Electromagnetic Induction

Faraday's law

The induced EMF in a closed circuit equals the negative rate of change of magnetic flux through it:

EMF = -dΦ/dt

where Φ is magnetic flux:

Φ = ∫ B · dA = B · A · cos θ  (uniform field, flat surface)

θ is angle between B and the surface normal.

For a coil with N turns:

EMF = -N · dΦ/dt

Lenz's law

The negative sign in Faraday's law is Lenz's law. The induced current creates a magnetic field that OPPOSES the change in flux. This is energy conservation — induced effects resist the motion or change that caused them.

Examples:

• Magnet pushed into coil → induced current creates B opposing magnet → magnet feels resistance.
• Magnet pulled OUT of coil → induced current creates B in same direction as magnet's leaving B (trying to pull it back).

Generators

A coil of N turns, area A, rotates in uniform B field at angular frequency ω. Flux through it:

Φ(t) = N · B · A · cos(ωt)

EMF:

EMF = -dΦ/dt = N · B · A · ω · sin(ωt)

This produces sinusoidal AC voltage. Power-grid frequency (60 Hz US, 50 Hz EU) = generator rotation rate.

Motional EMF

A conductor of length L moving at velocity v through B field induces:

EMF = B · L · v

(Force on free electrons in the moving conductor — they pile up at one end, creating EMF.)

JavaScript — induction calculations

// Magnetic flux through a coil
function magneticFlux(B, area, angle_deg = 0) {
  return B * area * Math.cos(angle_deg * Math.PI / 180);
}

// Induced EMF from changing flux (numerical derivative)
function inducedEMF(flux_t1, flux_t2, dt, num_turns = 1) {
  return -num_turns * (flux_t2 - flux_t1) / dt;
}

// Sinusoidal generator EMF
function generatorEMF(N, B, A, omega, t) {
  return N * B * A * omega * Math.sin(omega * t);
}

// 100 turn coil, 0.5 T B field, 0.01 m² area, 60 Hz (ω = 377)
const omega = 2 * Math.PI * 60;
console.log(`Peak EMF: ${(100 * 0.5 * 0.01 * omega).toFixed(0)} V`); // ~188 V

// Motional EMF
function motionalEMF(B, L, v) {
  return B * L * v;
}

// 1 m bar moving at 5 m/s through 0.1 T field
console.log(`Motional EMF: ${motionalEMF(0.1, 1, 5)} V`);  // 0.5

// Transformer voltage ratio
function transformerRatio(N_primary, N_secondary, V_primary) {
  // V_2/V_1 = N_2/N_1
  return V_primary * N_secondary / N_primary;
}

// 120 V primary, 100 turns; secondary needs 12V
console.log(`Need ${100 * 12 / 120} turns secondary for 12V`); // 10

// Self-inductance of a solenoid
const mu_0 = 4 * Math.PI * 1e-7;
function solenoidInductance(N, area, length) {
  return mu_0 * N * N * area / length;
}

// 1000 turns, 1 cm² cross-section, 10 cm long
console.log(`Solenoid L: ${(solenoidInductance(1000, 1e-4, 0.1) * 1000).toFixed(2)} mH`);
// 1.26 mH

// LR circuit time constant
function lrTimeConstant(L, R) {
  return L / R;
}

// 100 mH, 10 Ω
console.log(`LR τ: ${(lrTimeConstant(0.1, 10) * 1000).toFixed(0)} ms`); // 10

// Energy stored in inductor
function inductorEnergy(L, I) {
  return 0.5 * L * I * I;
}

console.log(`100 mH inductor with 1A: ${inductorEnergy(0.1, 1)} J`); // 0.05 J

// Eddy current power loss (rough)
function eddyCurrentPower(B_peak, freq, conductivity, thickness, volume) {
  // P ~ (π·B·f·t)² · σ · V / 6 — for sheet
  const factor = Math.PI * B_peak * freq * thickness;
  return factor * factor * conductivity * volume / 6;
}

// Aluminum sheet, 1 mm thick, in 0.1 T 60 Hz field, 0.01 m³
console.log(`Eddy loss: ${eddyCurrentPower(0.1, 60, 3.5e7, 0.001, 0.01).toFixed(0)} W`);

Where induction matters

• Power generation. All electric generators (steam, hydro, wind, nuclear) use Faraday's law.
• Transformers. Critical to power grids. Voltage transformation for transmission.
• Induction motors. ~80% of all industrial motors. Rotating B field drags conducting rotor.
• Wireless charging. Inductive coupling for phones, EVs.
• MRI and NMR. RF coils detect induced signals from precessing nuclear spins.
• Induction heating and cooking. Eddy currents heat metal pots; cooktop stays cool.
• Magnetic braking. Eddy currents in spinning metal disks slow rotation; used in roller coasters, trains.

Common mistakes

• Forgetting the negative sign (Lenz). Sign tells you direction of induced current — opposing the change. Sign matters for energy/circuit analysis.
• Confusing flux and field. B is field (T); Φ is flux through area (Wb = T·m²). Changing flux induces EMF — could be from changing B, area, or angle.
• Using static analysis for varying B. Time-varying B creates non-conservative E field. Need full Faraday's law.
• Forgetting the area's normal direction. Flux uses B·dA where dA is the surface normal vector. Sign of EMF depends on orientation.
• Treating transformers as static. Only AC works (need changing flux). DC doesn't produce continuous EMF in secondary (only at switch-on/off).
• Ignoring back-EMF. A motor produces back-EMF as it spins. Limits steady-state current and is why motors draw less current at full speed than at startup.