ε₀ = 8.854 × 10⁻¹² F/m (permittivity of free space).

Direction — along line between charges. Like charges repel; unlike attract.

Numerical magnitudes

Configuration	Force
Two 1 C charges 1 m apart	9 × 10⁹ N (≈ 1 million tonnes!)
Two electrons 1 nm apart	2.3 × 10⁻¹⁰ N
Proton-electron 1 Å apart (Bohr radius)	~8.2 × 10⁻⁸ N
Two protons 1 fm apart (in nucleus)	~230 N (countered by strong force)
Static charge on shoe vs ground after walking	~10⁻⁵ C; pico-Newtons
1 mC at 1 m (typical lab demo)	9000 N — substantial!

Vector form for multiple charges

For more than two charges, total force is the vector sum of pairwise forces:

F⃗_on_q = Σ_i k · q · q_i · (r_i_to_q) / |r_i_to_q|³

This is the principle of superposition for electric forces.

vs Gravity

Aspect	Coulomb	Gravity
Form	F = k·q₁q₂/r²	F = G·m₁m₂/r²
Constant	k = 9 × 10⁹	G = 6.67 × 10⁻¹¹
Sign	Attractive OR repulsive	Always attractive
Charges/masses	Both signs cancel	Always positive (no anti-mass)
Strength (proton-electron)	F_e = 8.2 × 10⁻⁸ N	F_g = 3.6 × 10⁻⁴⁷ N
Ratio at atomic scale	~10³⁹	—
Why dominates at large scale	Cancels (matter neutral)	Always adds (only attractive)

JavaScript — Coulomb force

const k = 8.99e9;  // N·m²/C²
const e_charge = 1.602e-19;  // C

// Force between two point charges
function coulombForce(q1, q2, r) {
  return k * q1 * q2 / (r * r);
}

// Two electrons 1 nm apart
console.log(`e-e at 1 nm: ${coulombForce(-e_charge, -e_charge, 1e-9).toExponential(2)} N`);

// Force vector between two charges at positions r1, r2
function coulombForceVector(q1, q2, pos1, pos2) {
  const r = [pos2[0]-pos1[0], pos2[1]-pos1[1], pos2[2]-pos1[2]];
  const r_mag = Math.sqrt(r[0]**2 + r[1]**2 + r[2]**2);
  const F = k * q1 * q2 / (r_mag * r_mag);
  // Direction from 1 to 2
  return r.map(c => F * c / r_mag);
}

// Total force on a charge from many others
function totalForce(test_charge, test_pos, charges, positions) {
  return charges.map((q, i) => coulombForceVector(test_charge, q, test_pos, positions[i]))
                .reduce(([sx, sy, sz], [fx, fy, fz]) => [sx+fx, sy+fy, sz+fz], [0, 0, 0]);
}

// Single 2 µC charge at origin, force on 1 µC at 1 m
console.log(coulombForceVector(2e-6, 1e-6, [0,0,0], [1,0,0]));

// Compare to gravity
function gravityForce(m1, m2, r) {
  const G = 6.674e-11;
  return G * m1 * m2 / (r * r);
}

// Proton-electron at 1 Å
const p_mass = 1.673e-27;
const e_mass = 9.11e-31;
const F_e = coulombForce(e_charge, -e_charge, 1e-10);
const F_g = gravityForce(p_mass, e_mass, 1e-10);
console.log(`Coulomb/Gravity ratio: ${(Math.abs(F_e/F_g)).toExponential(2)}`); // ~10⁴⁰

// Charges in a dielectric
function coulombInMedium(q1, q2, r, dielectric_constant) {
  return coulombForce(q1, q2, r) / dielectric_constant;
}

// Sodium and chloride ions in water (κ ≈ 80) vs vacuum
const F_vac = coulombForce(e_charge, -e_charge, 0.3e-9);  // ~3 Å apart in salt
const F_water = coulombInMedium(e_charge, -e_charge, 0.3e-9, 80);
console.log(`Vacuum: ${F_vac.toExponential(2)} N. Water: ${F_water.toExponential(2)} N`);
// Water reduces the attraction by 80×, allowing salt to dissolve

// Charge on a small object: count electrons removed
function chargeFromElectrons(num_electrons) {
  return -num_electrons * e_charge;  // negative if electrons added
}

console.log(`1 µC = ${(1e-6 / e_charge).toExponential(2)} electrons`);
// ~6.2e12 electrons (huge number for tiny charge!)

Where Coulomb's law shows up

• Atomic structure. Electrons orbit nuclei via Coulomb attraction. Sets atomic sizes, chemistry, optics, electrical properties.
• Chemistry. Ionic bonding (Na⁺Cl⁻), covalent bonding (electron sharing), van der Waals forces — all Coulomb at heart.
• Materials science. Crystal structures, dielectric constants, electrical conductivity all derive from Coulomb interactions.
• Electronics. Capacitors, transistors, all charge-based devices — Coulomb force is the fundamental driver.
• Plasmas. Hot ionized gas — Coulomb interactions between charged particles. Stars, lightning, fusion reactors.
• Particle physics. Detector design (e.g., wire chambers), beam dynamics in accelerators, scattering experiments.
• Biology. Membrane potentials, ion channels, neural signals — biochemistry largely about Coulomb interactions in water.

Common mistakes

• Forgetting it's a vector. Force has direction. For multiple charges, vector-sum forces.
• Using like charges with negative product. q₁q₂ > 0 (same sign) → positive (repulsive). q₁q₂ < 0 (opposite) → negative (attractive). Watch the signs.
• Wrong distance units. SI uses meters. Don't mix cm or mm without conversion.
• Treating it as the only force. At nuclear distances, strong nuclear force dominates over Coulomb between protons. Coulomb is one of four fundamental forces.
• Ignoring shielding/dielectrics. In materials, effective force is reduced by dielectric constant. Vacuum Coulomb is upper bound.
• Not checking neutrality. Macroscopic objects are usually nearly neutral. Most Coulomb forces in biology and chemistry are between net-neutral arrangements with displaced charges.