Illustrated Physics
physics, drawn
Electromagnetism

The magnets that put their whole field on one side

Lay bar magnets in a row, each turned a quarter turn from the last, and the row's field appears almost entirely on one side, with almost nothing on the other — although every magnet in it, alone, is perfectly symmetric. Turn the magnetisation smoothly instead of in steps and the weak side has no field at all, exactly. Nothing is shielding it. The two halves of the pattern each make a symmetric field, and their fields line up on one side and cancel on the other, because a magnetisation that turns as it goes can only make the field that decays in one direction.
16 min read 5 figures The shape decidesFields, not forces

Assumes: The loop that behaves like a needle · One number for every point, and nothing at all is lost

A flexible fridge magnet clings to a steel door on one face and barely to anything on the other. Turn it over and it slides off. The obvious explanation — that the back is coated with something that blocks the field — is wrong: there is nothing on the back. The sheet is magnetised in stripes, and the pattern of the stripes puts nearly all the field on one side.

The trick was noticed by John Mallinson in 1973, who was working on magnetic recording at Ampex and was thinking about how the pattern of magnetisation along a tape presents its field to a reading head; he called it a magnetic curiosity, and independently developed by Klaus Halbach at Berkeley in 1980 for the magnets that steer and wiggle electron beams in particle accelerators. Its explanation needs nothing but the superposition of fields that the loop that behaves like a needle used to build every magnet out of small current loops, and it ends at a statement that sounds impossible: a row of magnets whose field on one side is not weak but exactly zero.

A row of magnets turning as it goes

Take a row of identical blocks of permanent magnet and magnetise them in turn up, sideways, down, sideways the other way, up again — each a quarter turn from the last. Every block alone has a north pole and a south pole and a field on all sides of it. The row has a field almost entirely above it.

A row of magnets with its field on one side. Two wavelengths of a Halbach row: blocks of permanent magnet, 4 to a wavelength, each magnetised a 90° turn from the last (block arrows), with the magnetic field drawn as arrows above and below, their lengths proportional to its strength. Just above the row the field is 5 times stronger than the same distance below it. The magnetisation turning one way along the row puts the field on that side; turned the other way, it would put it underneath. With finer blocks the weak side empties further, and a continuously turning magnetisation leaves it with no field at all.
Fig. 1 Two wavelengths of a Halbach row: blocks magnetised up, left, down, right and up again (blue arrows), with the magnetic field above and below drawn as red arrows of length proportional to its strength. A twentieth of a wavelength from the row, the field above is five times the field below; the sense of the turning decides which side is strong.

The pattern repeats every four blocks, and that repeat length is its wavelength λ\lambda. Reverse the sense of the turning — up, sideways to the right instead of the left — and the strong field moves to the other side. The magnet is not symmetric between its two faces because the pattern is not: a magnetisation that turns clockwise as it moves along the row looks anticlockwise from the other side, and the two faces see different patterns.

Two symmetric halves

The cleanest way to see what is happening is to take the row apart. Keep only the blocks magnetised up and down, leaving gaps where the sideways blocks were: that is an ordinary row of alternating magnets, north up, south up, north up, and its field is the same above and below by symmetry. Keep only the sideways blocks: again a row of alternating magnets, now lying on their sides, and again a field the same size above and below.

Two half-arrays that agree on one side and cancel on the other. The peak field against position above (right) and below (left) a row a quarter-wavelength thick, in units of μ₀M₀, for its two halves taken separately — only the blocks magnetised up and down, alternating; only the blocks magnetised sideways, alternating — and for both together, the Halbach row. Each half alone is symmetric: 0.275 above and 0.275 below at a twentieth of a wavelength for the up–down blocks. Their fields have the same size, but the sideways half's field is shifted a quarter wavelength forward above the row and backward below it. Above, the two shifts line up and the fields add, to 0.549; below they are half a wavelength apart and cancel, to 0.0002.
Fig. 2 The peak field against distance above (right) and below (left) a row a quarter-wavelength thick, in units of μ0M0\mu_0M_0: only the up-and-down blocks, only the sideways blocks, and both together. Each half alone is symmetric, 0.275 on each side at a twentieth of a wavelength; together they add to 0.549 above and cancel to 0.0002 below.

The two half-rows make fields of exactly the same size, and that is not a coincidence: rotating every block of the up-and-down row by a quarter turn, and shifting it a quarter wavelength along, turns it into the sideways row. What differs is where their fields peak. Above the row, the sideways half’s field pattern is shifted a quarter wavelength forward relative to the up-and-down half’s; below it, a quarter wavelength backward. A quarter wavelength each way in field pattern means the two fields above are in step and the two below are half a wavelength out of step. They add above and cancel below.

This is superposition and nothing else. No part of the row absorbs the field below or steers it upward; each block produces exactly the field it would produce alone, and the fields of all the blocks simply sum. Below the row they sum to almost nothing because the contributions of the up-and-down blocks and the sideways blocks are equal and opposite there, point by point.

Where the missing flux goes

A field on one side only raises an obvious worry about flux. The field with no ends established that magnetic field lines never begin or end: every line that leaves a surface must return to it somewhere. An ordinary bar magnet sends its lines out of the north face, round through the space on both sides, and back into the south face. If the Halbach row’s weak side has no field, the lines leaving its strong face must find their way back without passing through the space below.

They do so by returning through the magnet itself and through the strong side. Above the row, the field lines form a chain of arches, each leaving the strong face over a block magnetised upward and re-entering it over a block magnetised downward half a wavelength along. Inside the magnet the lines run sideways, through the blocks magnetised sideways, from the bottom of each arch to the bottom of the next. The sideways blocks are doing the job that a sheet of iron does under an ordinary row of magnets — carrying the return flux — except that they are sources of field in their own right, adding to the arches rather than merely closing them. That is why a Halbach row’s strong side is twice as strong as the field of its up-and-down blocks alone with nothing behind them: the sideways blocks do not merely close the circuit, they put their own share of flux into the arches. Iron behind an ordinary alternating row also closes the circuit and also strengthens the front, which is why most magnet assemblies have a steel backing plate; the Halbach row does the same job with magnet instead of steel, which costs more material and saves the weight and the saturation limit of the iron.

The same accounting explains why the weak face can be close to a steel object without much effect. With no field leaving it, there is no flux for the steel to short-circuit and no force to pull the steel in. A fridge magnet’s back face will not hold a sheet of paper to the door; its front face, with the full field of the arches concentrated within a stripe width of the surface, holds a few sheets and then gives up as the field’s exponential decay over a few millimetres takes it away.

Why the weak side can be exactly empty

The quarter-wavelength shift has a deeper reason, and it gives the exact result. Outside the magnet there are no currents, so the field there can be written as the gradient of a potential that satisfies Laplace’s equation — the same equation that one number for every point found governing electrostatic potential, and that the sheet resistance that forgets the shape used through its link to functions of a complex variable. In two dimensions, a field that varies along the row as cos⁡kx\cos kx, with k=2π/λk = 2\pi/\lambda, and satisfies Laplace’s equation must vary across it as e−kye^{-ky} or e+kye^{+ky}. Above the row the physical solution is the one that dies away upward, and below the row the one that dies away downward.

Now write both as functions of the complex position x+iyx + iy. The solution that dies upward is a function of x+iyx + iy alone, eik(x+iy)e^{ik(x+iy)}; the one that dies downward is a function of x−iyx - iy. A field pattern of either kind has a definite handedness: as one moves along xx, the field vector turns in one sense for the first and the opposite sense for the second. A magnetisation that rotates steadily as it goes, M=M0(−sin⁡kx,cos⁡kx)M = M_0(-\sin kx, \cos kx), has a handedness too, and it can source only the field of matching handedness. It makes the upward-decaying field and has nothing with which to make the downward one. The field below a continuously rotating magnetisation is not small. It is zero, at every point, for any thickness of magnet — Mallinson’s result.

The argument also says what does not matter. The thickness of the row changes how strong the field is, through the factor 1−e−kd1 - e^{-kd}, but not which side it is on. The material does not matter, so long as it is magnetised in the pattern and keeps it. And the exact zero needs only two ingredients, a rotation that is steady and a row that is long: it is a property of the pattern’s handedness, not a delicate balance of sizes that a small error would spoil, so that the leakage of a real row comes almost entirely from how coarsely the rotation is approximated and from where the row ends.

How the two sides fall away

On the strong side, the field of the continuous row falls as e−2πy/λe^{-2\pi y/\lambda}, by a factor of 535 for each wavelength of distance, and its size at the surface is μ0M0(1−e−kd)\mu_0M_0(1 - e^{-kd}) for a magnet of thickness dd: a row a quarter-wavelength thick delivers 79 per cent of the most a thick one could.

How the two sides fall away. The peak field above a Halbach row (solid) and below it (dashed), in units of μ₀M₀, against distance from the row's surface in wavelengths, on a logarithmic axis, for rows of 4 and 8 blocks per wavelength a quarter of a wavelength thick; the grey line is a continuously turning magnetisation, μ₀M₀(1 − e^(−kd))e^(−ky). On the strong side the field falls as e^(−2πy/λ), a factor of 535 per wavelength, whatever the blocks. On the weak side, a tenth of a wavelength away, four blocks leave 9.4 per cent of the strong side's field and eight blocks 0.25 per cent: what leaks is the blocks' coarseness, in harmonics that are shorter and die faster, so the weak side is emptier the further from it one looks.
Fig. 3 The peak field above (solid) and below (dashed) a Halbach row, against distance in wavelengths on a logarithmic axis, for rows of four and eight blocks per wavelength; grey, the continuous row. The strong side falls as e−2πy/λe^{-2\pi y/\lambda} whatever the blocks. A tenth of a wavelength below, four blocks leave 9.4 per cent of the strong field and eight blocks 0.25 per cent.

A real row is built of blocks, not a smooth rotation, and the steps leak. A row of four blocks per wavelength approximates the continuous rotation by a staircase, and the staircase contains, besides the smooth rotation, small rotations at shorter wavelengths — a quarter, a fifth, a seventh of the main one — some of which turn the wrong way and so put their fields on the weak side. But these leaks have short wavelengths, and a field of wavelength λ/3\lambda/3 dies away three times as fast as the main one. Close to the weak face the leak is noticeable; a tenth of a wavelength away, four blocks leave less than a tenth of the strong field there, and eight blocks a few parts in a thousand. Further away, the weak side empties faster still.

How many blocks a wavelength needs. For Halbach rows a quarter-wavelength thick, the peak field a tenth of a wavelength above (circles) and below (squares), as a fraction of the continuous row's field above, against the number of blocks per wavelength, on a logarithmic axis; the curve is sin(π/n)/(π/n), the share of the magnetisation's turning the blocks capture. Two blocks a wavelength — plain alternating magnets — are symmetric, with 0.73 on each side. Four blocks put 0.92 of the ideal field on the strong side and 0.111 on the weak; eight, 0.98 and 0.0041. Most practical rows use four or eight: the strong side is nearly finished by four, and the weak side keeps emptying.
Fig. 4 The peak field a tenth of a wavelength above (circles) and below (squares) rows a quarter-wavelength thick, as a fraction of the continuous row’s strong side, against blocks per wavelength; grey, sin⁡(π/n)/(π/n)\sin(\pi/n)/(\pi/n). Two blocks, plain alternating magnets, give 0.73 on each side. Four give 0.92 above and 0.111 below; eight, 0.98 and 0.0041.

The strong side converges quickly: its field is the continuous value times sin⁡(π/n)/(π/n)\sin(\pi/n)/(\pi/n), the fraction of the smooth rotation a staircase of nn steps captures, which is 0.90 for four blocks and 0.97 for eight. The weak side falls by more than an order of magnitude for each doubling. Two blocks per wavelength is just a row of alternating magnets with no sideways ones — symmetric, and with the same field on both sides — so the whole effect comes from the blocks in between. Most practical arrays use four or eight blocks a wavelength: by four, the strong side has nearly all it can get, and every further block buys emptiness on the weak side.

Where one-sided fields are used

A fridge magnet is the cheapest example: a rubber sheet loaded with ferrite powder, magnetised by running it past a patterned magnetising head, so that it carries a stripe pattern with a period of a few millimetres. Its strong side faces the door, and the field dies away within a millimetre or two, which is why a few sheets of paper under it are enough to make it slide.

The serious uses exploit the strong side’s concentration. In a particle accelerator’s undulator, two Halbach rows face each other across a gap a centimetre wide, each with its strong side inward, and an electron beam passing between them is wiggled from side to side by the field, emitting the bright, nearly monochromatic light of a synchrotron source. The light’s wavelength is set by the row’s period shortened twice over — once because the electron sees the period contracted by its motion, and once by the Doppler shift of light emitted forward — so a period of a few centimetres produces X-rays from electrons of a few gigaelectronvolts, the same compression the flash a circling charge sends once a turn found in the radiation of a single bend. Electric motors built with Halbach rotors get a nearly sinusoidal field in the air gap and almost no field inside the rotor, which can therefore be made light and non-magnetic. And a moving Halbach row over a conducting track induces eddy currents that repel it; the drag that falls as the magnet speeds up followed how a moving magnet’s drag over a conductor turns into lift at speed, and the Inductrack design for magnetic levitation of trains uses Halbach rows to put all the magnet’s field into the track below and none into the passengers above. That levitation works only while the row moves, as nothing can be held still by a static field requires of any stable suspension by permanent magnets.

A field inside a ring that exceeds the magnet’s own

Wrap the row round into a ring and the same idea confines the field inside. A ring whose magnetisation turns twice as it goes once around — pointing up at the top, inward at one side, and so on — produces a uniform field in its bore and none outside it.

A ring of magnets with a field above its own remanence. The uniform field inside a Halbach cylinder — a ring whose magnetisation turns twice as fast as the angle around it, so that the field falls entirely inside — against the ratio of its outer to inner radius, for a magnet of remanence 1.4 T: B = Bᵣ ln(rₒᵤₜ/rᵢₙ) for a continuous ring, reduced by sin(2π/n)/(2π/n) for n segments. The field outside is zero and the field inside grows with the logarithm of the ring's size, passing the remanence of the material itself when the outer radius is e = 2.72 times the inner: 1.54 T at a ratio of 3, 3.22 T at 10. Rings of eight or sixteen segments give 90 and 97 per cent of it. Fields above about 4 T are not reached in practice, because the segments nearest the bore feel a reverse field larger than their coercivity.
Fig. 5 The uniform field inside a Halbach cylinder against the ratio of its outer to inner radius, for magnets of remanence 1.4 T: B=Brln⁡(rout/rin)B = B_r\ln(r_{\text{out}}/r_{\text{in}}) for a continuous ring, reduced by sin⁡(2π/n)/(2π/n)\sin(2\pi/n)/(2\pi/n) for nn segments. It passes the remanence at a ratio of ee, reaching 1.54 T at 3 and 3.22 T at 10; eight and sixteen segments give 90 and 97 per cent of it.

The field inside grows as the logarithm of the ring’s thickness, without limit in principle, and it passes the remanence of the material itself — the field inside a long bar of the magnet — once the outer radius is ee times the inner. Permanent-magnet rings producing two tesla in their bore are used for compact nuclear magnetic resonance spectrometers and for magnetic refrigerators, with no power supply and no cooling. The uniformity comes from the same handedness argument as the row’s emptiness: a magnetisation turning twice per revolution sources only the field pattern that turns once per revolution inside the ring — a uniform field — and nothing that grows outward. The two coils one radius apart reached a uniform region by cancelling the field’s curvature with a carefully chosen spacing; a Halbach ring has no curvature to cancel, at least in the ideal of a continuous, infinitely long ring.

The growth stops, in practice, near four tesla, and the reason is the subject of the magnet that reverses where its crystal is weakest. The segments nearest the bore sit in a strong field that points almost against their own magnetisation, and when that field exceeds their coercivity, they demagnetise. A ring designed for five tesla made from magnets of 1.4 tesla remanence would need its inner segments to withstand a reverse field of several tesla, beyond neodymium-iron-boron at room temperature; the strongest rings built use high-coercivity grades near the bore and cool the whole assembly to raise the coercivity further.

What the drawings leave out

Every field here is computed in two dimensions: the blocks are taken to be infinitely long perpendicular to the page, and each block is replaced by the magnetic charge on its faces, which is exact for a uniformly magnetised block in vacuum. Real blocks have finite length, and the field at their ends is three-dimensional; real rows have ends, whose stray fields fall only as a power of distance rather than exponentially and so eventually dominate the weak side far from the row. The blocks are taken to be perfectly magnetised in their nominal directions, with a permeability of one; real neodymium magnets have a relative permeability of about 1.05 and are magnetised to within a degree or two, both of which leak a little field to the weak side.

The figures also cannot show the forces between the blocks, which are what make a Halbach row hard to build. Neighbouring blocks magnetised at right angles push and twist each other strongly, and an array has to be assembled block by block in a fixture and glued, because left to themselves the blocks would rearrange into an ordinary alternating row. The domain of the drawings is a two-dimensional, perfectly magnetised, linear array, observed from a fraction of a wavelength to half a wavelength away.

Still open: how well a three-dimensional field can be shaped

The one-sided row is the simplest member of a family. A ring can be designed to produce a quadrupole field inside, or a sextupole, or a field that rotates with angle; spheres of magnetisation can produce a uniform field in a spherical cavity; and with modern optimisation, arrays of many small blocks are designed to produce a specified field over a volume, for instance the very uniform field a magnetic resonance image needs. How close permanent magnets can come to the fields that superconducting coils make — whether the fringe field can be made to vanish as completely in three dimensions as the weak side does in two, and at what cost in material and coercivity — is an engineering question being pushed by the demand for small, unpowered scanners.

The two-dimensional fact underneath is exact. A magnetisation that turns as it goes along a row can source only the field that decays on one side, because the field outside is a function of x + iy or of x − iy and the turning picks one: so a row of blocks a quarter turn apart puts 0.92 of the ideal field on its strong side and a tenth of that on its weak side, eight blocks leave a few thousandths, and a smooth rotation leaves nothing. Every magnet in the row has two poles and a field all round it; the row, by superposition alone, has a field on one side.

Part 9 of 9

This essay is one argument about Magnetism. The others:

Electrons that reach the anode, and electrons that turn back. Paths of electrons leaving a cathode 4 mm across at rest, in a cylindrical diode with an anode 8 mm across held at 4000 V, with no space charge, for axial magnetic fields of 0.80, 0.98, 1.02 and 1.30 times Hull's cut-off field of 142.2 millitesla; computed by integrating the Lorentz force. Below the cut-off the field only bends the path, and the electron still reaches the anode — at 0.98 of the cut-off it arrives almost tangentially. Just above it the electron turns back just short of the anode and returns to the cathode, then repeats the arc further round; at 1.30 times the cut-off it turns back well inside. Nothing reaches the anode, and the current stops completely within a few per cent of the field. The field at which no electron reaches the anode Part 8 — Put a hot wire inside a metal tube, make the tube positive, and electrons stream across from wire to tube. Add a magnetic field along the axis and they curve on the way, and still arrive — until, at one definite field, every one of them turns back a hair's breadth short of the tube and the current stops dead. The field at which it happens can be found without knowing anything about how the voltage is distributed between the wire and the tube, because one quantity the electron carries is fixed by the magnetic field alone. The microwave oven runs just inside that line.

Where this goes next

A torque, and no force at all. A loop of 1 turn enclosing 20 cm² and carrying 10 A has a magnetic moment of 0.02 A m². In a uniform field of 0.05 T the torque on it is m B sin θ, drawn here against the angle between the moment and the field: zero when they are aligned, largest at 0.001 N m across, and zero again when they are opposed. The second curve is the energy, −m·B, whose minimum is the aligned position and whose maximum is the opposed one — which is why a compass needle settles one way round and not the other. The net force is zero at every angle on this axis, exactly and not approximately: the force is I dl × B summed round the loop, the sum of dl round any closed path is zero, and a constant B comes outside the sum. The inset shows the four forces on a rectangular loop; the pair across the axis is the couple, and the pair along it cancels. The loop that behaves like a needle Far enough away, a current going round in a circle is indistinguishable from a bar magnet, and one number describes both. That number tells a uniform field how to turn the loop and gives it no way to pull on it at all — which is why two magnets attract by the fourth power of the distance and not the second. The tilt that starts the reversal, in a well of soft crystal. A planar defect three wall parameters wide (three times √(A/K)) in which the crystal's anisotropy falls to zero, drawn as the anisotropy K(x) across it; the curve is the shape of the first tilt of the magnetisation to become unstable as a reverse field is raised, the lowest mode of −A d²/dx² + K(x), and the dashed line is its eigenvalue. The equation is Schrödinger's, with the exchange stiffness A in place of ħ²/2m and the anisotropy as the potential, and the reversal begins in its ground state. Here the ground state lies at 0.373 K, so the magnet reverses at 37 per cent of the anisotropy field. The mode reaches out of the well a distance of about √(A/K) on each side, because exchange will not let the tilt change abruptly. The magnet that reverses where its crystal is weakest A perfect crystal of neodymium-iron-boron should hold its magnetisation against a reverse field of 7.7 tesla. The best magnets made of it give way at one to three. The shortfall, known for eighty years as Brown's paradox, is not a failure of the crystal but of its edges: reversal begins wherever the crystal is weakest, in a layer a few nanometres thick. Working out how weak that layer can afford to be turns out to be a problem already solved in quantum mechanics — the nucleation field is the ground-state energy of a particle in a well shaped like the defect. The drag peaks and then gives way to lift. The two forces on a magnet moving parallel to a thin conducting sheet, as fractions of the repulsion a perfect mirror image would exert, against the magnet's speed divided by the sheet's characteristic speed w = 2/μ₀σd, on a logarithmic axis. Drag grows in proportion to the speed at first, peaks at 1.27 w with 0.300 of the full repulsion, and then falls as w/v. Lift starts as the square of the speed, equals drag at v = w, where both are 0.293, and approaches the full repulsion at high speed: 0.900 at 10 w. The same eddy currents cause both. What changes with speed is how much of the image they form has time to sink away before the magnet has passed. The drag that falls as the magnet speeds up Drop a magnet down a copper pipe and it creeps, because the currents it stirs up in the metal pull back on it in proportion to its speed. Push it faster and the proportion fails: the drag rises to a peak and then falls away, while a second force, pointing straight up, grows to replace it. A magnet moving fast enough over a sheet of aluminium is not braked but held up, with lift thirty times its drag at the speed of a fast train, and the reason is that at high speed the metal has no time to forget the magnet has passed.

The objects named here

The third axis, after the field and the reading path: the things themselves, and every essay that touches each one.

CoercivityHalbach arrayLaplace equationMagnetic fieldMagnetisationPermanent magnetRemanenceSuperposition