Hysteresis

Ferromagnetism drew the loop and named its landmarks — remanence, coercivity, the virgin curve's three acts, the Barkhausen crackle of jumping walls. That was the portrait. This article is the engineering anatomy the portrait deferred: what a datasheet's two different coercivities mean and why hard-magnet suppliers must quote both; what sets the coercivity's size, from the milliampere-per-metre softness of transformer laminations to the mega-ampere stubbornness of Nd₂Fe₁₄B — five orders of magnitude between materials whose atomic moments are nearly identical; how much a loop costs per cycle and per second at 50 Hz; and how to find the operating point of a real magnet in a real magnetic circuit with a ruler — the load-line construction that turns the second quadrant of the B–H plane into the permanent-magnet industry's entire catalogue. Two theoretical results bracket everything. At the feeble-field floor, Rayleigh's 1887 law describes walls bowing reversibly in their pinning valleys and delivers the first quantitative loss formula — loop area growing as the cube of the field amplitude. At the ceiling, the Stoner–Wohlfarth model asks what coercivity would be if walls did not exist at all and reversal had to proceed by coherent rotation of every moment at once: $H_K = 2K/\mu_0M_s$ , the anisotropy field. Real materials live strictly between the two — soft magnets a thousand times below the ceiling because walls sneak the magnetization around it, hard magnets straining toward it and mysteriously stopping at a fifth of it (Brown's paradox, named and prosecuted below). The loop's area is quoted here and spent here — at 50 Hz it is a heating bill every transformer pays — but its derivation from magnetic work is owed to, and delivered by, Magnetic Energy in Materials, next.

Key takeaways

One physical loop, two standard plots: the M–H loop (materials physics: $M_r$ , intrinsic coercivity $H_{ci}$ where $M$ reverses) and the B–H loop (engineering: $B_r$ , normal coercivity $H_{cB}$ where $B$ reaches zero). Since $B = \mu_0(H+M)$ can vanish while $M$ is still saturated, $H_{cB} \le B_r/\mu_0$ always, while $H_{ci}$ can be far larger — hard-magnet datasheets must quote both.
Coercivity is not a property of the atoms; it is a property of the defects. The Stoner–Wohlfarth coherent-rotation ceiling $H_K = 2K/\mu_0M_s$ ( $4.5\times10^4$ A/m for iron) versus measured soft-iron $H_c \sim 1$ – $100$ A/m shows walls bypass rotation; hard magnets reach only 20–30% of their $H_K$ (Brown's paradox).
In feeble fields, walls bow reversibly in pinning valleys: Rayleigh's law $\mu(H) = \mu_i + \nu H$ , with minor-loop loss growing as $H_{\max}^3$ — the quantitative floor of hysteresis.
Cycling costs $W = \oint H\,dB$ = loop area per unit volume per cycle (derivation owned by Magnetic Energy in Materials); Steinmetz's empirical law $W \propto B_{\max}^{1.6}$ prices transformer cores: 40 J/m³ at 50 Hz is 2 kW per cubic metre of standing heat.
A magnet in an open circuit self-biases into its second quadrant: the shape's demagnetizing factor draws a load line $B = -\mu_0\frac{1-N}{N}H$ through the origin, and the operating point is its intersection with the demagnetization branch. $(BH)_{\max}$ marks the sweet spot; its energy meaning is owed to Magnetic Energy in Materials.
KEY IDEA: the loop is a defect-controlled energy landscape traversed irreversibly — Rayleigh's cubic law at the bottom, Stoner–Wohlfarth's anisotropy field at the top, $\oint H\,dB$ of heat per lap, and a load line picking the one point on it where a real magnet actually sits.

The core insight

Hysteresis quantified: two coercivities distinguish where B and where M reverse; pinning energetics (Rayleigh's law) set the reversible-to-irreversible crossover and the cubic small-field loss; the Stoner–Wohlfarth anisotropy field bounds all coercivity from above; and the load-line construction locates a real magnet's operating point on its second-quadrant demagnetization curve.

Where this lives today

Hysteresis engineering splits into a war on the loop and a war for it. Against it: amorphous and nanocrystalline soft alloys (Metglas, Finemet) with almost no magnetocrystalline anisotropy and hence almost no pinning contrast — loop areas of a joule per cubic metre, enabling the solid-state transformers of EV fast chargers; electrical-steel makers still chase watts per kilogram at 50 Hz across gigawatts of installed grid. For it: the NdFeB supply chain — and its dysprosium additions purchased specifically to hold $H_{ci}$ at motor temperatures — remains a geopolitical pressure point, with "heavy-rare-earth-free" magnet metallurgy an open materials race that is, at bottom, an assault on Brown's paradox: closing the factor-of-four gap between real coercivity and the Stoner–Wohlfarth ceiling would remake the industry without discovering a single new compound. And first-order-reversal-curve (FORC) diagrams — dense maps of nested minor loops — have become the standard fingerprinting tool for everything from meteorite paleomagnetism to counterfeit-magnet detection: the loop, read carefully enough, is a biography of the microstructure.

Origins · 1 minLord Rayleigh and how this was discovered

Lord Rayleigh

1842–1919 · Terling Place, Cambridge

John William Strutt, third Baron Rayleigh, gave hysteresis its first law from an estate laboratory in Essex. In 1887, measuring iron in fields a thousandth of saturation, he found the permeability rising linearly with amplitude and the minor-loop loss growing as its cube — the parabolic-branch law derived above, still the standard description of every magnetic material's feeble-field regime and still called the Rayleigh regime. It was a characteristic performance: find the clean limit of a messy phenomenon and extract its exact law. The same instinct gave the $1/\lambda^4$ scattering law that colors the sky (met again in this track's Rayleigh Scattering), the resolution criterion that bears his name in every optics course, the theory of surface waves that seismologists use daily, and the argon discovery — from an 0.5% density anomaly he refused to ignore — that won the 1904 Nobel Prize. He ran The Theory of Sound's two volumes, the Cavendish professorship after Maxwell, and a working farm, and published to the year of his death. The feeble-field law was a two-page aside; there are worse careers than Rayleigh's asides.

History

1880

— Warburg measures the area of the magnetization loop and identifies it as heat dissipated per cycle

1885

— Ewing names the phenomenon hysteresis, from the Greek for lagging behind

1887

— Rayleigh extracts the feeble-field law: linear permeability rise, parabolic branches, cubic loss

1892

— Steinmetz fits core loss to B to the power 1.6, the empirical law transformer designers still use

1919

— Barkhausen's amplified crackle reveals the loop's microscopic mechanism: discrete wall jumps

1948

— Stoner and Wohlfarth solve coherent rotation, bounding coercivity by the anisotropy field

1957

— Brown formalizes the paradox: real magnets switch far below the theoretical ceiling, indicting nucleation

The physical picture

Take the loop Ferromagnetism drew and ask what the material is doing at each point — not as narrative this time, but as energetics. A domain wall (its width and price are Magnetic Domains and Domain Walls' results; we spend them) sits in a crystal seeded with defects. Each defect locally perturbs the wall's energy: an inclusion the wall passes through saves wall area (walls like to intersect non-magnetic particles), a strain field raises or lowers the local anisotropy. Plot wall energy against wall position and you get a rugged one-dimensional landscape — hills and valleys with heights set by defect strength and spacings set by defect density. Now everything on the loop becomes mechanics in that landscape. Small applied field: the wall leans against its valley wall, climbing partway — displaced but trapped, and it slides back when the field is removed. This is the reversible foot of the virgin curve, and Rayleigh's law is its equation of motion. Larger field: the wall is pushed over its local crest and falls into the next valley — irreversibly, releasing the drop's energy as spin waves and micro-eddy currents (heat), and registering one Barkhausen click (the phenomenon Ferromagnetism introduced; we count its cost). The steep middle of the loop is an avalanche of such falls. Saturation: no walls remain; only coherent rotation against anisotropy is left, and it is smooth and reversible again. Read backwards, the landscape explains why coercivity is microstructure, not chemistry. Pure, well-annealed iron — few defects, shallow valleys — has walls that glide almost freely: $H_c$ of a few A/m. The same iron, cold-worked and dirty, pins walls in deep valleys: $H_c$ of hundreds. And a material engineered so that grains are too small to host any wall at all (Magnetic Domains and Domain Walls' single-domain limit) forces reversal by full rotation, the most expensive route in the catalogue — which is the design principle of every modern hard magnet.

Figure

Wall in a pinning landscape - reversible bowing in a valley, irreversible Barkhausen drop over a crest

Two loops, two coercivities

The community plots the same physics twice, and the difference is a standing exam trap. The materials physicist plots $M$ against $H$ ; the device engineer plots $B = \mu_0(H+M)$ against $H$ . In soft materials ( $M \gg H$ everywhere that matters) the two loops are visually identical scaled by $\mu_0$ . In hard magnets they diverge spectacularly in the second quadrant — the quadrant where permanent magnets live, with positive $B$ but negative (demagnetizing) $H$ . Follow the descending branch. As $H$ goes negative, a hard magnet's $M$ barely budges — pinning holds every wall, so $M \approx M_r$ across the whole quadrant. But $B = \mu_0(H + M)$ falls linearly regardless, simply because $H$ does. It reaches zero at

H_{cB} \le \frac{B_r}{\mu_0} \qquad (1)

where: $H_{cB}$ — the normal coercivity, where the flux density $B$ crosses zero; $B_r = \mu_0M_r$ — the remanence. The inequality is structural: $B = 0$ requires only $H = -M$ , and $M$ can be no larger than it was at remanence — so the B–H loop's zero crossing can never sit farther out than $B_r/\mu_0$ , no matter how stubborn the material is. The magnetization's own reversal happens elsewhere: at the intrinsic coercivity $H_{ci}$ , where $M$ crosses zero on the M–H loop. For Nd₂Fe₁₄B grades, $B_r \approx 1.2$ – $1.4$ T pins $H_{cB}$ near $10^6$ A/m, while $H_{ci}$ ranges from barely above that (cheap grades) to three times it (high-temperature grades). Why quote both? Because they answer different failure questions. $H_{cB}$ tells the circuit designer where the magnet stops pushing flux. $H_{ci}$ tells them when the magnet is destroyed — when the reverse field (from a stalled motor, a facing magnet, or its own shape at high temperature) has actually flipped the magnetization rather than merely opposing it. A magnet driven past $H_{cB}$ recovers when the abuse stops; driven past $H_{ci}$ , it comes back demagnetized. The gap between the two numbers is the safety margin, and it shrinks with temperature — which is why datasheets tabulate both against $T$ , and why Curie Law and Curie Temperature's collapse is a design input, not a curiosity.

The feeble-field floor: Rayleigh's law

At the bottom of the loop, before any wall has jumped, hysteresis has an honest quantitative theory — the oldest in the subject.

Derivation ARayleigh's law - reversible bowing plus the first taste of loss

The goal is the small-field magnetization law $B = \mu_iH + \nu H^2$ and its consequence, minor-loop loss growing as the cube of the field amplitude — the quantitative floor of hysteresis, due to Rayleigh (1887). The strategy is to model the wall as a membrane in the pinning landscape of the physical picture: a parabolic valley gives the reversible linear term; the first, rare crest-crossings give a correction whose statistics — many independent barriers with a smooth distribution of heights — produce the quadratic term. Watch what is honest here: the derivation of the linear term is mechanics; the $\nu H^2$ term's form follows from the barrier statistics, while its coefficient is empirical, material by material.

Step 1. The reversible term. A wall of area $S$ and energy $\sigma_w$ (price from Magnetic Domains and Domain Walls) sits in a valley of the pinning landscape with curvature $\alpha$ : displacing it by $x$ costs $\tfrac12\alpha x^2$ per unit area. An applied field exerts a pressure on a 180° wall of $2\mu_0M_sH$ (the field gains $2\mu_0M_sH$ per unit volume swept, so the wall feels that per unit displacement per unit area). Equilibrium:

2\mu_0M_sH = \alpha x \quad\Rightarrow\quad x = \frac{2\mu_0M_sH}{\alpha}

Step 2. Convert displacement to magnetization. Each unit of wall displacement reverses $2M_s$ of moment per unit wall area; with total wall area per volume $S_V$ , the magnetization response is $M = 2M_sS_Vx = (4\mu_0M_s^2S_V/\alpha)H$ . Linear, reversible, field-independent slope: this is the initial permeability,

\mu_i - \mu_0 = \frac{4\mu_0^2M_s^2S_V}{\alpha}

Why soft materials have huge initial permeability: lots of wall area, shallow valleys (small α) — both microstructural, neither atomic. The 10³–10⁴ initial slopes of transformer steel are read from exactly this formula's logic.

Step 3. The irreversible correction. Raise $H$ slightly and the weakest crests within reach begin to be crossed. If barrier heights are smoothly distributed, the number of crests a wall can cross grows linearly with the driving pressure, and each crossing contributes magnetization proportional to the pressure again — hence a term quadratic in $H$ :

B = \mu_iH + \nu H^2 \qquad (2)

where: $\mu_i$ — initial permeability (reversible bowing); $\nu$ — the Rayleigh constant (T·m/A per A/m, i.e. an irreversibility density), tabulated for soft materials.

Step 4. Cycle it. Traversing a minor loop of amplitude $H_{\max}$ , the ascending and descending branches follow $B = (\mu_i + \nu H_{\max})H \pm \tfrac{\nu}{2}(H_{\max}^2 - H^2)$ — Rayleigh's parabolic branches. The enclosed area (the loss, by the $\oint H\,dB$ rule quoted below) integrates to

W_{\rm cycle} = \frac{4}{3}\,\nu\,H_{\max}^3 \qquad (3)

Why cubic: the loop's width is the irreversible part, $\propto H_{\max}^2$ , and its height sweep is $\propto H_{\max}$ — area is width times height. Halve the drive amplitude and hysteresis loss falls eightfold; this steep scaling is why signal transformers and magnetic sensors, operating at tiny amplitudes, can use hysteretic materials at all. *Verified: branch equations and the 4/3 coefficient reproduce Rayleigh's 1887 result as presented in Chikazumi and Bozorth; the parabolic branches close exactly at ±H_max (checked by substitution); equation (2)'s limit ν→0 recovers linear reversible response, consistent with Magnetic Permeability's μ_i etiquette.*

The ceiling: Stoner–Wohlfarth and Brown's paradox

Now the opposite question: how large can coercivity possibly be? Remove every wall from the problem — a single-domain particle (below Magnetic Domains and Domain Walls' critical size) with uniaxial anisotropy $K$ , magnetized up, field applied straight down. Reversal must proceed by coherent rotation: every moment turning together, climbing the full anisotropy barrier.

Go deeper · Derivation B · 1 minThe anisotropy field - coercivity's absolute ceiling

Derivation BThe anisotropy field - coercivity's absolute ceiling

The goal is the switching field of an ideal single-domain uniaxial particle under antiparallel field — the Stoner–Wohlfarth bound $H_K = 2K/\mu_0M_s$ , the largest coercivity the material's anisotropy can support. The strategy is a stability analysis of the energy as a function of the magnetization angle: the magnetized-up state sits in a local minimum that the reverse field progressively flattens; switching happens when minimum and maximum merge. Watch the factor of 2 — it comes from the second derivative of $\sin^2\theta$ , not from any geometry.

Step 1. Write the energy per volume with $\theta$ the angle of $\mathbf{M}$ from the easy axis (up), field $H$ applied down (at $180°$ ):

u(\theta) = K\sin^2\theta + \mu_0M_sH\cos\theta

The Zeeman term is $+\mu_0M_sH\cos\theta$ because alignment with the downward field means $\theta = \pi$ .

Step 2. Equilibria: $u'(\theta) = \sin\theta\,(2K\cos\theta - \mu_0M_sH) = 0$ . The magnetized-up state $\theta = 0$ is always an equilibrium; its stability is the question.

Step 3. Second derivative at $\theta = 0$ : $u''(0) = 2K - \mu_0M_sH$ . The up state remains a minimum while $u''(0) > 0$ and destabilizes at

H_K = \frac{2K}{\mu_0M_s} \qquad (4)

where: $H_K$ — the anisotropy field. At this field the barrier has flattened to nothing and the whole moment rolls over at once.

Step 4. Put in numbers. Iron: $H_K = 2(4.8\times10^4)/(\mu_0\cdot1.7\times10^6) = 4.5\times10^4$ A/m. Nd₂Fe₁₄B ( $K \approx 4.9\times10^6$ J/m³, $M_s \approx 1.28\times10^6$ A/m): $H_K \approx 6.1\times10^6$ A/m. Verified: equation (4) is the antiparallel-field corner of the Stoner–Wohlfarth astroid (1948); the general-angle switching field dips to H_K/2 at 45°, a refinement recorded but not needed here; numbers recomputed for this edition.

Hold the two iron numbers side by side. Ceiling: $4.5\times10^4$ A/m. Measured coercivity of soft iron: 1–100 A/m — three to four orders below its own ceiling. The resolution is walls: real iron is far above the single-domain size, and a wall lets the magnetization reverse a sliver at a time, paying only the local pinning barrier instead of the global anisotropy barrier. Coherent rotation is the price of reversing everything at once; the wall is an installment plan. Soft magnetism is not weak anisotropy so much as efficient walls. Hard magnets are built to foreclose the installment plan — single-domain grains, grain-boundary phases that decouple neighbors, pinning centers engineered at the wall's own width. Yet even the best Nd₂Fe₁₄B grades reach $H_{ci} \approx 1$ – $2\times10^6$ A/m: 20–30% of their $6.1\times10^6$ ceiling. This persistent shortfall is Brown's paradox: nominally wall-free magnets still reverse early, because reversal nucleates — at surface asperities, grain corners, and stray soft phases where the local ceiling is far below the bulk one — and one nucleated reverse domain, once born, sweeps the grain. Sixty years of magnet metallurgy is, in one sentence, the war on nucleation sites.

Erasing: minor loops, AC demagnetization, and heat

The loop is memory; erasure is therefore a protocol, not an event. Simply removing the field strands the material at remanence. Driving to $-H_{ci}$ overshoots into reverse magnetization. To reach the demagnetized state — the flux-closed domain equilibrium of Ferromagnetism — you must shrink the loop around the origin: apply an alternating field of slowly decreasing amplitude. Each half-cycle traces a minor loop slightly smaller than the last; the state spirals inward through a nested sequence of minor loops and lands at $M = 0$ with domains restored. This is what a demagnetizing wand does, why bulk tape erasers hum, and why degaussing coils in old CRT monitors fired a decaying AC pulse at power-on. The thermodynamic alternative is total: heat past the Curie point and every trace of the loop vanishes — order itself is dissolved and renegotiated on cooling (Curie Law and Curie Temperature owns the collapse). Between the two extremes lies the loop's slow enemy, already met in Magnetic Domains and Domain Walls: thermal creep over pinning barriers, which erodes remanence logarithmically over years and sets the archival lifetime of every magnetic record.

The loss bill and Steinmetz's law

Every lap of the loop dissipates

W_{\rm cycle} = \oint H\,dB \qquad (5)

where: the integral runs once around the closed B–H trajectory and equals the enclosed area, in J/m³ per cycle. This article spends equation (5); its derivation from the work done against induced EMFs — and the accounting of exactly where the heat lands — is owned by Magnetic Energy in Materials, the next article, which pays this quote in full. Two readings matter here. First, the sign: the area is always traversed so that $W > 0$ — the loop is a one-way turnstile, which is why the curve cannot retrace itself (a reversible material has zero enclosed area). Second, the scale: engineering silicon steel encloses roughly 30–50 J/m³ at $B_{\max} = 1$ T. For design, Steinmetz's empirical law (1892) compresses a century of loop measurements:

W_{\rm cycle} \approx k\,B_{\max}^{\,n}, \qquad n \approx 1.6\ \text{to}\ 2 \qquad (6)

where: $k$ — a material constant; $n$ — the Steinmetz exponent, honestly empirical (Rayleigh's cubic law governs only the feeble-field toe; real transformer cores run near the knee of the loop, where the exponent settles near 1.6). Power loss is then $P = f\,W_{\rm cycle}$ — frequency times area — plus eddy-current losses, a separate bill with its own scaling ( $\propto f^2$ ), owned by Eddy Currents down the chapter and controlled by lamination, as Ferromagnetism previewed.

The load line: where a real magnet actually operates

A permanent magnet in open circuit does not sit at remanence. Its own shape demagnetizes it: the magnetization plants poles on the end faces, and those poles drive an internal $H$ opposing $M$ — the demagnetizing field $H_{\rm in} = -NM$ with shape factor $N$ , the doctrine established in Magnetic Boundary Conditions and spent here as a tool.

Go deeper · Derivation C · 2 minThe load line - a ruler-and-pencil solution for the operating point

Derivation CThe load line - a ruler-and-pencil solution for the operating point

The goal is the operating point $(H_m, B_m)$ of a magnet of demagnetizing factor $N$ in open circuit: a straight line through the origin of the B–H plane whose intersection with the material's second-quadrant demagnetization branch is the answer. The strategy is two equations in two unknowns — the universal identity $B = \mu_0(H+M)$ plus the shape law $H = -NM$ — solved by eliminating $M$ ; the material's loop then supplies the third relation graphically. Watch the signs: in the magnet's own frame $H$ is negative while $B$ and $M$ are positive, which is precisely what "second quadrant" means.

Step 1. Eliminate $M$ . From $H_m = -NM$ , $M = -H_m/N$ . Substitute into $B_m = \mu_0(H_m + M)$ :

B_m = \mu_0\left(H_m - \frac{H_m}{N}\right) = -\mu_0\,\frac{1-N}{N}\,H_m \qquad (7)

where: $N$ — demagnetizing factor along the magnetization (sphere $1/3$ , values from Magnetic Boundary Conditions). This is a straight line through the origin with negative slope $-\mu_0(1-N)/N$ : the load line. Long needle ( $N \to 0$ ): slope $\to -\infty$ , a nearly vertical line hugging the $B$ axis — the magnet barely demagnetizes itself. Flat disc magnetized through its face ( $N \to 1$ ): slope $\to 0$ , the line hugs the $-H$ axis and the magnet nearly kills its own flux.

Step 2. Intersect with the material. The magnet's state must also lie on its demagnetization curve — the second-quadrant branch of the B–H loop. The operating point is the intersection. For a hard magnet with a straight branch, $B = B_r + \mu_0H$ (rigid $M = M_r$ : the wall-free assumption, honest for NdFeB well below $H_{ci}$ ), the intersection is algebra:

B_m = \frac{1-N}{1-N+N}\,\ldots\ \text{— do it concretely for a sphere, } N = \tfrac13: \quad -2\mu_0H_m = B_r + \mu_0H_m \ \Rightarrow\ H_m = -\frac{B_r}{3\mu_0},\quad B_m = \frac{2}{3}B_r

Step 3. Read the product. The energy figure of merit at the operating point is $|B_mH_m|$ ; sweeping $N$ sweeps the operating point along the branch, and $|BH|$ peaks — for the rigid-branch magnet — at $B = B_r/2$ , giving $(BH)_{\max} = B_r^2/4\mu_0$ . The load-line slope there is $-\mu_0$ , corresponding to $N = 1/2$ . What that product means — flux-times-field delivered to a gap per cubic metre of magnet, hence why you buy magnets by $(BH)_{\max}$ — is the gap-budget theorem of Magnetic Energy in Materials, named here and delivered there. Verified: equation (7) reproduces the standard load-line slope (Jackson §5.11 form); sphere numbers recomputed: B_r = 1.2 T gives H_m = −3.18×10⁵ A/m, B_m = 0.80 T, |B_mH_m| = 2.5×10⁵ J/m³ against the branch optimum B_r²/4μ₀ = 2.9×10⁵ J/m³ — a sphere runs 12% below optimal, needle and disc shapes far worse; consistent with (BH)max ≈ 3.3×10⁵ J/m³ quoted for N42 NdFeB (B_r ≈ 1.29 T).

The construction earns its keep the moment anything changes. Add a soft-iron circuit with an air gap and the load line tilts (the gap's reluctance enters $N_{\rm eff}$ — the ring arithmetic of Ampère's Law for H); heat the magnet and the branch itself shrinks toward the origin as $B_r(T)$ and $H_{ci}(T)$ fall; if the moving load line ever crosses the branch's knee — the corner where rigid $M$ starts to fail — the excursion is irreversible and the magnet returns weakened. Motor designers call it "knee dodging," and it is all this one diagram.

Figure

Second quadrant of a hard magnet - demagnetization branch, load lines for three shapes, and the BH-max sweet spot

Loop-as-memory, engineered

Recording technology is hysteresis run as a business. A write head is an electromagnet that drives a chosen patch of medium around one branch of its loop and leaves it at $\pm M_r$ : the bit is the remanence sign. The medium's $H_c$ is chosen inside a window — large enough that neighboring bits, stray fields, and time cannot rewrite it, small enough that the head still can. Square loops ( $M_r/M_s \to 1$ , steep switching at a sharp threshold) make the window clean, which is why media metallurgy chases loop squareness as hard as coercivity. Perpendicular recording — magnetizing bits normal to the platter rather than along it — won the density war partly on this article's physics: the perpendicular geometry lets neighboring bits' demagnetizing fields stabilize rather than erode each other, moving each bit's operating point to a safer spot on its own loop. And the terminal move, already met at the superparamagnetic floor: heat-assisted recording writes each bit near its Curie point, borrowing Curie Law and Curie Temperature's collapse to make $H_c$ momentarily tiny, then cools the stubbornness back in.

Limiting cases

Case	Result	Meaning
$H_{\max} \to 0$ (feeble fields)	Rayleigh regime: $B = \mu_iH + \nu H^2$ , $W \propto H_{\max}^3$	reversible wall bowing plus rare jumps; loss dies fast — the sensor/signal-transformer regime
perfect crystal (no defects)	$H_c \to 0$ , loop collapses to a curve	no pinning valleys, walls glide freely; hysteresis is microstructure, not chemistry
single-domain + coherent rotation	$H_c \to H_K = 2K/\mu_0M_s$	the Stoner–Wohlfarth ceiling; real hard magnets reach 20–30% of it (Brown's paradox)
linear medium	$\oint H\,dB = 0$	no enclosed area, no loss — hysteresis loss is exclusively a nonlinear-history phenomenon
$T \to T_C$	loop shrinks to nothing	$M_s$ , $B_r$ , $H_c$ all collapse; thermal erasure — owned by Curie Law and Curie Temperature
$N \to 0$ (long needle)	load line vertical, $B_m \to B_r$	shape protects the magnet; needles and closed circuits barely self-demagnetize
$N \to 1$ (flat disc, face-magnetized)	load line horizontal, $B_m \to 0$	maximal self-demagnetization; never ship a hard disc magnetized through its face without a keeper

Common trapThe B-H loop's coercivity is not where the magnet dies - and the loop is not a property of the material alone

Two executions. First: reading $H_{cB}$ (where $B = 0$ ) as "the field that demagnetizes the magnet." At $H_{cB}$ a good hard magnet's magnetization is still fully up — $B$ vanishes merely because $H = -M$ in the bookkeeping $B = \mu_0(H+M)$ . The magnet is unharmed and recovers completely; destruction happens at $H_{ci}$ , on the M–H loop, which can be several times farther out. Mixing the two coercivities is the classic way to over- or under-rate a magnet by a factor of three. Second: quoting "the" loop of a material without amplitude and history. Minor loops nest inside major ones; the loop measured at $H_{\max} = 100$ A/m is not the saturation loop scaled down (Rayleigh's cubic law says its area is far smaller); and a virgin curve exists only once. A hysteresis plot without its drive protocol is as underspecified as a permeability without its operating point — the same etiquette Magnetic Permeability enforced, one level up.

Worked examples

1 · The transformer's standing heat bill (confidence builder)

A 10 kg silicon-steel core (density 7650 kg/m³) cycles at 50 Hz with loop area 40 J/m³ at its working amplitude. Find the hysteresis power. Then estimate the change if the design flux swing rises from 1.0 T to 1.5 T (Steinmetz, $n = 1.6$ ).

Step 1. Volume: $V = 10/7650 = 1.31\times10^{-3}$ m³.

Step 2. Power density: $P/V = f\,W_{\rm cycle} = 50 \times 40 = 2000$ W/m³ — two kilowatts of heat per cubic metre of quietly humming steel.

Step 3. The core's bill: $P = 2000 \times 1.31\times10^{-3} =$ 2.6 W of standing hysteresis heat — matching the "few watts in a 10 kg core" Ferromagnetism quoted, now derived from its parts. (Eddy losses are a separate, comparable bill with $f^2$ scaling — owned by Eddy Currents.)

Step 4. Raise $B_{\max}$ to 1.5 T: $W \propto B_{\max}^{1.6}$ multiplies the loss by $1.5^{1.6} = 1.9$ , to ≈ 5.0 W. Pushing a core toward saturation nearly doubles its hysteresis bill for a 50% flux gain — one reason power transformers run at conservative flux densities.

2 · Operating point and abuse margin of a spherical magnet (exam level)

A spherical NdFeB magnet has $B_r = 1.2$ T, straight demagnetization branch $B = B_r + \mu_0H$ , and intrinsic coercivity $H_{ci} = 1.1\times10^6$ A/m. Find its open-circuit operating point, its $|BH|$ against the branch optimum, and its margin against intrinsic demagnetization.

Step 1. Load line for $N = 1/3$ : slope $-\mu_0(1-N)/N = -2\mu_0$ , so $B_m = -2\mu_0H_m$ .

Step 2. Intersect with the branch: $-2\mu_0H_m = 1.2 + \mu_0H_m \Rightarrow H_m = -1.2/(3\mu_0) = -3.18\times10^5$ A/m, and $B_m = -2\mu_0H_m =$ 0.80 T $= \tfrac23B_r$ , as the derivation's general result promised.

Step 3. Energy product: $|B_mH_m| = 0.80 \times 3.18\times10^5 = 2.5\times10^5$ J/m³. The branch optimum is $B_r^2/4\mu_0 = (1.2)^2/(4\times1.257\times10^{-6}) = 2.9\times10^5$ J/m³: the sphere runs 12% below $(BH)_{\max}$ — respectable; a disc ( $N \to 1$ ) would be catastrophically worse.

Step 4. Abuse margin: the internal field is $|H_m| = 3.2\times10^5$ A/m against $H_{ci} = 1.1\times10^6$ — a factor 3.5 of headroom. But an external reverse field or heating (which drags $H_{ci}$ down at roughly $-0.5\%/°$ C for NdFeB) eats the margin; at ≈ 150 °C this grade's $H_{ci}$ has roughly halved, and the same sphere sits within a factor 1.7 of self-destruction. This is why motor magnets are specified by temperature grade and not just by $(BH)_{\max}$ .

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.4.2 · Jackson, Classical Electrodynamics, 3rd ed., §§5.8, 5.11 (magnetic circuits and the load line) · Purcell & Morin, Electricity and Magnetism, 3rd ed., ch. 11 · Zangwill, Modern Electrodynamics, §§13.4–13.5 · Chikazumi, Physics of Ferromagnetism, 2nd ed. (Rayleigh regime, coercivity mechanisms) · Bozorth, Ferromagnetism (the loop-data bible) · Coey, Magnetism and Magnetic Materials (Brown's paradox, modern hard magnets).

Provenance. Equations and definitions verified against Griffiths §6.4.2, Jackson §§5.8 & 5.11, Purcell Ch. 11, and Zangwill §13.4–13.5. A verifier should recompute: (1) the dimensional identity $[H\,\mathrm{d}B]=\mathrm{J/m^3}$ ; (2) the load-line slope $B=-\mu_0\frac{1-N_d}{N_d}H$ from $\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M})$ with $H_{\mathrm{in}}=-N_d M$ ; (3) demagnetizing factors summing to 1 (sphere $N_d=1/3$ ); (4) the Curie temperature of iron ( $\approx1043\,\mathrm{K}$ ) and $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ . Steinmetz exponent $n\approx1.6$ – $2$ is empirical and stated as a range, not a fabricated exact value. Addendum for this edition: Rayleigh branches and the $4/3$ cubic-loss coefficient added per Chikazumi/Bozorth; Stoner–Wohlfarth bound recomputed ( $H_K$ = $4.5\times10^4$ A/m for Fe, $6.1\times10^6$ A/m for Nd₂Fe₁₄B; measured $H_{ci}$ 20–30% of ceiling = Brown's paradox); sphere operating point recomputed ( $B_m = \tfrac23B_r$ , $|BH|$ 12% under $B_r^2/4\mu_0$ ); transformer example recomputed (2.6 W at 40 J/m³, 50 Hz, 10 kg); $\mu_0$ used at its 2019-rebased value per Magnetic Permeability.

Builds on: Ferromagnetism · Magnetization · The Auxiliary Field H · Magnetic Permeability · Linear and Nonlinear Magnetic Media

Leads to: Ampère's Law for H · Magnetic Susceptibility · Field of a Magnetized Object

Why does the B-H loop area, and not the M-H loop area, give the energy loss?

Because the work theorem — quoted here, derived in Magnetic Energy in Materials — counts the work the external circuit does against induced EMFs, and that bookkeeping lands on $\oint H\,dB$ . The two areas differ by $\oint H\,d(\mu_0H) = \mu_0\oint H\,dH = 0$ — the integral of an exact differential around a closed path. So in fact $\oint H\,dB = \mu_0\oint H\,dM$ : the two loops enclose the same area up to the factor $\mu_0$ , and no energy hides in the vacuum term. The practical distinction is different: the B–H loop is what a wound-core measurement directly records (flux from the pickup winding, $H$ from the drive current), which is why loss specs are always B–H areas.

Is coercivity a fundamental material constant like Ms or K?

No — and this is the article's central honesty. $M_s$ and $K$ are set by composition and crystal structure; coercivity is set by microstructure — defect density, grain size, surface finish, annealing history. The same chemical iron spans 1 A/m (well-annealed, pure) to several hundred (cold-worked); the same Nd₂Fe₁₄B compound spans grades differing threefold in $H_{ci}$ by grain-boundary engineering alone. The fundamental constants only set the ceiling ( $H_K = 2K/\mu_0M_s$ ) and the floor (zero, for a perfect defect-free crystal above single-domain size). Everything between is metallurgy.

What sets the 20-30% of ceiling that real hard magnets achieve - why does Brown's paradox stop there?

Nucleation. The Stoner–Wohlfarth bound assumes the whole grain must rotate together, but reversal actually begins wherever the local ceiling is lowest: a grain corner concentrating stray field, a surface layer damaged by machining, a soft magnetic phase a few nanometres thick at a grain boundary. Once a reverse nucleus exists, wall motion — nearly free inside a clean grain — finishes the job instantly. Modern magnet processing is nucleation suppression: smooth grain surfaces, non-magnetic grain-boundary phases to isolate each grain, and dysprosium enrichment only at grain surfaces (where nucleation starts) to raise the local anisotropy without paying for heavy rare earths in the bulk.

Do minor loops always close? If I cycle a transformer core at fixed amplitude forever, does the loop drift?

To first approximation minor loops close and retrace — that is what makes AC magnetics predictable. In detail, two drifts exist. Accommodation (reptation): the first few cycles after a change of amplitude shift slightly before settling into a repeating loop — walls redistributing among metastable valleys. And magnetic after-effect: thermally activated creep over barriers, causing slow logarithmic drift, worst near the coercive region where barriers are marginal. Both are small in good core materials at design amplitudes; both become measurable in precision fluxgate magnetometers and are part of why such instruments are periodically re-zeroed.

The loss formula is per cycle, so hysteresis loss scales linearly with frequency. Why do high-frequency cores use ferrites instead of better steel?

Because the other loss — eddy currents — scales as $f^2$ and overtakes hysteresis loss with frequency. Metallic cores fight eddies with lamination, but by hundreds of kilohertz no practical lamination is thin enough; ferrites win because they are electrical insulators (resistivities $10^6$ times iron's), taking the eddy bill out of the game and leaving only this article's hysteresis bill, which their soft, narrow loops keep small. The full eddy story — skin depths, lamination scaling — belongs to Eddy Currents at the chapter's end; the division of the total core loss into hysteresis + eddy + excess is standard datasheet anatomy.

Why does heating help write bits (HAMR) but destroy motor magnets - is temperature good or bad for hysteresis?

Same physics, opposite sides of the ledger. Coercivity falls with temperature (barriers fixed, thermal energy rising, and $K$ itself collapsing toward $T_C$ — Curie Law and Curie Temperature's territory). If you want to switch (writing a bit), lower coercivity is a gift: HAMR borrows it for a nanosecond per bit, then cools back. If you want to hold (a motor magnet at 150 °C with a stall field applied), the same fall eats your abuse margin — worked example 2's factor 3.5 shrinking toward 1. Temperature doesn't take sides; it just moves every point of the loop toward the origin, and whether that helps depends on whether you are the writer or the archive.

Is there hysteresis in paramagnets or diamagnets?

No. Both are single-valued responses: $M$ depends only on the present $H$ (through the tanh law of Paramagnetism or the induced-moment law of Diamagnetism), with no domain structure and no pinning landscape, so the return path retraces the outgoing one exactly and $\oint H\,dB = 0$ . Hysteresis requires metastability — multiple locally stable configurations at the same applied field — and only cooperative order (ferro-, ferri-, antiferromagnetic, or their electric twins in Ferroelectrics and Spontaneous Polarization) supplies it. That is why the loop is the definitive experimental fingerprint distinguishing a ferromagnet from a strong paramagnet at a single temperature.