Short versionStart with nothing in the gap. Whatever happens later is measured against what empty space does, so it is worth knowing what empty space does: exactly what you tell it, and nothing more.
1 · Empty spaceStart with nothing
An applied field H — made by running current through a coil — passes through empty space. The flux density you get back, B, is simply proportional to it:
μ₀ is the permeability of free space, a constant. Sweep H up, back through zero, and negative. The plot is a straight line through the origin, in both directions, every time. Space has no memory and no limit. It is the most boring material on this page and that is precisely why we start with it.
Circuit analogyThink of H as voltage — the drive — and B as current — the response. Empty space is a fixed resistor: the current follows the voltage in exact proportion, and μ₀ is its conductance. The analogy holds through everything that follows.
1 · Empty spaceThe reference line
Everything that follows gets measured against this line. Put a material in the field and it does one of two things. It adds to the field while the field is applied. Or — and this is the whole business — it carries a field of its own once the applied one is gone.
Short versionIron is not un-magnetized. It is already magnetized in every direction at once, which adds up to nothing. All a field does is talk it into agreeing with itself.
2 · Soft ironPut iron in the gap
Iron is made of magnetic domains: regions already magnetized as hard as the atoms allow, but pointing every which way, so the block as a whole shows nothing at the surface.
Raise H and the domains swing into line with the field. Every one that turns adds its own magnetization to the flux through the block, so B climbs far faster than it ever did in empty space. You are not creating magnetism here. You are organizing it.
2 · Soft ironSaturation
Once every domain is aligned, there is nothing left to turn. The iron is saturated — it has given you everything it has. Keep pushing H and B still rises, but only as fast as empty space would: the curve runs parallel to the free-space line, offset by whatever the iron contributed.
That offset is a property of the material and a hard ceiling. No amount of current buys more of it. This matters more than it sounds: saturation is why motor and transformer designers spend so much of their lives worrying about steel.
Circuit analogyIron is a far better conductor than empty space — a small drive produces a large response — until saturation, when the material's share of the current is fully drawn and only the free-space conductance remains.
2 · Soft ironRemove the field
Now take the field away. The domains relax — but not all the way back. B does not return to zero. What is left at H = 0 is the remanence, the residual flux density.
In soft iron it is small, and a knock or a stray field will shake it out. In a permanent magnet, that same leftover IS the product. The whole difference between soft and hard magnetic materials is how stubbornly they hang on to it.
Circuit analogyAt zero voltage a little current is still flowing. Nothing is driving it. It is stored in the material.
In practiceI got my start in magnetics on the other end of this problem: working out how to erase hard drives and floppy disks. Erasure is this same experiment run deliberately and badly. You drive the material around the loop with a strong alternating field, shrinking the amplitude each cycle, so it ends up back at the origin with nothing left over. Doing it properly is harder than it sounds, and the failure mode is not "mostly erased" — it is a disk that reads as blank to the operating system and perfectly well to anyone with a lab. My first real lesson in magnetics was that remanence is remarkably hard to get rid of once you have put it there. That is good news if you are selling magnets, and less good news if you are decommissioning a data center.
Short versionThe B curve is two things added together: what the material is doing, and what the field would have done on its own. Subtract the second and you can see the first.
3 · Isolating the materialSubtract the free space
The B curve mixes two things: what the material contributes, and what the applied field would have produced anyway. To see the material on its own, subtract the free-space line:
J is the magnetic polarization — the material's own magnetization, in tesla. In CGS units you will see the same quantity written 4πM, which trips up more people than it should. Different symbol, different unit system, same physical thing.
Circuit analogyJ is the material's share of the current — the total, minus what the fixed resistor would have passed on its own.
3 · Isolating the materialTwo plots, two questions
Subtracting μ₀H shears the plot. The free-space line flattens onto the H axis, where it belongs, and the iron curve stops climbing and goes level at saturation. That plateau is the saturation polarization, Js — a property of the material, NOT of the test rig.
From here we track both curves, because they answer different questions. B is what a magnetic circuit actually sees. J is what the material is actually doing. Confusing the two is the single most common mistake people make reading these plots, and it is an expensive one.
Short versionSame experiment, different material. Everything up to saturation looks familiar. What happens on the way back down is the entire difference between a lump of steel and a product.
4 · A hard magnetThe same experiment, sintered NdFeB
Swap the iron for a sintered neodymium-iron-boron magnet and run it again. The domains align as the iron's did, J rises, J saturates. So far, nothing new.
Now take H back to zero, and there it is: the domains stay put. J barely moves. B at H = 0 — the remanence — is very nearly the full saturation value. The iron gave most of it back. This does not.
Circuit analogyThe magnet has become a source. It delivers current with no voltage applied.
4 · A hard magnetDrive the field negative
The magnet resists, and it resists for a long time. J holds its plateau against a field pointing the wrong way, while B — which still includes μ₀H — declines down a nearly straight line. Only when the reverse field reaches the intrinsic coercivity does the magnetization give up, collapse and flip.
Circuit analogyA reverse voltage across a source reduces the current it delivers, but the source itself is untouched — until the reverse voltage is large enough to damage it.
Sweep back the other way and the mirror image closes the loop. The width of that loop is what hard means. It is not a statement about the material's mechanical hardness, which confuses people constantly — it is simply how much field it takes to turn the magnet around.
Short versionA magnet in service is never in the first quadrant. Keep the one quadrant that matters and you have the plot every manufacturer publishes.
5 · The second quadrantOverlay B and J
Lay the two plots on top of each other so B and J share axes. In the first quadrant they diverge by exactly μ₀H. At H = 0 they meet, which is worth remembering — it is why remanence is a single number rather than two.
5 · The second quadrantWhere a magnet actually lives
Here is the thing that makes the rest of it click. A magnet in service is never in the first quadrant. Once it has been magnetized and the magnetizing field is gone, its own flux is positive while the field acting on it is negative — its own demagnetizing field, plus whatever the circuit or a motor winding throws at it. So keep that quadrant and throw the rest away.
What is left is the demagnetization curve, the plot published for every commercial grade. Everything on the first page of a datasheet is a point on these two lines. Once you can see that, you can read any magnet datasheet ever printed.
Circuit analogyA source in use always has a load across its terminals, pulling its output below the open-circuit value. The demagnetization curve is the magnet's output characteristic: current delivered against reverse voltage.
Short versionFour numbers, taken one at a time. Remanence is how much you get, the two coercivities are how hard it is to take away, and the energy product is how much work it will do.
6 · RemanenceBr
Where both curves meet the vertical axis. This is the flux density the magnet delivers with no external field applied, in a closed circuit — the most it will ever give you.
It sets how much flux a design can pull out per unit of magnet area, and it is the first number in every grade comparison, usually the only one a non-specialist looks at. While it is the right place to start, it is a poor place to stop.
Circuit analogyBr is the short-circuit current: the most the source delivers, into a load of zero resistance.
7 · CoercivityHcb
Where the B curve crosses zero: the reverse field at which the magnet's NET flux density falls to nothing. Note carefully what has and has not happened. The material is still fully magnetized — J is still up at its plateau — but the applied field is now exactly cancelling it out.
Take the reverse field away and the magnet is completely fine. Nothing has been lost. This is a circuit condition, not damage.
For NdFeB at room temperature the B curve is nearly straight from Br down to Hcb, with a slope close to μ₀. You will also see it written HcB.
Circuit analogyHcb is the counter-voltage that brings the terminal current to zero. The source is still intact — its own drive is simply canceled.
8 · Intrinsic coercivityHci
Where the J curve crosses zero: the reverse field that actually reverses the material's magnetization. This is the real measure of how hard a magnet is to demagnetize, and it is the number that should worry you.
Between Hcb and Hci the magnet is still a magnet, doing its job, recoverable. Past Hci it is not, and it does NOT come back when you remove the field. That is the difference between a bad afternoon and a warranty claim.
Hci (also written HcJ) falls as temperature rises, which is why grades are specified by it and why the letter suffixes exist at all — N, M, H, SH, UH, EH, in rising order of coercivity and rising order of price. Heavy rare earths, dysprosium and terbium, are what buy you those letters.1
In practiceThis is the number I spend the most diligence time on, and it is almost always quoted at room temperature. The room-temperature figure is close to useless on its own. What matters is Hci at the top of the duty cycle, in the shape the magnet is actually used in, with the reverse field the circuit actually applies — and by then you have usually lost a third of it.
Two things follow. First, when somebody hands you a grade and a single coercivity number, the useful question is not "is that high?" but "at what temperature, and what does the knee look like there?" Second, the cheapest way to buy coercivity is grain boundary diffusion, which walks the heavy rare earth in along the grain boundaries rather than alloying it through the bulk — same protection, far less dysprosium, and a meaningfully different cost structure.2 If a supplier cannot tell you which of the two they are doing, that tells you something about the supplier.
Circuit analogyHci is the reverse voltage that damages the source itself. Below it the magnet recovers when the load is removed; past it, it does not.
9 · Energy product(BH)max
At every point on the B curve, B times H is an energy density. Slide along the curve and that product rises, peaks and falls again. The peak is the largest rectangle that fits under the curve, and it has a name.
(BH)max is the figure of merit for how much magnetic energy a unit volume can supply to a circuit. It is also the number hiding in the grade name: an N52 delivers roughly 52 MGOe.
In practiceBecause it is in the name, (BH)max is the number everybody shops on, and it is the number most often used to sell something that will not survive the application. A very high energy product with a low coercivity is a magnet that performs beautifully on the bench and then quietly gives up the first time it gets hot with a reverse field on it.
When I am reviewing a magnet selection, the grade number is where I start and never where I finish. Br tells you what you get. Hci tells you whether you keep it. If somebody leads a pitch with the energy product and has to go looking for the coercivity, that is worth noticing.
Circuit analogyCurrent times voltage is power. (BH)max is the source's maximum-power point — the load that draws the most work from it.
10 · Four numbers, one curveWhat the datasheet is telling you
Br sets the flux. Hcb sets how much reverse field the circuit can apply before the magnet's net output reaches zero. Hci sets how much it can apply before the loss is permanent. (BH)max sets how much work a given volume can do. That is the whole datasheet.
Datasheets print these at room temperature and again hot. The shape of the curve when hot — and specifically the knee that appears in J, where the curve stops being straight and falls away — is where the real questions start. A magnet operated past its knee does not recover, and the knee moves with temperature. Every serious argument I have ever had about a magnet specification has eventually turned out to be an argument about where the knee is.
Circuit analogyA magnet is a source. Br is its short-circuit current, Hcb the counter-voltage that nulls its output, Hci the reverse voltage that damages it, and (BH)max its maximum-power point.