Electric Field

Definition

The electric field at a point is the force per unit positive test charge:

E = F / q  (units: V/m or N/C — equivalent)

For any object placed at that point, force on it is F = qE. The field exists whether or not a test charge is present.

Around a point charge

For a point charge Q at the origin, electric field at distance r:

|E| = k · Q / r²  (radial, away from positive Q; toward negative)

Direction: along radial line from Q. Magnitude falls as 1/r².

Superposition

For multiple charges, total field is vector sum of individual fields:

E_total = E₁ + E₂ + E₃ + ...

Field lines and visualization

Feature	Meaning
Field line direction	Direction of E (force on positive test charge)
Field line density	Magnitude of E (more lines per area = stronger)
Field lines start/end	Begin at +, end at − (or at infinity)
Lines can't cross	E has unique direction at each point
Equipotential surfaces	Perpendicular to field lines

Conductors and shielding

In a conductor at static equilibrium:

• E = 0 inside (free charges redistribute until they cancel external fields).
• E perpendicular to surface (parallel component would move surface charges).
• Charge accumulates on outer surface.
• Faraday cage — exterior fields don't penetrate interior.

Electric dipole

Two equal opposite charges separated by distance d. Dipole moment:

p = q · d  (vector, pointing from − to +)

At distance r ≫ d, dipole field falls as 1/r³ (faster than monopole). Examples — water molecule, neutral atom polarized by external field.

JavaScript — electric field calculations

const k = 8.99e9;

// E from point charge at distance r
function pointFieldMagnitude(Q, r) {
  return k * Q / (r * r);
}

// 1 nC charge at 0.1 m
console.log(`E from 1 nC at 0.1 m: ${pointFieldMagnitude(1e-9, 0.1).toFixed(2)} V/m`); // ~899

// E vector from a point charge at origin, at position (x, y, z)
function pointFieldVector(Q, position) {
  const [x, y, z] = position;
  const r = Math.sqrt(x*x + y*y + z*z);
  if (r < 1e-12) return [0, 0, 0];
  const E = k * Q / (r * r * r);
  return [E * x, E * y, E * z];
}

// Total field from many charges at given point
function totalField(charges, positions, observation) {
  return charges.reduce(([sx, sy, sz], Q, i) => {
    const dx = observation[0] - positions[i][0];
    const dy = observation[1] - positions[i][1];
    const dz = observation[2] - positions[i][2];
    const r = Math.sqrt(dx*dx + dy*dy + dz*dz);
    if (r < 1e-12) return [sx, sy, sz];
    const E = k * Q / (r * r * r);
    return [sx + E*dx, sy + E*dy, sz + E*dz];
  }, [0, 0, 0]);
}

// Two equal charges at ±1 cm; field at midpoint should cancel
const charges = [1e-9, 1e-9];
const positions = [[-0.01, 0, 0], [0.01, 0, 0]];
console.log(totalField(charges, positions, [0, 0, 0]));  // ~[0, 0, 0]

// Dipole field at distance r along axis
function dipoleAxisField(p, r) {
  // E ≈ 2k·p/r³ along axis
  return 2 * k * p / (r * r * r);
}

// Water molecule dipole moment p = 6.2 × 10⁻³⁰ C·m
console.log(`Water dipole at 1 nm: ${dipoleAxisField(6.2e-30, 1e-9).toExponential(2)} V/m`);

// Force on a charge in a field
function forceOnCharge(q, E_vec) {
  return E_vec.map(c => q * c);
}

// Field strength for various things
const fields = {
  'Sunlight at Earth': 700,                          // V/m
  'Static-shocked sweater': 1e6,                    // V/m
  'Lightning bolt (peak)': 1e8,                     // V/m
  'Air dielectric breakdown': 3e6,                  // V/m
  'Hydrogen atom (electron at Bohr radius)': 5e11   // V/m
};
console.log(fields);

// Faraday cage shielding (idealized)
function shielding(external_E, inside_distance_from_wall) {
  // For ideal conductor: zero E inside (regardless of position)
  return 0;
}

// Capacitor: field between parallel plates
function capacitorField(charge, area) {
  const epsilon_0 = 8.854e-12;
  // E = σ/ε₀ where σ = Q/A
  return charge / (area * epsilon_0);
}

console.log(`100 nC on 0.01 m² plates: ${capacitorField(100e-9, 0.01).toFixed(0)} V/m`);

Where electric fields show up

• Capacitors and electronics. Charge storage, signal transmission, filters.
• Atomic physics. Electron orbitals, atomic spectra, chemistry.
• Materials. Dielectric properties, ferroelectrics, conductivity.
• Lightning and atmospheric. Electrostatic discharges; storm physics.
• Particle accelerators. Electric fields accelerate charged particles to high energies.
• Biological membranes. Cell membranes establish potential differences via ion gradients.
• Faraday cages. Shielding sensitive electronics from external interference.

Common mistakes

• Confusing E field with force. E is per unit charge; force is qE.
• Forgetting it's a vector. Field has direction. Add vectorially for multiple sources.
• Treating field lines as physical objects. They're visualizations only. Real fields are continuous and exist everywhere.
• Ignoring superposition. Total field = sum of individual fields. Each source contributes; don't forget any.
• Using point-charge formula inside a charge distribution. Inside a uniformly-charged sphere, only enclosed charge matters (Gauss's law). Outside, full charge counts.
• Forgetting conductor static behavior. Inside a static conductor, E = 0. Charges arrange on surface to ensure this.