Halbach Array: The Magnet Stack With One Blank Side

The Trick: Rotate the Magnetisation, Don't Flip It

Lay four identical magnet blocks in a row. In the familiar arrangement you flip them — north-up, north-down, north-up, north-down — and the field is equally strong above and below. A Halbach array instead rotates the magnetisation a quarter turn at every step: up, then along the row to the right, then down, then left, then up again. Only the magnetisation direction changes — and the sense of that rotation is what decides which face wins, so running it the other way (up, left, down, right) simply swaps the strong side.

Follow the flux for that ordering. Above the row it runs from the north pole on top of the up-block to the south pole on top of the down-block half a wavelength along — left to right. The horizontal block sitting between them is magnetised in exactly that direction, so it carries that flux inside the magnet and almost none of it has to arch through the air above: the top face goes quiet. Underneath, the return flux runs the other way, from the down-block back to the up-block, against the horizontal block's magnetisation — it gets no internal path and must loop out through the space below. Repeat that at every joint and the flux is evicted from one face and stacked onto the other, where it is not just stronger but smoother — a clean sinusoidal sheet rather than a picket fence of poles. (For this ordering the strong face is the underside, which is exactly how an Inductrack array is bolted on: strong face down.)

The cleanest way to see why the cancellation is exact is to swap the magnets for their equivalent bound surface currents. A uniformly magnetised block behaves like a sheet of current wrapped round its sides. When the magnetisation rotates smoothly along the row, those sheets merge into one sinusoidal distribution running through the slab: a surface current K(x) ∝ sin(2πx/λ) on the top face, its opposite on the bottom face, and the bound volume current in between. No single current sheet is one-sided — on its own it produces mirror-image evanescent fields above and below — but add the three contributions for a slab of thickness d and they cancel term for term on the back face while reinforcing on the front. It is the magnetostatic version of a phased array: the sources are placed so their fields interfere destructively in one direction. A Halbach array is a staircase approximation to that distribution, and λ — one full turn of the magnetisation — sets the length scale of everything that follows.

The Governing Relations, With the Numbers Reasoned Through

For an ideal, infinite planar array of thickness d whose magnetisation rotates continuously with wavelength λ, the peak field just outside the strong face is

B₀ = Bᵣ (1 − e^(−2πd/λ)), and at height y above that face B(y) = B₀ e^(−2πy/λ).

Two things follow. First, thickness saturates: at d = λ/2 you already have 1 − e^(−π) = 95.7% of what the material can give, so blocks taller than half a wavelength are wasted magnet. Second, the field is evanescent, not radiative — a pure exponential with decay length λ/2π. One wavelength of gap costs e^(−2π) = 1/535. Hence both the array's power and its tyranny: a fridge magnet striped at 3 mm is dead 3 mm from the door, while a maglev array with λ of order half a metre still reaches across a 3 cm gap.

Real arrays are staircases. Fourier-analyse M blocks per wavelength and every harmonic vanishes except orders n = 1 + pM, with amplitude M·|sin(πn/M)|/(π|n|); for the fundamental that is the segmentation factor sin(π/M)/(π/M). The classic four-block, 90°-step array scores sin(π/4)/(π/4) = 0.900. And here is the punchline most explanations miss: a plain alternating N–S–N–S row is not a rival technology, it is simply the M = 2 case, scoring 2/π = 0.637. The celebrated √2 advantage is nothing more exotic than 0.900/0.637 = 1.414. Going finer buys little: M = 8 gives 0.974 and M = 16 gives 0.994, so doubling the block count past four wins 8% for twice the machining, sorting and assembly risk.

Put N42 sintered NdFeB into the formula — remanence Bᵣ ≈ 1.30 T, energy product 42 MGOe ≈ 334 kJ/m³ — with d = λ/2 and four blocks per period: 1.30 × 0.957 × 0.900 ≈ 1.1 T at the face, from something you can hold in one hand. The back face is not literally zero. For M = 4 its leading harmonic is n = −3, coefficient 0.300 against the front's 0.900 — a third of the strong field at the surface — but it decays as e^(−6πy/λ), three times faster. At a gap of λ/4 the front holds 0.187 Bᵣ and the back 0.0027 Bᵣ: about 1.4%. The blank side is a very good approximation, not a theorem.

Inductrack: Levitation That Only Works Once You Are Moving

Richard F. Post, a fusion physicist at Lawrence Livermore National Laboratory, spent the 1990s turning this into a levitation scheme he called Inductrack: a Halbach array bolted under a vehicle, strong face down, sliding over a passive track of shorted litz wire — no track power, no cryogenics, no control electronics. Motion sweeps the sinusoidal field past each loop, inducing an EMF at frequency ω = 2πv/λ.

What happens next is decided by one dimensionless quantity, ωL/R, the ratio of the loops' inductive to resistive impedance. Below a transition speed

v_c = Rλ / (2πL) — typically 1–2 m/s for a good litz track —

the induced current is in phase with the EMF, dissipates in the wire and yields mostly drag. Above it the current lags by nearly 90°, becomes a near-perfect image of the array, and the force becomes lift. The lift-to-drag ratio is simply L/D = ωL/R = v/v_c, climbing linearly with speed: at 1–2 m/s the array is a brake, at 100 m/s a suspension with a ratio of order 100:1. Inductrack vehicles therefore need wheels for taxiing, and lift off at jogging pace.

High-speed lift is capped by magnetic pressure, of order B₀²/2μ₀ — for a 1.1 T face, ~0.48 MPa, from which Post's quoted 40 tonnes-force per square metre of array falls out once gap decay is folded in. The suspension is also naturally stiff: lift falls as e^(−4πy/λ), so for λ = 0.5 m it drops ~2.5% per extra millimetre of gap, with an undamped heave frequency ω = √(4πg/λ) ≈ 16 rad/s, about 2.5 Hz — which is why real designs must add damping rather than trust the magnets alone. A Livermore test track demonstrated the effect in the late 1990s; General Atomics later built a ~120 m Inductrack guideway in San Diego under the Federal Transit Administration's Urban Maglev programme, and NASA's Marshall Space Flight Center studied the same arrays for launch assist.

How It Is Built, and How the Field Is Actually Measured

Assembly is the hard part. Every block sits in the reverse field of its neighbours, so the stack is simultaneously trying to shear itself apart and demagnetise itself. The operating point must stay above the knee of the second-quadrant B–H curve: N42's intrinsic coercivity of ~875–955 kA/m (11–12 kOe) sounds ample until you heat it. Remanence falls about −0.12% per °C and intrinsic coercivity four to five times as fast (roughly −0.5 to −0.6% per °C), so a standard N-grade block is limited to ~80 °C in service even though the Curie point of Nd₂Fe₁₄B is ~310 °C; H, SH and UH grades trade remanence for 120, 150 and 180 °C.

Measurement is refreshingly direct. A three-axis Hall probe on a motorised mapping bench gives B(x, y) to a few parts in a thousand; an NMR teslameter, calibrated against the proton gyromagnetic ratio, pins absolute field to ~10 ppm. The signature test is to plot ln B against height y: an ideal array gives a straight line of slope −2π/λ, and any curvature is harmonic contamination from segmentation or misalignment. Before assembly each block's moment is measured in a Helmholtz coil and fluxmeter; because commercial blocks scatter ±2% in Bᵣ and a degree or two in easy-axis direction, accelerator groups sort their magnets by simulated annealing so errors cancel along the array — how free-electron-laser undulators reach rms phase errors below ~3°. On the kitchen table, a sheet of magnetic viewing film laid on a fridge magnet renders the stripes visible in a second — the cheapest field map in physics.

From Fridge Doors to Free-Electron Lasers

Undulators. Klaus Halbach's own motivation was accelerator hardware, still the array's most consequential home. Almost every modern synchrotron insertion device and X-ray FEL undulator — the ~30 mm-period, ~1.25 T devices of LCLS at SLAC, the 40 mm-period, 5 m segments of the European XFEL — is a Halbach or hybrid Halbach structure (superconducting undulators are the growing exception), wiggling electrons so they radiate coherently. Before rare-earth arrays that job took electromagnets and power supplies.

Cylinders. Wrap the same rotation round a circle and the strong side points inwards. The ideal Halbach dipole cylinder — the “magic ring” — gives a uniform transverse field in its bore and essentially nothing outside, of magnitude B = Bᵣ ln(R_out/R_in). A wall ratio of e ≈ 2.72 buys one full Bᵣ, about 1.3 T, with no current and no cryogen. That equation drives benchtop NMR spectrometers at 43–90 MHz (1.0–2.1 T) and research low-field MRI at ~50–100 mT; pushed with nested, shaped and graded material it has yielded permanent-magnet dipoles of ~5 T — the record work of Masayuki Kumada and colleagues in Japan in the early 2000s.

Motion. Halbach rotors put the flux in a motor's airgap instead of its back iron, cutting rotor mass; magnetic gears and couplings exploit the same thing. Student Hyperloop pods — MIT's winning 2016 SpaceX entry among them — levitated passively on Halbach arrays over an aluminium plate.

Your kitchen. Flexible fridge-magnet sheet is the low-grade cousin: ferrite powder in a rubber binder, magnetised from one side into stripes of roughly 2–5 mm period. It is only an approximation to a true Halbach pattern, but shows both signatures — markedly stronger on one face, dead within about one stripe period. Slide two across each other and you feel the periodicity as a row of ridges.

Look-Alikes, Limits, and What Is Still Open

Halbach levitation is routinely confused with three different physics. It is not the Meissner effect or flux pinning: the YBCO puck hovering over a track expels and traps flux, is stable at rest, and needs liquid nitrogen. It is not the superconducting EDS of Japan's L0 series — 603 km/h on the Yamanashi test line, 21 April 2015 — whose cryogenic onboard coils drive figure-of-eight null-flux coils in the guideway. And it is not the attractive EMS of Transrapid and the 431 km/h Shanghai maglev, where electromagnets are pulled up under a steel rail across a ~10 mm gap by a kilohertz feedback loop; kill the power and the vehicle drops.

The deepest misconception is about Earnshaw's theorem (Samuel Earnshaw, 1842) — extended to magnetisable bodies by Werner Braunbek in 1939 — which forbids stable static levitation of anything with μ > 1. A Halbach array does not repeal it, and it does not use Braunbek's loophole either: his exemption is diamagnetism, μ < 1, which is how a graphite flake or a superconductor hovers at rest. Inductrack takes a different exit entirely — the force is not static at all but comes from time-varying, dissipative currents induced by relative motion. Stop the vehicle and it settles onto its wheels.

Other real limits: below v_c the array is a heater, dumping I²R into the track; the exponential gap sensitivity makes ride quality a damping problem, not a magnet problem; and the design is hostage to neodymium and dysprosium supply, and to temperature. Open work is materials and geometry — dysprosium-free and MnBi-type magnets that gain rather than lose coercivity when hot, additively manufactured blocks with continuously graded magnetisation that would beat the sin(π/M)/(π/M) penalty outright, switchable nested cylinders, and hybrid tracks pairing Halbach arrays with high-temperature-superconducting tape.

The same magnet mass, arranged four ways. The alternating N–S row is not a rival to the Halbach array — it is the two-block-per-period Halbach array, and the √2 gain is just 0.900/0.637.

Arrangement	Blocks per wavelength M	Strong-face factor sin(π/M)/(π/M)	Back face
Single uniformly magnetised slab	—	no periodic sheet; field returns round the edges	Equal to the front — fully two-sided
Alternating N–S–N–S row	2	2/π = 0.637	Identical to the front, by symmetry
Classic Halbach, 90° steps	4	0.900 (√2 × the N–S row)	Third-harmonic residue, ~1.4% of the front at λ/4 gap
Fine-segmented Halbach	8	0.974	Seventh-harmonic residue — negligible
Ideal continuous rotation	∞	1.000	Exactly zero for an infinite array