Electromagnetism

The field that points against the magnet it is in

There are two magnetic fields in use and the difference between them is which currents a loop is allowed to count. The consequence nobody expects on being told the definitions: inside a permanent magnet H points the other way from B. It has to — a loop inside the magnet threads no wire, so its H circulation is zero, and the only arrangement left has H running backwards.

18 min read 6 figures Fields, not forces Approximations that lie

Assumes: The law that is always true and rarely useful · The field that wraps a current

The repair that makes Ampère’s law exact ends with Ampère’s law repaired: the circulation of the magnetic field round a closed path counts every current threaded through it, conduction and displacement together, and the sum is exactly $\mu_0 I$ at every length of wire.

“Every current threaded” is the phrase that has to be taken seriously once there is matter in the loop. A lump of iron contains electrons in orbit and electrons with spin, every one of which is a current loop behaving like a needle, and when the material is magnetised those currents no longer cancel. They are real currents — they produce real magnetic fields, and they are threaded through a real loop.

So the circulation of $\mathbf{B}$ round a path inside magnetised material counts them, and there are an enormous number of them.

Two fields, and the only difference is bookkeeping

The response is to define a second field that counts only the currents somebody put there on purpose:

\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \mathbf{M},

where $\mathbf{M}$ is the magnetisation — the magnetic moment per unit volume, which is the density of all those atomic currents. Subtracting it removes exactly the bound current from the circulation, and what is left is

\oint \mathbf{H}\cdot d\boldsymbol{\ell} = I_{\text{free}},

with no material constant in it. That is a useful law, because the free current is the thing an ammeter reads.

And then a consequence follows immediately that most treatments state and few draw.

Fig. 1 A uniformly magnetised sphere, with the field lines of B on the left and of H on the right, both from the exact solution — uniform inside, a dipole outside. Outside they are the same picture up to a constant. Inside they are opposite: B is 0.67 tesla along the magnetisation and H is 267 kiloamps per metre against it.

Take a closed path lying entirely inside a permanent magnet. No wire passes through it, so the free current threaded is zero, so the circulation of $\mathbf{H}$ round that path is zero. But $\mathbf{B}$ certainly circulates — its lines are closed loops and some of them lie inside the magnet. A field with no circulation and a field with circulation cannot be parallel everywhere.

The arrangement that satisfies both is the one in the figure. Inside the magnet, $\mathbf{H}$ points from the north face to the south — against the magnetisation, against $\mathbf{B}$ — and outside it runs the other way, so that a path going up through the magnet and back round the outside gets zero in total.

That is why it is called a demagnetising field. It is the magnet’s own $\mathbf{H}$ , it opposes the magnetisation that produced it, and it is the reason a short fat magnet is weaker than a long thin one — a shape that pushes its poles closer together produces a larger opposing field between them.

The lines that end, and the lines that do not

The two panels differ in a second way that is easier to see than to say.

The $\mathbf{B}$ lines are closed. Every one of them leaves the north face, goes round outside, comes back in at the south face and continues up through the magnet to where it started. None of them begins anywhere and none of them ends. That is not a property of this magnet; it is the statement that isolated magnetic charge has never been found, and it is exact.

The $\mathbf{H}$ lines begin on the top face and end on the bottom. They have sources, the sources sit on the surface, and their strength is the component of the magnetisation across it. That is the point of the second field beyond the bookkeeping: $\mathbf{H}$ in a current-free region has no circulation and does have divergence, so it behaves exactly like an electrostatic field with charges on the pole faces — which makes every method from electrostatics available.

Both statements can be held at once without contradiction. $\mathbf{B}$ has no sources and does have circulation; $\mathbf{H}$ has sources and, away from free current, no circulation. They are two different fields and the one thing they share is that neither is more real than the other.

Fig. 2 The same construction for a much weaker magnetisation — a ferrite rather than a rare-earth alloy. The pictures are identical, because the field of a uniformly magnetised sphere is linear in its magnetisation and the geometry contains no scale. What changes is only the numbers: 0.21 tesla inside and 83 kiloamps per metre against it.

What a core is for, counted

The other reason for two fields is practical, and it is the whole of transformer and motor design.

What is threaded through the loop, once and then again. A ring core wound with 50 amp-turns, with a circular path taken round inside it. For each of three core materials the figure shows what that path threads: the free current in the winding, which is the same in all three, and the bound current — the atomic circulations of the material, aligned by the field and adding up to a sheet of current round the core's surface. The circulation of H counts the first and the circulation of B over μ₀ counts both. The bound current exceeds the free by the relative permeability less one, which here is up to 4999, and that is the entire content of a magnetic core: it is a way of getting a few thousand times more threaded current than was paid for in copper. — Fig. 3 A ring core wound with fifty amp-turns, and what a path taken round inside it threads. The free current is fifty in every case. The bound current — the atomic circulations aligned by the field, adding up to a sheet of current round the core’s surface — is the relative permeability less one times that, which here reaches nearly a quarter of a million amp-turns.

A magnetic core is a device for getting a few thousand times more threaded current than was paid for in copper. The winding supplies fifty amp-turns; the material supplies a quarter of a million; and the field is what the total produces.

Written with $\mathbf{H}$ the calculation is one line and contains only the fifty. The material enters through the relation between $\mathbf{B}$ and $\mathbf{H}$ — which for a soft material is a permeability, a number between a few hundred and a hundred thousand — and which for a superconductor is the extreme case, since a field expelled entirely means B is zero inside and H is exactly minus the magnetisation — and that division of labour is what makes the second field worth having.

It is also where the trouble starts, because for a permanent magnet there is no such number. The figure at the top of this essay is the proof: $\mathbf{B}$ and $\mathbf{H}$ point in opposite directions inside it, and no positive permeability relates two vectors that disagree about their direction. A permanent magnet has to be described by its actual curve of $\mathbf{B}$ against $\mathbf{H}$ , in the quadrant where they have opposite signs, and the operating point is found by intersecting that curve with the demagnetising line the shape imposes.

Why the two are so persistently confused

The names do not help and neither do the units, and it is worth being explicit about both, because the confusion is not carelessness — it is built into the vocabulary.

$\mathbf{B}$ is measured in tesla and $\mathbf{H}$ in amps per metre. That second unit is the giveaway: $\mathbf{H}$ is a current per unit length, which is precisely what its defining law says it is — a circulation per unit path is a current threaded, so a field per unit length is a current. A solenoid of a thousand turns per metre carrying an amp produces $H = 1000$ A/m inside it, with no material constant anywhere. The unit is the law.

$\mathbf{B}$ is measured in tesla because it is the field that appears in the force law: the force on a moving charge is $q\mathbf{v}\times\mathbf{B}$ and there is no version of it with $\mathbf{H}$ in it. A compass needle, a loudspeaker coil and a mass spectrometer all respond to $\mathbf{B}$ . That is the strongest argument for calling $\mathbf{B}$ the magnetic field and it is the one most working physicists use.

Which leaves the names, which run the other way round. $\mathbf{H}$ was named “the magnetic field” and $\mathbf{B}$ “the magnetic flux density” or “magnetic induction” — a nomenclature inherited from the nineteenth century, when $\mathbf{H}$ was conceived as the field produced by magnetic poles and $\mathbf{B}$ as what it induced in matter. That picture was abandoned and the names were not. Most textbooks now call $\mathbf{B}$ the magnetic field and are careful to say which convention they are using, and older references and a good deal of engineering practice do not.

The Gaussian unit system adds a last twist. In it $\mathbf{B}$ and $\mathbf{H}$ have the same dimensions and are equal in vacuum, measured in gauss and oersted respectively — two names for one quantity where there is no matter, and two genuinely different quantities where there is. That is exactly the arrangement most likely to make the distinction look like a formality, and it is why a great deal of magnetics literature reads as though the two were interchangeable.

None of it changes the physics in the first figure. Whatever the two fields are called and whatever units they are in, one of them circulates inside the magnet and the other does not, and they point opposite ways.

The millimetre that takes the field away

The magnetic-circuit calculation makes one prediction that surprises everybody the first time.

One millimetre of air against three hundred of iron. The field in the gap of a ring core of mean path 300 millimetres wound with 50 amp-turns, against the length of the gap, for three core materials. The whole calculation is one line: the circulation of H round the path is the free current, and since B is the same in the core and in the gap, H in the gap is larger than H in the core by the relative permeability. So the gap, however short, takes nearly all of the available circulation. At the highest permeability drawn, a gap of one millimetre leaves 5.7 per cent of the field the closed core had. Nothing about the iron changed; the missing circulation is being spent on a millimetre of air. An ordinary core saturates near 1.8 tesla, which is above every field drawn here, so the whole family is inside the linear regime and the calculation applies throughout. — Fig. 4 The field in the gap of a ring core of mean path three hundred millimetres wound with fifty amp-turns, against the gap’s length. The whole calculation is one line: the circulation of H round the path is the free current, and B is the same in the core and the gap, so H in the gap exceeds H in the core by the relative permeability. A one-millimetre gap leaves under six per cent of the field the closed core had.

Put the numbers in. The circulation is $H_c \ell_c + H_g \ell_g = NI$ . The flux is the same all the way round, so $B$ is the same in both; but $H = B/\mu_0\mu_r$ in the core and $B/\mu_0$ in the gap, so $H_g$ is $\mu_r$ times $H_c$ . With $\mu_r = 5000$ , a gap of one millimetre is worth five metres of iron.

Three hundred millimetres of core and five thousand millimetres of equivalent path in the gap: the gap takes 94 per cent of the circulation and the iron takes six. Almost all of the magnetomotive force somebody paid for in copper is being spent on a millimetre of air.

Which is why a motor’s air gap is machined to tenths of a millimetre and is the single most expensive tolerance in it; why a transformer core is built from interleaved laminations with no gap anywhere in the magnetic path; and why the standard method for making an inductor whose inductance is stable is to put a deliberate gap in it, since a gap dominated circuit depends on a length that does not change rather than on a permeability that does.

One millimetre of air against three hundred of iron. The field in the gap of a ring core of mean path 300 millimetres wound with 200 amp-turns, against the length of the gap, for three core materials. The whole calculation is one line: the circulation of H round the path is the free current, and since B is the same in the core and in the gap, H in the gap is larger than H in the core by the relative permeability. So the gap, however short, takes nearly all of the available circulation. At the highest permeability drawn, a gap of one millimetre leaves 5.7 per cent of the field the closed core had. Nothing about the iron changed; the missing circulation is being spent on a millimetre of air. The dashed line is where an ordinary core saturates, above which none of this applies at all. — Fig. 5 The same curves over a shorter range of gap and a four-times-larger winding, which is where a real design sits. The saturation line is now crossed by the highest-permeability core at gaps under a tenth of a millimetre: below that the figure is drawing a field the iron cannot carry, and nothing on this page applies there.

The circuit the calculation is a circuit of

The gap arithmetic is usually done by a formal analogy, and the analogy is exact enough to be worth stating and limited enough to be worth bounding.

Write the magnetomotive force as $NI$ , the flux as $\Phi = BA$ , and the reluctance of a piece of path as $\ell/\mu A$ . Then the circulation law becomes

NI = \Phi \sum_k \mathcal{R}_k,

which is Ohm’s law with the winding as the battery, the flux as the current and the reluctances in series. Reluctances in a branched core add in parallel exactly as conductances do, for the same reason: the flux divides and the magnetomotive force across each branch is the same.

The gap result falls straight out of it. The gap’s reluctance is $\ell_g/\mu_0 A$ and the core’s is $\ell_c/\mu_0\mu_r A$ , so their ratio is $\mu_r \ell_g/\ell_c$ — five thousand times one over three hundred, which is seventeen. Seventeen parts in eighteen of the total reluctance is a millimetre of air.

Three things the analogy does not carry, and each has bitten somebody:

There is no dissipation. A resistance dissipates power and a reluctance stores energy. Nothing is lost to a reluctance, and the energy put into a gap comes back out when the current falls — which is why a gapped inductor stores its energy in the gap rather than in the iron, and why the gap volume, not the core volume, sets how much energy an inductor can hold.

Flux leaks and current does not. A wire confines current absolutely; a core does not confine flux, because the surrounding air has a finite permeability rather than an infinite reluctance. Across a gap the flux bulges outward — fringing — which lowers the gap’s effective reluctance below $\ell_g/\mu_0 A$ by ten or twenty per cent for a gap comparable to the pole dimensions, and by much more for a long one.

And the reluctance depends on the flux. A resistance is a constant over an enormous range of current and a permeability is not. Push the flux toward saturation and the core’s reluctance rises steeply, so a magnetic circuit is nonlinear in the one part that was supposed to be negligible, and the failure arrives as a sharp knee rather than gradually.

Why a field line leaves a pole face straight out

The last consequence of the two-field bookkeeping is the one that makes magnetic circuits work as a design method at all.

Flux leaves a pole face very nearly straight out. The angle a field line makes with the normal in the air outside a core, against the angle it makes with the normal inside it, for three core materials. Two boundary conditions produce the whole curve: the part of B across the surface is continuous, and the part of H along it is, when no free current runs there — so the tangent of one angle is the tangent of the other divided by the relative permeability. A line running at 89 degrees to the normal inside the highest-permeability core here leaves the surface at 0.66 degrees, which is very nearly straight out. That is what guiding the flux amounts to: not that the iron attracts the field, but that a line inside it may lie at any angle at all and will still emerge from a face almost perpendicular, so the shape of the pole face and nothing else decides where the field goes. — Fig. 6 The angle a field line makes with the normal outside a core against the angle it makes inside, for three materials. Two boundary conditions produce the curve, and a line lying at 89 degrees to the normal inside a high-permeability core leaves the surface within a hundredth of a degree of straight out.

The two boundary conditions are the ones the two fields separately satisfy. The normal component of $\mathbf{B}$ is continuous because $\mathbf{B}$ has no sources; the tangential component of $\mathbf{H}$ is continuous because $\mathbf{H}$ has no circulation where there is no free current. Dividing the second by the first gives

\frac{\tan\theta_2}{\tan\theta_1} = \frac{\mu_2}{\mu_1},

with the angles measured from the normal. For iron against air that ratio is thousands, so a line at almost any angle inside comes out almost perpendicular.

It is worth saying what that does and does not mean. Nothing attracts the field into the iron. What happens is that a line inside the iron forgets the angle it had: whatever it was doing in there, it emerges from a face nearly straight out, so the direction of the field in the gap is set by the orientation of the pole face and by nothing else. A designer shapes the faces and the field does what the faces say.

Two accounts of what a magnet is, one of which was right

The bookkeeping in this essay is a choice between two pictures, and both were proposed within a few years of each other by people who could not settle between them.

Poisson’s account, from 1824, treats a magnet as containing magnetic charge — north stuff at one end and south stuff at the other, obeying an inverse-square law exactly like electric charge. It is a complete and self-consistent theory. It gets the field outside a magnet exactly right, it makes the pole-face picture literal, and it is where the magnetic scalar potential comes from.

Ampère’s account, from 1820, treats a magnet as containing circulating currents — no magnetic charge anywhere, only electric current going round in small loops, with a bar magnet being a solenoid whose winding is made of atoms. It too is complete, and it too gets the field outside exactly right.

The two cannot be told apart from outside. A small loop of current and a short bar of magnetic charge produce identical fields at any distance large compared with their size, which is the general statement about dipoles and is why the argument could run for a century.

They differ inside. The current picture gives $\mathbf{B} = \tfrac23\mu_0 M$ inside the sphere, along the magnetisation; the charge picture gives $-\tfrac13\mu_0 M$ , against it — the same value $\mathbf{H}$ has, since in that picture $\mathbf{H}$ is the field and there is nothing else. So the question has an experimental answer as soon as anything can be put inside.

The measurement is the deflection of a beam of neutrons, which carry a magnetic moment and pass through iron, and it agrees with Ampère. Nothing anybody has found behaves like magnetic charge, and Ampère’s hypothesis — proposed with no knowledge of electrons and no way of testing it — turned out to name the right constituent a century before anybody could see one.

The scalar potential the charge picture produced survives anyway, as a computational device rather than an account of what is there, and it is the subject of the essay after this one.

The magnetisation is stipulated and the permeability is not a constant

Everything here assumes the magnetisation is given. The sphere is uniformly magnetised by stipulation, and the reason that case is worth drawing is that it is the only shape for which a uniform magnetisation produces a uniform internal field — an ellipsoid is the general case and a sphere is the easy one. A real bar magnet’s internal field is not uniform, its magnetisation is not either, and the two have to be solved for together.

The permeability in the circuit calculations is a constant and is not. A real core’s $\mathbf{B}$ against $\mathbf{H}$ curve bends over and saturates, its slope depends on where on the curve it is measured, and it depends on the history as well. Quoting one number for a core is quoting the slope near the origin, which is not where a working machine sits.

And the field is treated as static throughout. At any frequency the currents induced in a conducting core oppose the flux, the field is expelled from the interior over a skin depth, and the effective permeability falls. That is why cores are laminated or made of ferrite, and none of the arithmetic here says so.

A quarter of a million amp-turns with nothing to draw them on

They cannot show the bound current. The quarter of a million amp-turns in the third figure is a real current — electrons really are circulating — and it is drawn as a bar on a chart because there is no way to draw it in the material. It does not flow through wires, it does not dissipate anything, and it produces the same field a solenoid of that many amp-turns would.

Nor can they show why $\mathbf{H}$ is the field whose circulation is fixed and $\mathbf{B}$ the field whose divergence is. The two roles are not symmetric and there is no picture that makes the asymmetry look inevitable; it comes from the fact that magnetisation is a current density and not a charge density, so subtracting it fixes a curl rather than a divergence. A figure can show the consequence, which is that one set of lines ends on the surface and the other does not.

And they cannot show that the choice is a convention with alternatives. There is a second consistent formulation in which magnetisation is represented by fictitious magnetic charges rather than by bound currents, and it gives the same fields everywhere outside matter and a different $\mathbf{B}$ inside a magnet. Which is right is settled by experiment — by the force on a current loop moving through the material, which distinguishes them — and the current-loop description is the one that agrees.

Still open: which description a neutron sees

The two formulations differ in the field they assign inside magnetised matter, and the experiment that distinguishes them is the deflection of a beam of particles carrying a magnetic moment as it passes through. Neutron measurements of the field inside magnetised iron have been used to argue for the current-loop description since the 1950s and are generally taken to settle it.

What is less settled is the related question for the momentum and energy of the field in a magnetised medium, where the two candidate formulations disagree about how much momentum a field in matter carries — a dispute that has run for over a century and that has a direct counterpart for electric fields in dielectrics. Whether the disagreement is physical or a matter of how the total is divided between field and matter is argued about still, and the experiments that bear on it measure forces on bodies rather than fields inside them.

The habit worth carrying away is about defining a quantity by what it counts. Two fields that differ only in which sources they acknowledge will disagree wherever those sources are, and the disagreement is not an error in either. $\mathbf{B}$ and $\mathbf{H}$ point in opposite directions inside a permanent magnet, they are both correct, and the question “which is the real magnetic field” has no answer beyond asking which currents the question is about.

Part 4 of 5

This essay is one argument about Ampere law. The others:

$How much of the answer a finite wire gives back. The field beside a straight segment of wire, divided by what the infinite-wire formula would give, against the length of the segment in units of the distance to the field point, on a logarithmic horizontal axis. The field is integrated element by element along the segment rather than evaluated from a closed form. A wire ten times as long as the distance already gives 92.8 per cent of the infinite answer, and one as long as the distance gives 45 — which is the practical content, and the reason the infinite-wire result is used for laboratory wires without apology. The second column is the part that matters for the law rather than for the number. The circulation of this field round a circle of radius d is not μ₀I; it falls short by exactly the fraction the segment fails to subtend. What makes up the difference is the displacement current of the charge piling up at the segment's two ends, and the two terms, both integrated here, sum to μ₀I to within 2.2e-8 per cent at every length. So Ampère's law is not approximately true for an open circuit and exactly true for a closed one — it is exactly true always, and it is a computation only when symmetry supplies the direction and the magnitude along the loop. Symmetry is doing the work; the law is doing the bookkeeping.$ The law that is always true and rarely useful Part 3 — Ampère's law holds for every loop and every current. Apply it to a wire of finite length and it gives an answer that is wrong by half — until the displacement current of the charge piling up at the wire's two ends is put back in, whereupon the two terms sum to μ₀I exactly at every length. The law was never approximate. It was never a computation either.