Lenz's Law

Lenz's law

The induced current in a closed circuit flows in a direction that opposes the change in magnetic flux that produced it.

Mathematically, this is the negative sign in Faraday's law:

EMF = -dΦ/dt

The negative sign is Lenz's law in equation form.

Visualizing Lenz's law

Scenario	Flux change	Induced current direction	Effect
Magnet (N pole forward) pushed into coil	Increasing in coil	Counterclockwise (viewed from magnet) — creates N facing magnet	Repels magnet (resistance)
Magnet (N pole forward) pulled OUT of coil	Decreasing in coil	Clockwise — creates S facing magnet	Attracts magnet (resistance)
External B increasing through coil	Increasing	Opposes B	Reduces increase
External B decreasing	Decreasing	Reinforces B	Counters decrease

Energy conservation

Lenz's law is a manifestation of energy conservation. Consider:

• If induced current ADDED to the change → you'd get magnetic energy increasing without input. Push a magnet in slowly → coil pulls it in faster → free kinetic energy → free energy. NO.
• Lenz forces you to do WORK against the induced opposition. That work goes into the induced current's energy (which dissipates as heat in resistance, or stored as KE if motor turns).

The mechanical work to push a magnet through a coil EQUALS the electrical energy generated.

Real applications

System	How Lenz's law works
Magnet falling through copper tube	Eddy currents in tube oppose magnet's motion → terminal velocity ~few m/s
Roller coaster magnetic brakes	Conductors near track experience induced eddy currents; opposite force decelerates train
Train regenerative braking	Motor used as generator; energy harvested back to overhead lines or batteries
Induction motor rotor	Stator's rotating B induces currents in rotor; opposing torque drags rotor along
Transformer core losses	Eddy currents in core (laminated to minimize)
Levitating frog (high B field)	Diamagnetic response — induced currents in water in frog oppose strong B → repulsion

JavaScript — Lenz's law applications

// Magnetic braking force on a moving conductor
function magneticBrakeForce(B, length, velocity, resistance) {
  // Motional EMF = B·L·v; current = EMF/R; force = B·I·L
  const I = B * length * velocity / resistance;
  return B * I * length;  // F = ILB, opposing motion
}

// 1 m wide rail in 0.1 T field, moving 1 m/s, 1 Ω resistance
console.log(`Brake force: ${magneticBrakeForce(0.1, 1, 1, 1).toFixed(2)} N`);  // 0.01

// Magnet falling through tube: terminal velocity (rough model)
function terminalVelocityInTube(mass, B, alpha) {
  // F_drag = α·v at terminal: F_drag = mg
  // α depends on tube conductivity and magnet geometry
  return mass * 9.81 / alpha;
}

// Back-EMF in a motor
function backEMF(N, B, A, omega) {
  // Same as generator EMF
  return N * B * A * omega;
}

// At full speed, back-EMF nearly equals supply voltage
function motorCurrent(V_supply, V_back_EMF, R_motor) {
  // I = (V_supply - V_back_EMF) / R
  return (V_supply - V_back_EMF) / R_motor;
}

// Stalled motor (no back-EMF): I = V/R, large
console.log(`Stalled motor: ${motorCurrent(120, 0, 5)} A`);  // 24 A
// Spinning motor (back-EMF 100V): I = 20/5 = 4 A
console.log(`Spinning motor: ${motorCurrent(120, 100, 5)} A`);

// Eddy current power loss in a conductor
function eddyPower(B, area, frequency, conductivity) {
  // Rough; depends on geometry
  // Power ~ (B·f)² · σ · A
  return B * B * frequency * frequency * conductivity * area;
}

// Magnetic levitation force on a diamagnetic object (rough)
function diamagneticForce(susceptibility, B, gradB) {
  // F = -χ · V · B · ∇B / μ_0 (per unit volume)
  const mu_0 = 4 * Math.PI * 1e-7;
  return -susceptibility * B * gradB / mu_0;
}

// Water (χ ≈ -9.1e-6), in 16 T field with gradient 1000 T/m
console.log(`Diamagnetic levitation force: ${diamagneticForce(-9.1e-6, 16, 1000).toExponential(2)} N/m³`);
// Strong enough to levitate small objects

Where Lenz's law shows up

• Generators. Conservation of energy — work in equals electrical energy out.
• Motors. Back-EMF limits current; Lenz's law explanation.
• Magnetic brakes. Induced currents oppose motion → smooth, frictionless braking.
• Transformers. Energy transfer via mutual induction; Lenz's keeps it from being a "free energy" device.
• Induction stoves. Eddy currents in pots are induced; Lenz's law sets their direction.
• Diamagnetic levitation. Induced currents in conductors create repulsion (Meissner effect in superconductors).
• Educational physics. Classic demo: magnet through copper tube; falls slowly, dramatically.

Common mistakes

• Forgetting Lenz IS the minus sign. The negative sign in Faraday's law isn't a quirk — it's the law itself.
• Treating induced current direction as random. Always opposes the change. Use right-hand rule with this in mind.
• Confusing flux change with B field strength. Lenz cares about CHANGE in flux. If B is constant (no change), no induction.
• Thinking about static fields. Lenz only applies to time-varying flux. Static fields don't induce.
• Ignoring energy source. The energy for the induced current comes from somewhere — usually mechanical work being done against the induced opposition.
• Confusing motional and time-varying induction. Both are aspects of Faraday's law. Sign rule (Lenz) applies to both.