Illustrated Physics
physics, drawn
Concept

Coercivity — where it appears

The reverse field needed to bring a magnetised material back to zero magnetisation, and the number that divides a permanent magnet from a transformer core. It is not a property of a material alone: it depends on how long the field is applied, because a barrier of finite height is crossed eventually.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Where a magnet actually sits on its own curve. The second quadrant of a magnet's B–H curve, for a material with a remanence of 1.28 T, and the load lines four shapes of it impose. A magnet's own poles put it in a reverse field of N·M, so the working point is where the curve meets the line B = −μ₀(1−N)/N·H. A long thin magnet with N = 0.02 keeps 98 per cent of its remanence and a squat one with N = 0.7 keeps 30 per cent — the same material, cut differently.

The magnet that has to fight its own field

A bar magnet's own poles put it in a reverse field, so the same material cut short and fat is weak and cut long and thin is strong. And a magnetised material does not have a magnetisation — it has a magnetisation and a history, which is why the word for what it does is the Greek for coming late.

electromagnetism · Magnetisation Two energies with opposite slopes, and the width between them. The energy per unit area of a domain wall in iron, against the width the wall is assumed to have, split into the two terms that decide it. The exchange term falls as one over the width, because a reversal spread over more atoms turns through a smaller angle between each neighbouring pair. The anisotropy term rises in proportion to the width, because every atom inside the wall points away from an easy direction and pays for it. The sum has a least value, found here by scanning four hundred thousand widths: 92.9 nanometres and 4.46 millijoules per square metre. Neither constant is a dimension of the sample, so neither is the width: it is the first length in magnetism that belongs to the material. The assumed profile turns at a uniform rate, which is not the cheapest way to turn, so this width is 41 per cent above the conventional πδ of the exact profile, and this energy 11 per cent above the exact 4√(AK). A trial function can only ever overestimate, and eleven per cent is how much this one costs.

The first length that belongs to the substance

Every length in magnetism so far has been a length of the sample — a demagnetising factor is a shape, an avalanche cutoff is a sample's own restoring field. A domain wall's width is not. It is √(A/K), made of two material constants and nothing else, and across seven ordinary magnets it runs from two and a half nanometres to nine hundred.

electromagnetism · Magnetisation The barrier a reverse field takes away. The energy of a single-domain particle against the direction its moment points, in units of its anisotropy energy, for four strengths of reverse field along the easy axis. With no field the two directions are equally good and the barrier between them is exactly the anisotropy energy. A reverse field tilts the landscape and lowers the barrier out of the forward well as the square of the field: at 0.0 of the anisotropy field the barrier is 1.000 KV, at 0.1 of the anisotropy field the barrier is 0.810 KV, at 0.3 of the anisotropy field the barrier is 0.490 KV, at 0.6 of the anisotropy field the barrier is 0.160 KV. Each barrier is located by scanning two hundred thousand directions for the stationary points rather than by substituting the formula. The whole of the argument follows from the barrier being finite: a particle does not need the field that removes the barrier, only time enough to be shaken over what is left of it.

Nothing keeps a magnetisation for ever

A magnetised particle sits in a well with a barrier between it and the other direction, and a barrier of finite height is crossed eventually. So remanence has a lifetime, coercivity is a different number depending on how fast it is measured, and a grain below about twenty nanometres of iron forgets within a second at room temperature.

electromagnetism · Magnetisation 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.

electromagnetism · Magnetisation 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.

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.

electromagnetism · Magnetism

Named alongside it

The objects these essays reach for when they reach for this one.

MagnetisationDemagnetising fieldHysteresisAnisotropyDomain wallExchange lengthMagnetic anisotropyRemanenceSingle-domainArrhenius lawDissipationDomains

All concepts