1 · Empty spaceStart with nothing
An applied field H — produced by current in a coil — passes through empty space. The flux density that results, 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. Space has no memory and no limit.
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 is measured against this line. A material placed in the field either adds to it, or — in the case of a magnet — carries a field of its own after the applied one is gone.
2 · Soft ironPut iron in the gap
Iron is made of magnetic domains: regions already magnetized as fully as the atoms allow, but pointing in different directions, so that the block as a whole shows nothing.
As H rises, the domains swing into line with the field. Each one that turns adds its magnetization to the flux through the block, so B climbs far faster than it did in empty space.
2 · Soft ironSaturation
Once every domain is aligned there is nothing left to turn. The iron is saturated: it has contributed all it can. From here B rises only as fast as empty space would — the curve runs parallel to the free-space line, offset by the iron's saturation.
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
The domains relax — but not entirely. 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 easily lost. In a permanent magnet it is the product.
Circuit analogyAt zero voltage a little current is still flowing. Nothing is driving it; it is stored in the material.
3 · Isolating the materialSubtract the free space
The B curve mixes two things: what the material contributes, and what the applied field would produce anyway. To see the material alone, subtract the free-space line:
J is the magnetic polarization — the material's own magnetization, expressed in tesla. In CGS units the same quantity is written 4πM.
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 collapses onto the H axis, and the iron curve flattens at saturation instead of continuing to climb. The plateau is the saturation polarization, Js — a property of the material, not of the test.
From here we track both. B is what a magnetic circuit actually sees. J is what the material is doing.
4 · A hard magnetThe same experiment, sintered NdFeB
Replace the iron with a sintered neodymium-iron-boron magnet. Its domains align as the iron's did, and J rises to saturation.
Reduce H to zero and the difference appears: the domains stay put. J barely moves, and B at H = 0 — the remanence — is nearly the full saturation value.
Circuit analogyThe magnet has become a source. It delivers current with no voltage applied.
4 · A hard magnetDrive the field negative
The magnet resists. J holds its plateau against a field pointing the other way, while B — which includes μ₀H — declines along a nearly straight line. Only when the reverse field reaches the intrinsic coercivity does the magnetization collapse and reverse.
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 and the mirror image closes the loop. The width of the loop is what hard means: the field it takes to reverse the magnet.
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.
5 · The second quadrantWhere a magnet actually lives
A magnet in service is never in the first quadrant. Once it is magnetized and the magnetizing field is removed, 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 applies. Keep only that quadrant.
This is the demagnetization curve — the plot published for every commercial grade. Everything on a datasheet's first page is a point on these two lines.
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.
6 · RemanenceBr
Where both curves meet the vertical axis: the flux density the magnet delivers with no external field, in a closed circuit. It sets how much flux a design can extract per unit of magnet area, and it is the first number in every grade comparison.
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. The material is still magnetized — J is still positive — but the applied field exactly cancels it.
For NdFeB at room temperature the B curve is nearly straight from Br to Hcb, with a slope close to μ₀. Datasheets also write it 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 true measure of resistance to demagnetization. Between Hcb and Hci the magnet is still a magnet; past Hci it is not, and it does not recover when the field is removed.
Hci (also HcJ) falls with temperature. That is why grades are specified by it, and why heavy rare earths — dysprosium and terbium — are added to raise it.
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, the product of B and H is an energy density. Slide along the curve and it passes through a maximum — the largest rectangle that fits beneath the curve.
(BH)max is the figure of merit for how much magnetic energy a unit volume can supply to a circuit, and it is the number in the grade name: an N52 grade delivers roughly 52 MGOe.
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.
Datasheets print these at room temperature and again at elevated temperature. The shape of the curve at temperature — especially the knee that appears in J — is where the diligence questions begin.
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.