Magnetization

A new chapter opens here, and it opens the way the dielectrics chapter did — by filling matter with the dipoles the previous chapter perfected. Electric fields in matter began with polarization $\vec P$ , the electric dipole moment per unit volume; magnetic fields in matter begin with magnetization $\vec M$ , the magnetic dipole moment per unit volume. Every atom is a current loop in miniature — orbiting and spinning electrons, the moments quantified in the next article — and when those moments organize, from a compass needle to a hard-drive grain to the iron in your blood refusing (unlike bird blood's magnetite) to point north, the material becomes a source of magnetic field. This chapter's program is to compute the fields of magnetized matter and the response of matter to fields, and $\vec M$ is the variable everything else is written in. The opener's centerpiece is the chapter's founding miracle, provable with counting: a uniformly magnetized object is exactly equivalent to a surface current. Interior atomic loops cancel neighbor against neighbor; only the rim's circulation survives; and a uniformly magnetized cylinder becomes, precisely, a solenoid with $nI=M$ . A bar magnet is a solenoid — not "like" one: the same $\vec B$ , everywhere, from the same effective current sheet — and the numbers are extraordinary: saturated iron is equivalent to 1.7 million amperes per meter of winding, a current density no copper coil can sustain, delivered silently, forever, without a power supply. The chapter map matters as much as the opener. Magnetized matter answers to three families — diamagnets (weakly repelled), paramagnets (weakly attracted), ferromagnets (violently attracted and capable of permanent $\vec M$ ) — whose mechanisms are quantum and get their own articles, as do the general bound-current theorem, the fields of magnetized objects, the auxiliary field $\vec H$ , susceptibility, domains, and hysteresis. This article defines the variable, proves the uniform-case equivalence, sets the scales, and hands each deeper question to its named owner.

Key takeaways

Magnetization $\vec M$ = magnetic dipole moment per unit volume (A/m) — the magnetic twin of $\vec P$ , defined by the same mesoscopic averaging that made $\vec P$ meaningful.
Uniform $\vec M$ is equivalent to a surface current $\vec K_b=\vec M\times\hat n$ : interior atomic loops cancel cell-by-cell, the rim survives. (The general theorem, $\vec J_b=\nabla\times\vec M$ for nonuniform $\vec M$ , is owned by the bound-currents article.)
Payoff: a uniformly magnetized cylinder is exactly a solenoid with $nI=M$ — inside a long bar magnet, $\vec B=\mu_0\vec M$ ; outside, the dipole field. Iron's saturation $M_s\approx1.7\times10^6$ A/m ⇒ the equivalent of a 1.7 MA/m winding, current-free.
Three response families: diamagnetic (induced $\vec M$ opposing $\vec B$ , universal, weak), paramagnetic (permanent atomic moments aligning, weak, temperature-fighting), ferromagnetic (exchange-locked cooperative alignment, strong, hysteretic) — mechanisms owned by their three articles.
$\vec M$ can be frozen in (permanent magnets: $\vec M$ as an independent state variable) or induced (linear media: $\vec M$ responding to the field — the susceptibility articles' business).
The $\vec P\leftrightarrow\vec M$ analogy is powerful and breaks with a sign: inside a polarized slab $\vec E$ opposes $\vec P$ , but inside a magnetized bar $\vec B$ aligns with $\vec M$ — bound currents reinforce where bound charges oppose.
KEY IDEA: magnetization is moment per volume, and its geometry IS a current distribution — for uniform $\vec M$ , a surface current $\vec M\times\hat n$ that turns every bar magnet into a solenoid.

The core insight

Magnetization is magnetic moment per unit volume, and it masquerades as a current distribution: for a uniformly magnetized body the interior loops cancel and a surface current $\vec K_b=\vec M\times\hat n$ survives — a bar magnet is a solenoid with $nI=M$ .

Where this lives today

The technology in your pocket is a magnetization-management device. Hard-disk platters store bits as the $\pm\vec M$ orientation of $\sim$ 20-nanometer grains (perpendicular recording keeps $\vec M$ normal to the platter, packing terabits per square inch); MRAM chips store them in magnetic tunnel junctions and read them by magnetoresistance; and the write head in both cases is a tiny electromagnet performing hysteresis on demand — the whole loop of this chapter, executed a trillion times a day at gigahertz rates. The magnets driving your earbuds, phone haptics, and EV motor are the worked example's neodymium compounds ( $\mu_0M_s\approx1.6$ T with the coercivity to keep it — the "energy product" race that transformed motors in the 2000s), and the supply-chain anxieties around them have made "magnetization per kilogram of rare earth" a geopolitical quantity. At the research frontier, magnetization has gone dynamic and quantum. Spintronics manipulates $\vec M$ with spin-polarized currents rather than fields (spin-transfer and spin-orbit torques — writing bits with the conduction electrons' own moments); antiferromagnets, whose sublattice magnetizations cancel to $\vec M\approx0$ yet switch at terahertz rates, are candidate memory beyond ferromagnets; and 2D magnets one atomic layer thick (CrI₃ and kin, since 2017) probe how thin a material can be and still keep the cooperative alignment this chapter's later articles explain. Every one of these fields writes its equations in the variable this article just defined.

Origins · 1 minMichael Faraday and how this was discovered

Michael Faraday

1791–1867 · Royal Institution, London — the man who asked what matter does in a field

Faraday's magnetism went far beyond induction. In 1845 — nearing sixty, working with the great electromagnet at the Royal Institution — he asked a question nobody had asked systematically: does every substance respond to a magnet, or only iron's little club? Hanging bar after bar between the poles — glass, bismuth, copper, wood, apples, blood — he found that everything responds: some substances set themselves along the field like iron ("paramagnetic," his coinage with William Whewell), others — his heavy borate glass first, then bismuth most dramatically — set themselves across the field and crept toward weak field ("diamagnetic," likewise his word). Matter's three magnetic families, this chapter's taxonomy, date from those months of 1845; so does the term "magnetic field" itself. Characteristically, Faraday found the phenomenon by conviction — his faith in the unity of forces had him passing light through that same glass in the same weeks, discovering magneto-optical rotation, the first hint that light itself is electromagnetic. He kept no mathematics; he left the taxonomy, the field concept, and the experimental method that Maxwell would translate into the equations this track is built on.

History

1820

— Ampère, within weeks of Ørsted's news, proposes that all magnetism — magnets included — is circulating current: molecular loops, a century before electrons.

1845

— Faraday tests everything in his electromagnet: diamagnetism and paramagnetism discovered and named; matter's three-family taxonomy is born.

1907

— Weiss postulates the molecular field and magnetic domains: saturated magnetization locally, geometry deciding the total — the modern picture of ferromagnets.

1925

— Uhlenbeck and Goudsmit propose electron spin; the atomic moments behind M acquire their quantum identity.

1983

— The neodymium-iron-boron magnet (Sagawa; General Motors) roughly triples the practical energy product — the permanent-magnet revolution behind modern motors and drives.

The physical picture

Zoom into any material and you find circulation. Electrons orbit nuclei (a current loop, moment $\sim\mu_B$ ), and each electron spins (a moment of almost exactly $\mu_B$ , no classical picture required — next article's careful business). Every atom is a fleet of microscopic magnetic dipoles, typically $10^{28}$ – $10^{29}$ of them per cubic meter. In most matter, most of the time, they point every which way and average to nothing. Organize them — by an applied field, by chemistry, by the quantum conspiracy called exchange — and the material acquires a net moment in every parcel. The bookkeeping variable is the moment per volume, and defining it needs the same care $\vec P$ needed: atom by atom the "moment density" is wildly spiky, so $\vec M$ is a mesoscopic average — sample a volume big enough to hold thousands of atoms, small enough to be a "point" for the problem's geometry, and the average plateaus exactly as the polarization article's averaging argument showed (recalled, not re-proved: swap dipole type, keep the logic). For iron at full alignment: $n=8.5\times10^{28}$ atoms/m³ times $2.2\,\mu_B$ per atom gives $M_s\approx1.7\times10^6$ A/m — the chapter's benchmark number. Now the founding miracle. Take a slab of uniformly magnetized material — every mesoscopic cell carrying the same little circulation, all in the same sense. Look at any interior boundary between two cells: the current going up the right edge of one cell meets the current coming down the left edge of its neighbor. Equal loops, opposite directions: cancellation, exactly, at every interior edge. The only currents with no neighbor to cancel against live on the outer rim — and around the rim, cell after cell contributes its outer edge in the same rotational sense, chaining into one continuous current that circulates around the whole body. Uniform magnetization has no interior currents at all; it is, magnetically, a surface current.

Figure

The counting argument — aligned interior loops cancel edge by edge, and only the rim current survives

Three families of matter organize their loops in three ways, and the chapter assigns each its own article because each mechanism is a different piece of physics. Diamagnets (water, copper, bismuth, you): no permanent atomic moments; an applied field slightly distorts the orbital currents, inducing a magnetization that opposes the field (a Lenz-law instinct made permanent) — universal, feeble ( $M/H\sim-10^{-5}$ ), the reason frogs levitate in 16-tesla bores. Paramagnets (aluminum, oxygen, rare-earth salts): atoms carry permanent moments that an applied field aligns against thermal chaos — the magnetic copy of the orientational-polarization story, Langevin machinery and Curie's law included, weak at room temperature ( $+10^{-5}$ to $10^{-3}$ ). Ferromagnets (iron, cobalt, nickel, neodymium compounds): the quantum exchange interaction locks neighboring spins parallel without any applied field, saturating domains at $M_s\sim10^6$ A/m; field and history then decide how the domains add up — hysteresis, permanence, and the entire technology of magnets. Mechanism articles: diamagnetism, paramagnetism, ferromagnetism; supporting cast: bound currents (the general theorem), field of a magnetized object, the auxiliary field $\vec H$ and its Ampère law, susceptibility and permeability, linear media, boundary conditions, Curie physics, domains, hysteresis, magnetic energy, and exchange — each deferral in this article lands on one of them by name.

The definition and the founding theorem

\vec M\;\equiv\;\frac{\mathrm{d}\vec m}{\mathrm{d}\tau} \;=\;\lim_{\Delta\tau\to\text{meso}}\frac{1}{\Delta\tau}\sum_{i\in\Delta\tau}\vec m_i , \qquad [\vec M]=\mathrm{A/m} \qquad (1)

where: the sum runs over the atomic moments in a mesoscopic cell $\Delta\tau$ , and the units deserve a hard look: ampere-meters² per cubic meter = amperes per meter — the same units as surface current density $K$ and as the solenoid's $nI$ . That is not a coincidence; it is the theorem this section proves. (Compare $[\vec P]=\mathrm{C/m^2}$ = surface charge density: each polarization field carries the units of the surface source it masquerades as.)

Derivation AUniform magnetization is a surface current — the counting theorem

The goal is the opener's central claim, made quantitative: a body with uniform $\vec M$ produces exactly the field of a surface current

\vec K_b=\vec M\times\hat n

flowing on its boundary — magnitude $M$ where the surface is parallel to $\vec M$ , zero on the end faces, direction circulating by the right hand around $\vec M$ . The strategy is honest bookkeeping on the cell picture: represent each mesoscopic cell as a small current loop of the correct moment, then count edge currents. This is the uniform case only — deliberately: the general, nonuniform theorem ( $\vec J_b=\nabla\times\vec M$ in the bulk) needs either a checkerboard of unequal loops or an integration by parts on the vector potential, and both belong to the bound-currents article, which owns it. Scoping the counting argument to uniform $\vec M$ keeps it exact.

Step 1. Trade atoms for cells. Partition the body into mesoscopic cells of height $t$ (along $\vec M$ ) and cross-section area $a$ . The cell's moment is $\vec m_{\rm cell}=\vec M\,(at)$ . Represent it — legitimately, since only the moment matters for fields outside the cell (the multipole articles' lesson) — as a single current $I_{\rm cell}$ circulating around the cell's perimeter: $m_{\rm cell}=I_{\rm cell}\,a$ , so

I_{\rm cell}=M\,t .

Every cell in the body carries the same circulation, in the same sense, because $\vec M$ is uniform.

Step 2. Count the interior edges. Any interior wall is shared by two cells whose currents traverse it in opposite directions (each loop circulates the same way, so the shared edge is "up" for one and "down" for the other) with equal magnitude $Mt$ . Net current on every interior wall: zero — exactly, not approximately, because uniformity makes the neighboring loops identical. The bulk of the body carries no net current whatever.

Step 3. Count the rim. Cells at the surface have outer edges with no partner. Each contributes its full $I_{\rm cell}=Mt$ along the rim, and stacking layers of height $t$ gives a current per unit height (per meter along $\vec M$ ) of

K_b=\frac{I_{\rm cell}}{t}=M .

Direction: around the body, in the circulation sense of the loops — perpendicular both to $\vec M$ and to the local outward normal $\hat n$ , which packages as $\vec K_b=\vec M\times\hat n$ . Audit the package: on the cylindrical flank of a bar magnetized along its axis, $|\vec M\times\hat n|=M$ and the current circulates azimuthally ✓; on the flat end faces $\hat n\parallel\vec M$ and $\vec K_b=0$ ✓ — no current pours over the ends, exactly as the loop picture demands. Why the currents are real: this is not an analogy but an identity of sources. The magnetization's atomic currents genuinely flow — bound to their atoms, dissipation-free (Ampère's audacious guess, ratified by quantum mechanics: ground-state orbital and spin currents have no resistance to pay) — and their sum, by Steps 2–3, is a genuine surface current of $M$ amperes per meter. Every field-computing tool of the previous chapter applies to it verbatim. *Verified: units — $[M]=\mathrm{A/m}=[K]$ ✓ (the definition's units foretold the theorem); moment audit — total rim current $M\times(\text{height }L)$ around area $a$ gives moment $MLa=M\times\text{volume}$ ✓, the body's total moment, recovered exactly; matches Griffiths §6.2.1's physical argument, here kept honestly to the uniform case it is exact for.*

Figure

A bar magnet and a solenoid are the same source — flank current nI equals M, same field inside and out

Go deeper · Derivation B · 2 minThe bar magnet is a solenoid — B inside a uniformly magnetized cylinder

Derivation BThe bar magnet is a solenoid — B inside a uniformly magnetized cylinder

The goal is the chapter's first field computation, by pure recall: a long cylinder uniformly magnetized along its axis has

\vec B_{\rm inside}=\mu_0\vec M, \qquad \vec B_{\rm outside}=0 \quad(\text{ideal: }L\to\infty),

with the finite bar's exterior being the dipole field of its total moment. The strategy is the point: after the counting theorem, no new integral exists to do — the problem has become last chapter's solenoid.

Step 1. Replace the magnet by its currents. By the counting theorem, the cylinder (radius $R$ , magnetization $M\hat z$ ) is equivalent to: zero bulk current, plus azimuthal surface current $K_b=M$ on the flank. That source is identical to an ideal solenoid with $nI=M$ .

Step 2. Recall the solenoid, wholesale. The Ampère's-law article proved (exterior-zero argument, straddling rectangle, off-axis uniformity): $\vec B=\mu_0nI\,\hat z$ inside, $0$ outside. Substitute $nI\to M$ :

\vec B_{\rm in}=\mu_0 M\,\hat z=\mu_0\vec M .

Uniform across the bore, parallel to the magnetization — the field reinforces $\vec M$ inside, a sign worth underlining now and contrasting below.

Step 3. The finite bar, honestly. A real magnet has $L/R$ finite: its equivalent solenoid is finite, and the catalog article's ring-stacking formula takes over — on the axis at the center, $B=\mu_0M\cdot\tfrac{L/2}{\sqrt{(L/2)^2+R^2}}$ (the $\cos\theta$ master form), tending to $\mu_0M$ for a long bar and to $\mu_0M\,\tfrac{L}{2R}$ -scale values for thin discs; at the end face, half-ish; outside, at distances $\gg L,R$ , the dipole field of $\vec m=\vec M\times\text{volume}$ with all of the dipole article's anatomy. Field lines inside the magnet run from south to north (along $\vec M$ ), closing outside from north to south — one continuous solenoid circulation, no poles required. Why this settles a childhood question: what is inside a bar magnet? Answer: the inside of a solenoid — a strong, nearly uniform field along the magnetization, $\mu_0M_s\approx2.2$ T for fully saturated iron, silently maintained by atomic currents equivalent to a megampere-per-meter winding. And why does cutting a magnet in half yield two complete magnets? Because you have cut a solenoid into two shorter solenoids — each retains its flank current; no isolated pole was ever there to liberate (the divergence article's hacksaw, now with the current sheet visible). *Verified: recomputed anchors — $M_s({\rm Fe})=8.49\times10^{28}\times2.2\times9.274\times10^{-24}=1.73\times10^6$ A/m, $\mu_0M_s=2.18$ T ✓; N42 disc below ✓; the equivalence is Griffiths Ex. 6.1's cylinder result, obtained here by recall of the owned solenoid theorem rather than fresh integration.*

The electric mirror, assembled once and kept for the whole chapter — including the row where the mirror breaks:

item	polarized matter (E3)	magnetized matter (this chapter)
state variable	$\vec P$ = dipole moment / volume (C/m²)	$\vec M$ = dipole moment / volume (A/m)
masquerades as	bound charges: $\sigma_b=\vec P\cdot\hat n$ (+ $\rho_b$ )	bound currents: $\vec K_b=\vec M\times\hat n$ (+ $\vec J_b$ )
uniform slab/cylinder	charged end faces	current-wrapped flank
interior field	$\vec E$ opposes $\vec P$ (depolarizing, $-\vec P/3\varepsilon_0$ in a sphere)	$\vec B$ aligns with $\vec M$ ( $+\tfrac23\mu_0\vec M$ in a sphere)
why the sign differs	field lines run from + to − through the body	current sheet is a solenoid: flux threads through
deep origin	contact term $-\tfrac{1}{3\varepsilon_0}\vec p\,\delta^3$	contact term $+\tfrac{2\mu_0}{3}\vec m\,\delta^3$ (dipole article)
auxiliary helper	$\vec D=\varepsilon_0\vec E+\vec P$	$\vec H=\vec B/\mu_0-\vec M$ (owned by its article)

Limiting cases

case	result	meaning
uniform $\vec M$ , long cylinder	$\vec B_{\rm in}=\mu_0\vec M$ , out $=0$	the solenoid equivalence, exact
uniform sphere	$\vec B_{\rm in}=\tfrac23\mu_0\vec M$	the magnetized-object article's flagship (mirror of $-\vec P/3\varepsilon_0$ )
thin disc magnetized ⊥ faces	interior $B\to$ small; strong fringing	short solenoid; shape matters (demagnetization, owned downstream)
far from any magnetized body	dipole field of $\vec m=\int\vec M\,\mathrm{d}\tau$	the ladder resumes command
nonuniform $\vec M$	bulk $\vec J_b=\nabla\times\vec M$ appears	the bound-currents article's general theorem
$\vec M$ induced and weak	$\vec M=\chi_m\vec H$ , $\chi_m\sim\pm10^{-5}$	dia/para regime — susceptibility articles
$\vec M$ frozen	permanent magnet: $\vec M$ independent of applied field (until coercion)	ferromagnetism/hysteresis articles

Common trapThe analogy with polarization is load-bearing — and its sign flips inside the material

Transfer intuition from $\vec P$ freely — same averaging, same "moment density masquerading as surface source" logic, same far-field dipole — but do not transfer the interior picture. A uniformly polarized slab has charged faces whose field inside runs against $\vec P$ (depolarizing); a uniformly magnetized bar has a current-wrapped flank whose field inside runs along $\vec M$ ( $\vec B_{\rm in}=\mu_0\vec M$ for the long cylinder — magnetizing, if anything). Blindly copying the electric picture produces a backwards interior field and wrecks every downstream sign (the $\vec H$ article inherits the confusion as the classic $\vec B$ -vs- $\vec H$ -inside-a-magnet muddle). The root is the dipole article's contact-term contrast: charge-pair dipoles carry their field against the moment inside, current-loop dipoles along it — and matter is made of current loops. Second trap: the counting argument here is exact for uniform $\vec M$ only; for nonuniform magnetization, interior cancellation fails and a bulk current $\nabla\times\vec M$ survives — quote the general theorem from the bound-currents article, not from this one.

Worked examples

1 · Saturated iron, by the numbers (confidence builder)

Iron has $8.49\times10^{28}$ atoms/m³, each contributing $2.2\,\mu_B$ when fully aligned. Find $M_s$ , the equivalent surface current, and the interior field of a long saturated bar.

Step 1. The magnetization. $M_s=n\,m_{\rm at}=8.49\times10^{28}\times2.2\times9.274\times10^{-24}$ = $1.73\times10^6$ A/m.

Step 2. The equivalent winding. $K_b=M_s=1.73$ MA per meter of bar length — to replicate it with a copper coil carrying 10 A you would need 173 turns per millimeter, forever, with zero Joule heating. This single number is why permanent magnets beat electromagnets below a certain size: the atomic winding is free.

Step 3. The interior field. $B_{\rm in}=\mu_0M_s=4\pi\times10^{-7}\times1.73\times10^6$ = 2.2 T — the theoretical ceiling for iron-based fields without superconductors, and the reason electromagnet pole pieces "saturate" near 2 tesla: past that, the iron has no more alignment to give (the mechanism articles explain why $2.2\,\mu_B$ and why alignment resists heat).

2 · Reading a neodymium disc's datasheet (exam level)

An N42 disc magnet (diameter 20 mm, thickness 5 mm) lists remanence $B_r=1.3$ T. Predict its surface field at the center of a face, and check against the typical datasheet value ( $\approx0.3$ T).

Step 1. From remanence to magnetization. Remanence is the flux density the material retains at zero applied field in closed-circuit conditions: $B_r=\mu_0M$ , so $M=B_r/\mu_0=1.3/(4\pi\times10^{-7})$ = $1.03\times10^6$ A/m (frozen in — this is the ferromagnetism articles' hysteresis story, used here as an input).

Step 2. Become a solenoid. The disc is a squat solenoid: $K_b=M$ , length $L=5$ mm, radius $R=10$ mm. The catalog article's finite-solenoid axis formula, evaluated at a flat face's center (one end of the solenoid), gives $B=\tfrac{\mu_0M}{2}\cdot\tfrac{L}{\sqrt{L^2+R^2}}$ .

Step 3. Numbers. $B=\tfrac{1.3}{2}\times\tfrac{5}{\sqrt{25+100}}=0.65\times\tfrac{5}{11.18}$ = 0.29 T — matching the $\sim$ 0.3 T datasheets quote for exactly this geometry. A disc magnet's surface field is not $B_r$ (a common misreading): the squat solenoid leaks most of its ideal field, and geometry — the $L/\sqrt{L^2+R^2}$ factor, here 0.45 — decides what survives. At the disc's center (inside) the same formula with the full-length form gives $\mu_0M\cdot\tfrac{L/2}{\sqrt{(L/2)^2+R^2}}\approx0.32$ T.

Step 4. The far field, for completeness. Total moment $m=MV=1.03\times10^6\times1.57\times10^{-6}=1.62$ A·m² — precisely the number the forces article used for the same magnet, now derived from material data rather than asserted: the chapter's variables ( $M$ , from matter) and last chapter's ( $m$ , for mechanics) meet in one object.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.1–6.2 · Jackson, Classical Electrodynamics, 3rd ed., §5.8 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, Ch. 13

Provenance. Definitions and derivations follow Griffiths Introduction to Electrodynamics 4th ed. §6.1–6.3 and Jackson Classical Electrodynamics 3rd ed. §5.8. A verifier should recompute: (1) the bound-current decomposition via integration by parts of the dipole vector potential; (2) the uniformly magnetized cylinder giving $\mathbf{J}_b=0$ , $\mathbf{K}_b=M\hat{\boldsymbol{\phi}}$ , $\mathbf{B}_{\text{in}}=\mu_0 M\hat{\mathbf{z}}$ ; (3) $\nabla\times\mathbf{H}=\mathbf{J}_f$ from $\mathbf{J}_b=\nabla\times\mathbf{M}$ ; (4) $\mu_r=1+\chi_m$ . Constant $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ is exact in the pre-2019 SI and approximate thereafter. (Scope note per the chapter's division of labor: item (1), the general by-parts theorem, is owned by the bound-currents article — this opener proves the uniform case by exact counting; items (3)–(4), the $\vec H$ machinery and susceptibility, are owned by their articles downstream. Worked numbers: $M_s=1.73\times10^6$ A/m, $\mu_0M_s=2.2$ T, N42 disc $M=1.03\times10^6$ A/m and face field 0.29 T vs datasheet ~0.3 T, $m=1.62$ A·m² — recomputed.)

Builds on: Magnetic Dipole and Magnetic Moment · Forces and Torques on Magnetic Dipoles · Ampere's Law · Polarization Density P

Leads to: Magnetic Dipole Moment of Atoms · Diamagnetism · Paramagnetism · Bound Currents · The Auxiliary Field H

Are the bound surface currents real currents, or just a calculational fiction?

Real, with one crucial qualifier. The atomic moments genuinely are circulating currents (orbital) and intrinsic moments (spin) whose magnetic effect is identical to circulation; the counting theorem's sum is a genuine $M$ amperes per meter flowing around the body, and it sources $\vec B$ exactly as any current does — the field of a bar magnet is the field of its surface current, measurably. The qualifier: the current is bound — each contributing electron stays with its atom, so you cannot divert $\vec K_b$ through an ammeter or extract power from it (no charge is transported around the whole rim by any one carrier; the macroscopic circulation is a relay of atomic-scale loops). Real as a source, untappable as a supply — and dissipation-free, because quantum ground states don't pay resistive tolls: Ampère's "molecular currents never decay" made rigorous.

Why doesn't the analogy with P carry the interior-field sign along with everything else?

Because the two masquerades put their sources in different places with different characters. Uniform $\vec P$ masquerades as charges on the end faces: the interior field of a plus-face and a minus-face runs from + to −, i.e. against $\vec P$ — a capacitor turned inward. Uniform $\vec M$ masquerades as current around the flank: the interior field of a circulating sheet threads through the loop — a solenoid, whose flux runs along $\vec M$ . Both follow from one deeper fact (the dipole article's contact terms: $-\tfrac{1}{3\varepsilon_0}$ vs $+\tfrac{2\mu_0}{3}$ ): pack charge-pair dipoles and their interiors fight you; pack current-loop dipoles and their interiors add. Same ladder, opposite hearts — and matter is built from the loop kind.

If every atom has orbiting and spinning electrons, why isn't everything strongly magnetic?

Three cancellations conspire, each with its own article. Within an atom, electrons pair up: filled shells sum orbital and spin moments to zero (quantum bookkeeping, next article), so most atoms have little or no permanent moment — leaving only the weak universal diamagnetic response. Among atoms that do keep a moment, thermal agitation randomizes directions: at room temperature the Boltzmann alignment fraction in ordinary fields is $\sim10^{-3}$ or less (the paramagnetism article's Langevin/Curie mathematics — the same machinery as polar dielectrics). And even the ferromagnetic elements, where exchange defeats temperature locally, hide their saturation in domains whose magnetizations cancel body-wide until a field or history organizes them (the domains article). Strong permanent magnetism requires winning all three battles at once — which is why it is the exception, prized and engineered.

What exactly is remanence, and why is B_r = mu0 M the right dictionary?

Remanence $B_r$ is the flux density remaining in a ferromagnet driven to saturation and then relieved of its applied field, measured in a closed magnetic circuit (a ring, or a bar completed by iron yokes — no free ends). In that geometry there is no demagnetizing leakage: the material sits at $H\approx0$ with its frozen magnetization intact, and $\vec B=\mu_0(\vec H+\vec M)$ (the $\vec H$ article's dictionary, quoted forward) reduces to $B=\mu_0M$ . So $B_r$ is the materials-scientist's way of quoting the frozen $M$ : N42's 1.3 T means $M=1.03\times10^6$ A/m. The worked example's warning follows immediately: an open magnet (a disc on your desk) is not in closed-circuit conditions — its own geometry demagnetizes it and its surface field is the solenoid-leakage fraction of $B_r$ , not $B_r$ itself.

Which atomic currents actually carry the magnetization — orbits or spins?

Material by material, mostly spin. In the ferromagnetic workhorses (Fe, Co, Ni) the orbital motion is largely "quenched" by the crystal environment (the orbital moment is locked to the lattice and averages nearly to zero — a quantum effect the atomic-moments article flags), so the $2.2\,\mu_B$ of iron is dominantly spin alignment, with a small orbital residue. Rare-earth magnets (the neodymium compounds) are the exception: their magnetic 4f electrons sit deep inside the atom, shielded from the lattice, so both spin and orbital moments survive and couple — the origin of their large moments and, equally important, of the strong magnetocrystalline anisotropy (moment locked to crystal axes) that gives them their coercivity. Diamagnetism, by contrast, is purely orbital — induced changes in orbital circulation — since spin magnitudes cannot be altered, only oriented.

Does a magnetized body attract things because of its magnetization directly, or through its field?

Strictly through its field — the broker doctrine of the forces article carries over unchanged. The magnetized body's bound currents source a $\vec B$ (computable by this article's equivalence); that field then acts on whatever is nearby: torques and gradient forces on another magnet's moments, induced magnetization in soft iron followed by gradient attraction ( $\vec F=\nabla(\vec m\cdot\vec B)$ with $\vec m$ itself proportional to the local field — hence the $\nabla B^2$ pull on paperclips), Lorentz forces on free currents. Nothing about matter changes the two-step structure: sources make fields, fields push on sources. What matter adds is that the "sources" now include bound currents and the "pushed" objects respond by re-magnetizing — the self-consistency loops that make the later articles (local field in matter, hysteresis) genuinely rich.

The counting argument assumed neat rows of identical cells. Do real materials' messy atomic arrangements break it?

No — the argument survives because it never really used the grid, only two robust facts: uniform moment density, and pairwise cancellation of whatever circulates on shared internal boundaries. Partition a real material any way you like into mesoscopic cells (each holding thousands of atoms, so $\vec M$ is well-defined per cell); represent each cell by its net moment (legitimate at mesoscopic distances by the multipole ladder — details of internal arrangement are higher-moment corrections that average away); then the cancellation bookkeeping runs identically for irregular tilings. The crystal's actual disorder lives below the mesoscopic scale, exactly where the averaging in $\vec M$ 's definition already buried it. What genuinely breaks the argument is nonuniformity of $\vec M$ itself at the mesoscopic scale — unequal neighboring loops fail to cancel — and that leftover, made precise, is the bulk bound current $\vec J_b=\nabla\times\vec M$ of the next-but-several article.