The magnet that has to fight its own field

Cut four magnets from one billet. Make the first a long thin rod, the last a squat disc, and the other two in between. Magnetise them all to saturation with the same coil and then measure what each produces. The rod gives nearly the full flux density the material is capable of. The disc gives a third of it. Nothing has been done to the material.

The number $N$ in that construction is the same depolarising factor that decides the field inside a dielectric, computed from the same elliptic integral, obeying the same sum rule. Electric and magnetic polarisation are different physics and the geometry is identical, because the geometry is a statement about a divergence and a boundary rather than about what is doing the polarising.

Why a magnet opposes itself #

A magnetised body has, in the pole picture, north pole strength spread over the surface where the magnetisation emerges and south where it enters. Those poles make a field, and inside the body that field runs from north back to south — which is to say, against the magnetisation. The poles are a bookkeeping device rather than objects, since the field has no ends and no sources; what is real is the divergence of the magnetisation, and calling it a pole density is a way of importing the electrostatic machinery wholesale.

The depolarising factor along a spheroid’s axis, plotted against shape, was computed for a dielectric and applies here unchanged — the two problems are the same boundary-value problem with different symbols. That is why a long thin magnet is so much easier to magnetise than a squat one: the factor along the long axis is small, so the field the body turns against itself is small, and the same material behaves quite differently in two shapes.

So the interior field $H$ is negative when the magnetisation is positive, and a permanent magnet in free space is always operating in the second quadrant of its own $B$–$H$ curve. It is not a device that has been put in a reverse field by an experimenter; it is a device that is in one because of its own existence.

The consequences are practical and slightly counterintuitive. Shortening a magnet weakens it disproportionately, because $N$ rises steeply as an aspect ratio falls below about three. A keeper works, because a soft-iron bar bridging the poles provides a low-reluctance return path, drops the demagnetising field nearly to zero, and moves the working point back up the curve. And a magnet can be destroyed by its own shape: a material whose demagnetisation curve has a knee, like an alnico or a ferrite, can be pushed past that knee by a large enough $N$, and then it does not recover when the shape is restored.

A magnetised body is a stack of current loops with their interiors cancelling, so what is left is a surface current around the outside. That is the honest picture of where a magnet’s field comes from, and it explains the self-opposition directly: the surface current makes a field that runs backwards through the material even while it runs forwards outside it.

The shape that maximises the energy is not the longest one #

The four shapes in the opening figure suggest an obvious design rule — make it long and thin, get closer to the remanence, win. The rule is wrong, and seeing why is the most useful thing the load-line construction does.

What a permanent magnet is for is not a large $B$ inside itself. It is a field outside itself, and the energy in that external field turns out to be proportional to the product $BH$ at the working point, taken with a sign that makes it positive in the second quadrant. That product is the figure of merit, and it is what a magnet is graded by.

Now look at what the load line does to it. A very long thin magnet sits high on the curve with $B$ near the remanence and $H$ close to zero, so the product is small — nearly full flux and almost no field driving it. A very squat one sits low, with a large $H$ and a small $B$, and the product is small again. The maximum is somewhere in between, and for a material whose demagnetisation curve is a straight line — which the rare-earth magnets very nearly are — it is exactly halfway: at $B$ equal to half the remanence, with

$$(BH)_{\max} = \frac{B_r^2}{4\mu_0}.$$

For the 1.28 tesla remanence drawn in the opening figure that is about 330 kilojoules per cubic metre, which is the grade of a good neodymium magnet, and it is reached at a load line where $B$ over $\mu_0 H$ is one — a demagnetising factor of a half, which is a distinctly stubby shape.

So the best use of a given volume of magnet is not the shape that gets nearest to its own remanence. A magnet operating at 98 per cent of remanence is being wasted: it is a lot of material producing very little external field, because almost all of its flux is returning through itself rather than through anywhere useful. The design rule that comes out of the construction is the opposite of the one the first reading suggests, and it is why magnets in motors and loudspeakers are discs and blocks rather than needles.

What a loop is made of #

The demagnetisation curve above was taken as given. It is the second quadrant of a hysteresis loop, and a loop is worth building from something rather than drawing from memory.

The smallest object that has one is a single particle small enough to be uniformly magnetised, with a preferred axis. Its energy has two terms — an anisotropy that prefers the easy axis, and the field, which prefers its own direction — and the magnetisation sits at a minimum of the sum. Sweep the field and follow that minimum, and the loop appears without any further ingredient.

That is what hysteresis is, stated as generally as it can be: a system following a local minimum after that minimum has stopped being the lowest one. Nothing about magnetism is required. The same structure produces the metastable branches of a van der Waals isotherm, the buckling of a strut that stays buckled, the contact angle a drop shows depending on whether it advanced or retreated, and the two states of a light switch.

The angle dependence is the part nobody guesses. A field along the easy axis has to overcome the whole anisotropy. A field at forty-five degrees has a component along the axis that pulls and a component across it that torques, and the two together do the job with half the field.

What a material is made of #

A real material is not one particle. It is an enormous number of regions with a distribution of anisotropies, sizes and orientations, and — because they are close enough to feel one another — a distribution of interaction fields. Summing rectangular switches over such a distribution is the Preisach model, and it is enough to produce every qualitative feature of a real loop.

The initial curve is the one part of that figure which cannot be revisited. Once a sample has been taken round the major loop, no field history returns it to the virgin curve; the only route back is to demagnetise, which means an alternating field of decreasing amplitude — the same procedure the figure uses, and the same one a workshop uses on a screwdriver that has picked up a magnetisation it should not have.

That irreversibility is not a metaphor. A loop encloses an area, and the area is energy.

Every cycle of the loop dissipates that area as heat, per unit volume, and a transformer core running at fifty hertz goes round it fifty times a second for its whole working life. Hysteresis loss is one of the two terms in a core’s heating; the other is the eddy currents the changing flux drives in the iron itself, and the two scale differently with frequency, which is how they are told apart in a measurement.

The loop is not smooth #

Every loop drawn here is a continuous curve, and a real one is not. Wind a coil round a piece of iron, connect it to an amplifier and a loudspeaker, and slowly bring a magnet up to it: what comes out is not a smooth swelling tone but a hiss, like sand pouring.

That is Barkhausen’s observation, from 1919, and it is the most direct evidence there is for what the Preisach picture assumes. The magnetisation does not advance smoothly; it advances in a very large number of discrete jumps, each a domain wall breaking free of whatever was holding it and running to the next obstacle. Each jump is a sudden change of flux, each change of flux induces a voltage pulse in the coil, and the pulses arrive as noise.

Two things follow that are worth having. The jumps are the irreversibility: a wall that snaps forward does not snap back when the field is eased, and the energy released in the snap is dissipated rather than returned. So the noise is audible confirmation that the loop’s area is a loss, and a material with a narrower loop is a quieter one.

And the statistics of the jumps are measurable. Their sizes follow a power law over several decades, with no characteristic scale — the same signature that appears in a great many systems where something is dragged through a disordered landscape, from crack fronts to earthquakes to the wetting of a rough surface. Which is a hint that the phenomenon this essay describes is less about magnetism than the vocabulary suggests: what makes a loop is a state persisting after it stopped being optimal, and what makes it noisy is a landscape with obstacles in it at every size.

Why the area is an energy #

The claim that the enclosed area is a loss per unit volume per cycle deserves its one line of derivation, because it is what makes hysteresis a cost rather than a curiosity.

Whatever is driving the sample — a winding, a moving magnet — has to change the flux through it, and changing a flux against an induced emf takes work. Per unit volume, the work done in changing the flux density by $\mathrm{d}B$ while the internal field is $H$ is $H\,\mathrm{d}B$. Take the sample once round a closed cycle and the total work per unit volume is $\oint H\,\mathrm{d}B$, which is the area the loop encloses.

If the material were reversible the path out and the path back would coincide, the integral would be zero, and every joule put in on the rising branch would come back on the falling one. It is exactly the non-coincidence of the two branches — the fact that the state depends on the history — that makes the integral non-zero, and its sign is fixed: the return path always lies inside the outward one, so the work is always positive and always leaves as heat.

That is the general statement, and it is not about iron. Any hysteretic system dissipates its loop area per cycle, which is why a rubber band warms when it is stretched and released repeatedly, and why a damper built on a hysteretic material works at all.

How a loop is measured, and why the sample is a ring #

Everything above says a magnetic measurement is contaminated by the shape of what is being measured, which raises the obvious question of how anybody obtains a material’s curve at all. The answer is to use the one shape with no demagnetising field: a closed ring.

Ampère’s law taken round the centreline of a wound toroid gives $H$ directly from the winding current and the path length, with no geometry to correct for. That is why a toroid is the standard specimen for measuring a hysteresis loop — it has no ends, so the demagnetising factor is zero, and the measurement reports the material rather than the shape.

A toroid has no ends, so the magnetisation never emerges through a surface, so there are no poles and no demagnetising field: N is zero along the ring and the applied H is the internal H. Wind a primary to drive it and a secondary to sense the flux, integrate the secondary’s voltage, and the result is the material’s own loop with no geometry to be corrected for.

The sensing winding of a hysteresis measurement is a loop around the sample, and its voltage is the rate of change of flux — so the loop is obtained by integrating that voltage against the drive current. The measurement is therefore of a derivative, and the integration is where the drift and the calibration difficulties live.

Everything else — a rod, a horseshoe, a fridge magnet — has to be measured on a curve obtained this way and then placed on it by its own load line, which is exactly the construction the hero figure draws.

Hard and soft, and the trade behind the words #

The energy stored in a magnetic field is a density spread through space rather than a property of the source, and a magnet’s demagnetising field carries some of it — which is the other way of saying why a short magnet is weak: it is paying an energy for its own stray field. A material with a wide loop is called hard and one with a narrow loop soft, and the two are wanted for opposite jobs. A permanent magnet needs a wide loop, so that its working point is far from anything that could disturb it and so that a stray field cannot rearrange it. A transformer needs a narrow one, because the area is a loss it pays continuously.

Every magnetic device is a compromise between the field it is meant to produce and the losses that come with cycling it. A hard material keeps its magnetisation and is hard to change; a soft one changes easily and keeps nothing — and the area of the loop, which is the energy lost per cycle, is what a transformer designer minimises and a permanent-magnet designer wants large.

What separates the two is anisotropy — the energy difference between magnetising along the preferred direction and across it. Rare-earth magnets have enormous anisotropy from the crystal field on a 4f shell — the same shell structure whose exclusion rules decide what an atom looks like — so their loops are wide and their coercivities measured in megaamps per metre. Transformer silicon steel is grain-oriented precisely to reduce the effective anisotropy the flux meets, so its loop is narrow. The same design variable, turned two ways.

Where the model stops #

The Stoner–Wohlfarth particle is coherent and most particles are not. Above a critical size it is cheaper to form a domain wall than to hold a uniform magnetisation against its own demagnetising energy, and then the reversal happens by a wall sweeping through rather than by everything rotating together. That critical size is tens of nanometres for iron, and the coercivities the coherent model predicts are far larger than any bulk material shows — the discrepancy is called Brown’s paradox and it is the standard evidence that real reversal is nucleated at defects.

The Preisach model has no physics in it. It is a fitting framework: a distribution is chosen to reproduce a measured loop and then used to predict other loops. It does that well and it says nothing about why the distribution has the shape it has.

A magnetic moment is not a little bar magnet. Every dipole in this essay is a current loop or an intrinsic angular momentum, and the pole picture that makes the demagnetising argument easy is a device that fails the moment anything moves — a monopole would radiate differently, feel a different force in a moving frame, and violate the divergence law. The two pictures agree on every field they both compute and disagree about what exists.

Nothing here is a temperature. Every anisotropy falls as a material is heated and vanishes at the Curie point, so a magnet’s coercivity is strongly temperature-dependent and a magnet run too hot loses its magnetisation permanently. That the ordering itself disappears at a definite temperature is a phase transition, and belongs with the essays that own those.

And the demagnetising factor assumes uniform magnetisation. A real bar magnet is not uniformly magnetised — it is more strongly magnetised in the middle than at the ends — so its working point varies along its length and the single N is an average.

What the pictures cannot show #

No figure here shows a domain. The Preisach figure draws a sum over switches and there is no picture of what a switch is; the Stoner–Wohlfarth figure draws one particle and cannot show that a real sample contains 10¹⁵ of them with a distribution the figure had to be told.

Nor can any drawing show the thing that makes a loop a loop, which is that the state depends on the path. Every value of the field on those axes carries three or more values of the magnetisation, so the drawing is not a graph of a function, and the arrows that are conventionally added to say which way round it is traversed are carrying information the geometry does not.

Where this ladder goes next #

This is the first rung of the magnetisation anchor, and it establishes the two facts everything else rests on: a magnetised body sits in its own reverse field, and its state is a function of its history rather than of the field it is in.

The rungs above it are the ones this essay named and declined. The exchange interaction, which is why any of this happens — the reason a piece of iron has a magnetisation to begin with is a quantum-mechanical energy that has no classical counterpart, and it is far larger than any magnetic dipole interaction between the same moments. Domains, which exist because a uniformly magnetised body pays an enormous demagnetising energy and can avoid most of it by dividing itself. And the ordering temperature, where the whole business disappears at once.

The habit worth carrying away is the one the shape-dependence forces. Before attributing a device’s behaviour to its material, work out what its geometry would have done to any material. Half the difference between two magnets is usually the aspect ratio, half the difference between two capacitors is usually the fringing, and the fraction of a measurement that is about the sample rather than the substance is the first thing worth estimating and the last thing usually estimated.