Domains, Hysteresis & Choosing Magnetic Materials

Stroke a screwdriver tip along a bar magnet a dozen times, always the same way, and the screwdriver becomes a magnet — handy for retrieving dropped screws. Drop it hard onto concrete a few times and the trick is gone. Nothing was added or removed; the same iron atoms sit in the same steel. What the stroking built and the dropping wrecked was organisation — and organisation is the entire difference between a lump of iron and a permanent magnet. Dia-, Para- & Ferromagnetism ended with a promise: iron's atomic moments are locked to their neighbours in unanimous blocks called domains. This module opens the box. You will see why a nail with perfectly aligned domains inside each block can still show no magnetism at all, how an applied field magnetises it — walls between domains sliding, then whole blocks swinging round — and why the process refuses to run backwards cleanly. That refusal has a name, hysteresis ("lagging behind"), a shape — the famous B–H loop — and a price: energy burned as heat every single cycle, which is why transformer cores hum warm at 50 Hz. Best of all, hysteresis is a feature you can shop for. A thin loop makes soft magnetic material for electromagnets and transformer cores: easy to magnetise, easy to erase. A fat loop makes hard material for permanent magnets: stubborn both ways. One graph tells you which metal to buy — and reading it is the skill this module installs.

Key takeaways

A ferromagnet is divided into domains — regions of perfectly aligned atomic moments; an unmagnetised sample has domains pointing every which way, cancelling in bulk.
Magnetising = organising: first domain walls move (aligned domains grow), then domain directions rotate; at saturation, the sample is one aligned block and no further magnetisation is possible.
The response lags the cause: plot $B$ against the magnetising field $H$ through a full cycle and you get the hysteresis loop, not a single line.
Two landmarks on the loop: retentivity $B_r$ (field remaining when $H$ returns to zero) and coercivity $H_c$ (reverse field needed to force $B$ to zero).
The area of the loop measures energy dissipated as heat per unit volume per cycle — the running cost of cyclic magnetisation.
Soft materials (soft iron: tall thin loop, tiny $H_c$ ) make electromagnets and transformer cores; hard materials (steel, alnico, NdFeB: fat loop, huge $H_c$ ) make permanent magnets.
Key idea: magnetisation is domain politics with memory — the B–H loop is the ledger of that memory, and its shape is a buyer's guide to magnetic materials.

The core insight

Ferromagnets magnetise by moving domain walls and rotating domains — processes with friction and memory — so B lags H around a hysteresis loop whose retentivity and coercivity mark what the material remembers and how hard that memory is to erase, and whose area is the heat paid per cycle.

Where this lives today

The two loop archetypes split the modern world. The thin-loop empire lives in every transformer and motor: grain-oriented silicon steel — walls deliberately made easy to move — holds hysteresis loss to tens of J/m³ per cycle, and grid engineers still fight for every joule. Mu-metal, the ultimate soft material, lines rooms where brain magnetism is measured, because its walls move at the whisper of a field. The fat-loop empire is rearranging energy technology: NdFeB magnets with million-A/m coercivities put a kilowatt motor in a drone and gearless megawatt generators in wind-turbine nacelles. And the memory itself became memory: a hard disk stores bits as small magnetised regions of a moderately-hard film — coercive enough to survive, soft enough for a write head to flip — each bit's write tracing a private hysteresis loop. Much of civilisation's stored knowledge is, physically, retentivity defended by coercivity.

The physical picture

Start inside the unmagnetised nail. Dia-, Para- & Ferromagnetism established that below the Curie temperature, each iron atom's moment is locked parallel to its neighbours'. Naively that should make every nail a strong magnet. It doesn't — because total alignment would pour an enormous magnetic field into the surrounding space, and fields cost energy. Iron finds a cheaper arrangement: it splits into domains, typically micrometres across ( $10^{15}$ -odd atoms each). Inside any one domain the alignment is total — the lock-in never weakens — but different domains point in different directions, chosen so their external fields largely cancel. Locally saturated, globally silent. "Unmagnetised" iron is not disordered iron; it is perfectly organised iron in cancelling committees. Between two domains sits a domain wall: a thin transition layer (tens of nanometres — hundreds of atoms) across which the moment direction swivels from one domain's direction to the next's. Now apply a field, gently at first. Domains whose alignment already agrees with the field are energetically favoured, so they grow — the walls between them and their less-favoured neighbours slide outward, converting territory atom by atom, like a border moving rather than a population migrating. This wall motion is the cheap, early stage of magnetisation. Push harder and the second, more expensive stage engages: whole domains whose easy directions disagree with the field rotate their alignment toward it. Push hard enough and the sample becomes a single aligned block: saturation. Every moment enlisted; more field now adds nothing the material can give.

Figure

Magnetising a ferromagnet — domain walls move, then domains rotate

Here is the twist that generates everything else in this module: the walls don't slide smoothly. Real iron is full of imperfections — grain boundaries, impurities, internal stresses — and walls snag on them: a wall clings to a defect until the field builds enough pressure, then jerks free to the next snag. (Played through a loudspeaker, magnetising iron audibly crackles with these jumps — the Barkhausen effect, 1919.) Snags are friction, and friction has two consequences you know from mechanics: energy is lost as heat, and processes don't retrace when reversed. Remove the field, and the walls — snagged again on the way back — do not return home. The sample keeps some alignment. It remembers.

The B–H loop, walked once

The memory has a standard portrait. Put an iron sample inside a solenoid; the coil's current dials the magnetising field $H$ ( $H = nI$ , your knob from Magnetisation, Intensity & Susceptibility), and track the sample's $B$ through one full cycle of pushing, reversing and returning. The result is the most information-dense graph in school magnetism.

Figure

The hysteresis loop — anatomy of magnetic memory

Walk it with the domains in mind. The virgin curve (dashed): from the demagnetised state, $B$ climbs steeply as walls slide, then bends as rotation finishes the job — the knee from Magnetisation, Intensity & Susceptibility's snapshot, the curve on which any quoted $\mu_r$ of iron is one chord.

Switch-off: from saturation, wind $H$ back to zero. The walls snag on the way home; alignment survives; $B$ lands not at zero but at the retentivity (or remanence) $B_r$ . The sample is now a permanent magnet — this point is what "permanent magnet" means on a graph. Soft iron retains a lot ( $B_r$ over 1 T); so does steel.

Erase: to force $B$ to zero you must push backwards — apply a reverse field. The reverse push that just wipes the sample clean is the coercivity $H_c$ , the single most important number in magnet shopping. Soft iron: $H_c$ of a few A/m to ~100 A/m — a stray field can erase it. Hard steel: thousands. Modern neodymium magnets: around $10^{6}$ A/m — you cannot erase them without machinery built for the purpose.

Keep going and the reverse field saturates the sample the other way; cycling back completes the closed loop. Two facts about the finished portrait carry all the marks. First, the loop never retraces: for a given $H$ there are two possible $B$ values, depending on history — the definition of hysteresis. Second, the enclosed area equals the energy dissipated as heat, per unit volume, per cycle — every snap of every snagged wall pays irreversible heat, and the area is that bill totalled (stated and used here; the proof is university bookkeeping).

Derivation AWhy the iron splits into domains at all

We want to see why a ferromagnet below its Curie point — whose atomic moments insist on local alignment — nonetheless organises itself into many oppositely-pointing domains rather than one grand aligned block. The strategy is an energy audit between two configurations of the same bar: single-domain versus split. We use only one physical input, familiar from the whole track: magnetic fields store energy, and more field-filled space means more stored energy. Watch how the compromise between two costs predicts an intermediate domain size, which is exactly what microscopes find.

Step 1. Price the single-domain bar. Fully aligned, it is the strongest magnet its atoms allow — Bar Magnets & the Dipole Picture shows the result: a dense external field arcing pole to pole, filling the surrounding space. All of that external field is stored energy the configuration must pay for — of order half a joule per cubic centimetre for iron.

Step 2. Let the bar split into two oppositely-magnetised halves. Each half's external field now feeds almost directly into its neighbour's return path: flux that once ballooned through the room short-circuits locally, and the energy bill drops by roughly half. Split again: cheaper again. Nature discovered flux management long before transformer engineers did. Why the moments tolerate this: the anti-alignment happens only at the borders between domains. Deep inside each domain every atom still sits parallel to its neighbours, fully honouring the lock-in. Only the thin wall pays a disagreement penalty.

Step 3. Price the walls. Each wall is a layer where moments swivel gradually between the two domain directions — every atom pair across it is slightly misaligned, and misalignment costs lock-in energy. So each new wall adds a fixed energy per unit wall area. Splitting is not free.

Step 4. Find the compromise. External-field savings shrink with each further split (halving something already small), while wall costs mount steadily, one wall at a time. Below some domain size, a new wall costs more than it saves: splitting stops. The equilibrium is many domains of intermediate size — micrometres in ordinary iron, exactly the scale seen when domain patterns are imaged with fine magnetic powder. Verified: the audit predicts that particles below roughly a hundred nanometres can't afford even one wall and must stay single-domain — observed, and prized in recording media. It also explains "unmagnetised": cancelling domains is the cheapest state, so it is what iron relaxes toward when heated, hammered, or cycled with decaying AC.

Soft versus hard — the buyer's guide

Everything a designer needs is the loop's shape — tall-and-thin versus fat-waisted. The two archetypes:

property	soft magnetic material	hard magnetic material
loop shape	tall and thin	fat, wide waist
coercivity $H_c$	tiny (A/m to ~10² A/m)	huge (10³–10⁶ A/m)
retentivity $B_r$	high — but trivially erased	high and defended by $H_c$
loop area (heat per cycle)	small	large
magnetises / demagnetises	easily, instantly	reluctantly, permanently
examples	soft iron, silicon steel, mu-metal	hard steel, alnico, NdFeB, ferrite magnets
careers	electromagnet cores, transformer cores, relay armatures, magnetic shielding	permanent magnets: motors, speakers, compass needles, fridge doors

The assignments follow from the loop, and ICSE asks you to argue them. An electromagnet must switch: full strength with current, nothing without — soft iron's tiny $H_c$ (near-total self-erasure at switch-off) makes it ideal. A transformer core is remagnetised fifty times a second, forever; loop area is a heat bill paid at 50 Hz, so silicon steel is bred for the thinnest loop. A permanent magnet must hold its magnetisation against stray fields, vibration and time: what matters is coercivity, the moat around the memory — hard steel at minimum, alnico or neodymium in anything modern. Note the trap hiding in the table (and beloved of examiners): soft iron's retentivity is actually higher than steel's. Retentivity alone does not make a permanent magnet — what good is remembering, if any passing field can make you forget? The permanent-magnet virtue is coercivity. And the demagnetising menu is the loop read backwards. To wipe a magnet: heat past the Curie point (770 °C for iron — the lock-in itself collapses); hammer it (jolts kick snagged walls loose, letting domains relax toward cancellation); or — the clean method — cycle an alternating field of shrinking amplitude, spiralling the loop inward to zero: exactly what a demagnetising wand or degausser does.

Limiting cases

case	result	meaning
tiny $H$ , gentle cycling	slim inner loops, nearly reversible	walls rock in their snags without jumping — minimal loss
$H$ far past the knee	$B$ flat-lines at saturation	all domains enlisted; the material is spent
loop area → 0	ideal soft material	reversible magnetisation: transformer heaven, no memory
$H_c$ enormous	ideal hard material	memory defended against any plausible stray field
particle below ~100 nm	single domain — no walls at all	wall motion impossible; magnetisation only by rotation (recording media)
$T$ raised past $T_C$	loop collapses to a line through the origin	no domains, no memory: ferromagnetism itself has dissolved

Common trapHigher retentivity means a better permanent magnet

It does not — and the counterexample is the classic ICSE trip-wire: soft iron retains more field than steel when the magnetising field is removed ( $B_r$ is higher), yet permanent magnets are made of steel and its harder descendants, never of soft iron. The reason is the second landmark: soft iron's coercivity is minuscule, so its splendid remembered field is erased by the first stray field, jolt or warm afternoon that comes along. Steel remembers slightly less but defends the memory with a coercivity thousands of times larger. Permanent-magnet quality is retentivity guarded by coercivity — on the loop, a wide waist, not just a tall intercept. If an exam asks "why steel and not soft iron for permanent magnets," the one-word core of the answer is coercivity.

Worked examples

Board level — read the loops, assign the careers (ICSE pattern, 5 marks)

Materials P and Q have hysteresis loops measured on identical samples. P: retentivity 1.35 T, coercivity 40 A/m, small loop area. Q: retentivity 0.90 T, coercivity 5.5 × 10⁴ A/m, large loop area. (a) Which material should be used for the core of a school electromagnet, and why (two reasons)? (b) Which for the permanent magnet of a loudspeaker, and why? (c) A student argues P is the better permanent magnet "because it retains more magnetism." Correct the argument.

Step 1. Part (a): P. Reason one: its tiny coercivity (40 A/m) means it demagnetises almost completely the moment the current stops — an electromagnet must switch off. Reason two: its small loop area means little heat is wasted per magnetising cycle, so it can be switched (or driven by varying current) without cooking the core.

Step 2. Part (b): Q. A loudspeaker magnet must stay magnetised for years against vibration (its whole job is shaking a coil) and against the coil's own alternating field. Q's coercivity is over a thousand times P's: its remembered field of 0.90 T is defended. Loop area is irrelevant here — a permanent magnet is not cycled.

Step 3. Part (c): the student quoted retentivity, which only says how much field remains immediately after magnetising. Permanence is coercivity's business: P's larger memory (1.35 T > 0.90 T) can be wiped by a ~40 A/m stray field — commonplace near any current-carrying device — while erasing Q needs 5.5 × 10⁴ A/m, found only in a deliberate demagnetiser. Q is the permanent magnet; P is the better temporary magnet.

Competitive level — the transformer core's heating bill (legacy-JEE / ISC-stretch pattern)

A transformer core of volume $1.2 \times 10^{-3}\ \text{m}^3$ is taken around its hysteresis loop 50 times per second. The loop's area, in B–H units, corresponds to 350 J/m³ per cycle. (a) Find the power dissipated as hysteresis heat. (b) Find the energy wasted per day, in kWh, and its daily cost at ₹6.0 per kWh. (c) Replacing the core with a better silicon steel of loop area 40 J/m³ — by what factor does the hysteresis loss fall?

Step 1. Part (a). The loop area is heat per unit volume per cycle, so power = area × volume × frequency:

P = 350 \times 1.2\times10^{-3} \times 50 = \mathbf{21\ \text{W}}

A silent 21 W — the core runs warm doing nothing else at all.

Step 2. Part (b). Energy per day $= 21\ \text{W} \times 24\ \text{h} = 504\ \text{Wh} = \mathbf{0.504\ \text{kWh}}$ . (The kWh arithmetic is exactly Household Circuits' billing craft.) Daily cost $= 0.504 \times 6.0 = \mathbf{₹3.02}$ — about ₹1,100 a year, for one modest core, forever. Multiply by a national grid's worth of transformers and you see why core steel is a research field.

Step 3. Part (c). Loss is proportional to loop area at fixed volume and frequency: factor $= 350/40 = \mathbf{8.75}$ . The 21 W bill drops to 2.4 W. This is precisely why transformer laminations are silicon steel bred for thin loops, not ordinary iron. Timing note: 3 minutes; the entire problem is the single sentence "area × volume × cycles per second = power," plus kWh housekeeping. If you find yourself hunting for μ or χ, you've missed that the loop area was handed to you pre-integrated. (Note for completeness: real cores also lose energy to eddy currents — induced circulating currents, a separate bill you will meet in Electromagnetic Induction; lamination attacks that one.)

Exam traps

Axis discipline. The loop is $B$ (sample) versus $H$ (your coil's push, $= nI$ ). Retentivity is the B-intercept ( $H = 0$ ); coercivity is the H-intercept ( $B = 0$ ). Swapping them is the most common sketch error.

Retentivity ≠ permanence. Soft iron out-remembers steel at the moment of switch-off; steel out-keeps it. Permanent-magnet ranking goes by coercivity (see the warning above). Expect exactly this as a reason-assertion question.

The loop is history-dependent. One $H$ , two loop values of $B$ — which applies depends on where the sample has been. Treating $B(H)$ as single-valued inside the loop is structurally wrong.

Area units. Loop area on B–H axes has units $\text{T}\cdot\text{A/m} = \text{J/m}^3$ — energy per volume per cycle. The safe formula is $P = (\text{area}) \times V \times f$ ; dropping $V$ or $f$ is the standard slip.

Board scope honesty. Hysteresis was deleted from CBSE's Class-12 syllabus in the 2023 rationalization and is absent from the current JEE-Main and NEET syllabi — for those exams this module is background that makes Dia-, Para- & Ferromagnetism's "ferro has memory" line make sense. It remains squarely on ISC (loop sketch, landmark definitions, soft/hard uses) — and IGCSE tests the behavioural half only: temporary (soft-iron) versus permanent (steel) magnets, and electromagnet-versus-permanent-magnet choices, without the B–H loop language.

Vocabulary variants. Retentivity = remanence = residual magnetism; coercivity = coercive force. Match your board's wording and define on first use.

Origins · 1 minJames Alfred Ewing and how this was discovered

James Alfred Ewing

1855–1935 · Dundee → Tokyo → Cambridge

James Alfred Ewing was twenty-three when he sailed from Scotland to become one of Meiji Japan's imported professors of engineering, and it was in Tokyo — between founding Japan's first seismology society — that he began magnetising iron wires and plotting what came back. He found the response forever trailing the push: field off, magnetism lingering; field reversed, the past still showing through. In 1881 he named it hysteresis, from the Greek hysterein, "to lag behind." Back in Britain he mapped B–H loops methodically, built a model of coupled compass needles that reproduced domain behaviour decades before domains were confirmed, and understood that loop area meant wasted heat — the insight transformer designers still bill by. His later life read like a novel: Director of Naval Education, then head of Room 40, the Admiralty's First-World-War codebreaking bureau. The word he coined now describes lagging systems everywhere, from thermostats to economics.

History

1881

— James Alfred Ewing, back from Tokyo, names hysteresis and begins the systematic mapping of B–H loops.

1907

— Pierre Weiss proposes that ferromagnets are divided into spontaneously magnetised domains — organisation, not magnitude, is what a field changes.

1919

— Heinrich Barkhausen wires a coil around iron and hears magnetisation crackle: domain walls jump, they do not glide.

1931

— Francis Bitter photographs domain patterns directly, dusting fine magnetic powder onto polished iron — the committees made visible.

1983

— Sagawa's team in Japan announces Nd₂Fe₁₄B: neodymium magnets with coercivities near 10⁶ A/m, the fat loop that shrank every motor and speaker.

Going further

Texts. ISC Class 12 Physics, magnetism unit (hysteresis loop, retentivity, coercivity, uses of soft iron and steel) · IGCSE 0625, §4.1 (temporary vs permanent magnets, electromagnets) · NCERT Class 12 Ch. 5 (older editions carry the hysteresis section; retained here for ISC/legacy readers) · HC Verma Vol. 2, Ch. 37. For the university-level machinery — wall-width energetics, why coercivity is microstructure not chemistry, loop-area bookkeeping, permanent-magnet load lines — see Magnetic Domains and Domain Walls and Hysteresis (UG deep dive).

Provenance. All results standard. A verifier should recompute: transformer example $P = 350 \times 1.2\times10^{-3} \times 50 = 21$ W; per day $21 \times 24 = 504$ Wh $= 0.504$ kWh; cost $0.504 \times 6 = ₹3.02$ /day (≈ ₹1,104/yr); improvement factor $350/40 = 8.75$ , residual $21/8.75 = 2.4$ W; quiz numerical $200 \times 2\times10^{-3} \times 50 = 20$ W; anchors — soft iron $H_c \lesssim 10^2$ A/m, steel $\sim10^3$ – $10^4$ , NdFeB $\sim10^6$ A/m; iron $T_C = 770$ °C; domain scale μm, wall scale tens of nm.

Prerequisites. Dia-, Para- & Ferromagnetism. Leads to. Classify the Material — χ, μᵣ & Behaviour (E5's drill gym, coming up) · Eddy Currents and The Transformer (E6/E7 — where the loop's heating bill meets its induction twin).

If domains are already perfectly aligned inside, why isn't every piece of iron a magnet?

Because alignment is local and the directions are negotiated globally. Each domain is saturated within itself — the ferromagnetic lock-in never rests — but the directions of different domains are arranged to cancel, because that starves the external field and fields cost energy (the derivation's audit). "Unmagnetised" iron is like a parliament of disciplined parties with no majority: total internal order, zero net policy. Magnetising doesn't create alignment; it redistributes territory between alignments that already exist.

What physically happens when I stroke a screwdriver with a magnet to magnetise it?

Each pass drags a strong, oriented field along the steel, pressuring walls to move and domains to swing toward the field direction — and because the passes always run the same way, the gains accumulate. Steel's coercivity then defends the new arrangement when the magnet leaves. Dropping the screwdriver reverses the trick without any field: each impact jolts snagged walls loose, and freed walls relax toward the cheapest state — the cancelling committees. Heating past the Curie point is the nuclear option: it dissolves the lock-in itself.

Does a permanent magnet fade with time?

Only as fast as something pushes its domains back — stray fields comparable to its coercivity, jolts, or heat. A modern NdFeB magnet ( $H_c \sim 10^6$ A/m) loses of order 1 % over years; an old steel bar magnet, with far smaller coercivity, genuinely fades over decades — which is why lab magnets were stored with soft-iron "keepers" bridging their poles to close the flux loop. Near the Curie point, any magnet forgets in seconds.

Why does the magnetisation lag instead of just following H back down?

Friction in the walls. Moving a domain wall through real iron means dragging it past grain boundaries, impurities and stress points that snag it; the wall jumps free only when pushed hard enough, and each jump is irreversible — the energy spent tearing free radiates away as heat and cannot be refunded by retracing the path. So on the way back down, walls stick where they were pushed rather than gliding home: B stays high after H has left. Lag, memory and heat are three faces of the same snag-and-jump physics — which is why the Barkhausen crackle, the loop's openness, and the warm core all show up together.

What's the difference between hysteresis loss and eddy-current loss in a transformer?

Both heat the core, but by different physics with different fixes. Hysteresis loss is this module: domain walls snapping past defects, billed as loop-area × volume × frequency; the fix is material choice (silicon steel's thin loop). Eddy-current loss is Electromagnetic Induction's business: the changing flux induces circulating currents in the conducting core itself, which heat it resistively; the fix is geometry — laminating the core into insulated sheets that interrupt the current paths. A real transformer engineer budgets the two separately: hysteresis scales with $f$ , eddy currents with $f^2$ , so which one dominates depends on the operating frequency.

Are the domains real and visible, or a theoretical convenience?

Entirely real and repeatedly photographed. Bitter's 1931 technique — fine magnetic powder settling along domain boundaries on polished iron — made the walls visible in an ordinary microscope; modern magneto-optic imaging (the Kerr effect) films walls moving in real time as a field is applied, snags, jumps, and all. You can also hear them: a coil around an iron bar, fed to an amplifier, crackles with Barkhausen jumps as a magnet approaches. Domain sizes run from fractions of a micrometre to millimetres depending on the material and its treatment — big enough that "committee politics" is documentary description, not metaphor.