Magnetic Levitation

Three approaches to maglev

Type	Mechanism	Examples
EMS (Electromagnetic Suspension)	Attractive force between train and rail; needs feedback control	Transrapid (Germany, China)
EDS (Electrodynamic Suspension)	Repulsion via induced currents in track; uses superconductors	JR-Maglev (Japan)
Meissner-effect	Type I superconductor expels B; magnet floats above	Lab experiments, future maglev concepts
Diamagnetic	Repulsion of induced currents in non-magnetic materials	Levitating frogs, water
Spin-stabilized	Gyroscopic stability defeats Earnshaw	Levitron toy

Earnshaw's theorem

It's mathematically impossible to suspend a charged or magnetic object stably in 3D using purely static (non-changing) sources of the same field type. There's always at least one direction in which the object slides away.

Bypassed by:

1. Active control. Electromagnets adjusted in real time (1000+ Hz feedback) to maintain stability.
2. Time variation. Driving force at high frequency.
3. Diamagnetism. Materials with negative susceptibility find STABLE positions in field gradients.
4. Spin. Gyroscopic effects can stabilize.
5. Quantum. Superconductor flux pinning gives stable equilibria.

Real-world maglev systems

System	Type	Top speed	Status
Shanghai Maglev	EMS	431 km/h	Operational since 2004
JR-Maglev	EDS (superconducting)	603 km/h (test)	Tokyo-Nagoya planned 2027
Linimo (Nagoya)	EMS	100 km/h	Operational since 2005
Beijing S1	EMS	110 km/h	Operational
Hyperloop concepts	EDS in vacuum	~1000 km/h target	Various tests
Levitating Frog (Nijmegen)	Diamagnetic in 16 T	0 km/h (stationary)	Lab demo, 1997

JavaScript — magnetic levitation calculations

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

// Water (χ ≈ -9e-6) volume 1 cm³, in 16 T field, 1000 T/m gradient
const F_water = diamagneticForce(-9e-6, 16, 1000, 1e-6);
console.log(`Diamagnetic force on water cm³: ${F_water.toExponential(2)} N`); // ~1.1e-2 N
console.log(`Gravity on cm³ water: ${(1e-3 * 9.81).toExponential(2)} N`);     // 9.8e-3 N
// Comparable — levitation just possible

// Force per unit mass for diamagnetic levitation requirement
function levitationGradient(susceptibility, B, density, g = 9.81) {
  // F/V = ρg → χ·B·gradB / μ_0 = ρg
  // gradB = ρ·g·μ_0 / (χ·B)
  const mu_0 = 4 * Math.PI * 1e-7;
  return density * g * mu_0 / (Math.abs(susceptibility) * B);
}

// Levitate water in 1 T?
console.log(`gradB needed for water in 1 T: ${levitationGradient(-9e-6, 1, 1000).toFixed(0)} T/m`);
// ~1.4e6 T/m (impossibly high!)
// In 16 T: 9000 T/m — still very challenging
console.log(`In 16 T: ${levitationGradient(-9e-6, 16, 1000).toFixed(0)} T/m`);

// Maglev train: induced current and force
function maglevInducedCurrent(B, length, speed, R_track) {
  // Motional EMF, current
  return B * length * speed / R_track;
}

function maglevForce(B, I, length) {
  // Force on the inducing magnet
  return B * I * length;
}

// 1 T train magnet, 5 m track length, 100 m/s, 1 mΩ track loop
const I_maglev = maglevInducedCurrent(1, 5, 100, 1e-3);
console.log(`Induced current: ${I_maglev.toFixed(0)} A`);  // 500 kA
const F_maglev = maglevForce(1, I_maglev, 5);
console.log(`Force per unit: ${F_maglev.toFixed(0)} N`);   // 2.5 MN — massive

// Power consumption of maglev (mostly drag at high speed)
function maglevPower(speed_kmh, drag_coef = 0.20, area = 8, density = 0.5) {
  // Reduced air density at low pressure (Hyperloop), or normal
  const v = speed_kmh / 3.6;
  const F_drag = 0.5 * density * drag_coef * area * v * v;
  return F_drag * v / 1000;  // kW
}

console.log(`Maglev at 500 km/h normal pressure: ${maglevPower(500, 0.2, 8, 1.225).toFixed(0)} kW`);
console.log(`Hyperloop at 1000 km/h vacuum (0.001 atm): ${maglevPower(1000, 0.5, 8, 0.0012).toFixed(0)} kW`);

Where maglev matters

• High-speed trains. Japan SCMaglev, Shanghai, Linimo. Smooth, fast, frictionless.
• Mass transit. Shorter-range maglev (Nagoya, China S1) for urban transport.
• Hyperloop concepts. Maglev + vacuum tubes for very high speeds.
• Magnetic bearings. Frictionless bearings for high-speed pumps, flywheels, vacuum pumps.
• Wind turbines. Maglev bearings reduce maintenance, increase efficiency.
• Lab experiments. Levitating frogs, water droplets in superconductor experiments.
• Future concepts. Magnetic launch systems, space elevator concepts.

Common mistakes

• Believing magnets alone can levitate stably. Earnshaw's theorem prevents static stable equilibrium with permanent magnets only. Need active control, diamagnetism, or special configurations.
• Confusing maglev types. EMS (attractive, with feedback) vs EDS (repulsive, induction-based) — different physics.
• Underestimating Hyperloop challenges. Maglev tech ready; vacuum tube infrastructure is massive engineering challenge.
• Treating diamagnetism as weak. Yes, χ is small (~10⁻⁵) for water. But in strong fields with gradients, force is significant. Used in real lab experiments.
• Believing Levitron is permanent. Only works with rotation; air drag eventually slows spin and toy falls. Demo, not stable solution.
• Forgetting air drag at high speeds. Maglev at 500+ km/h — air drag dominates power consumption. Hyperloop addresses this with vacuum.