Bound Currents

A bar magnet lifts a chain of paper clips all day, every day, with no battery, no wires, no power bill. Fields come from currents — the whole magnetostatics chapter built $\mathbf{B}$ out of $\mathbf{J}$ — so where is the current in a piece of cold iron? The answer is one of the prettiest bookkeeping theorems in electromagnetism. A magnetized body is, microscopically, an array of atomic current loops. Deep in the bulk, adjacent loops circulate the same way, so along every shared edge their currents run in opposite directions and cancel. What survives is only the mismatch: wherever the magnetization $\mathbf{M}$ varies from cell to cell, the cancellation is imperfect and a net volume current remains; and at the surface, the outermost loops have no neighbours to cancel against, leaving a ribbon of surface current. Counting the survivors carefully gives two clean formulas — $\mathbf{J}_b = \nabla\times\mathbf{M}$ inside, $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ on the skin — and a licence: delete the matter, keep the currents, and every field the magnet makes is reproduced exactly. This is the magnetic twin of the bound-charge theorem of Bound Charges: Surface and Volume: polarization, a density of charge-pair dipoles, condenses into $\rho_b = -\nabla\cdot\mathbf{P}$ and $\sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}}$ ; magnetization, a density of current-loop dipoles, condenses into a curl and a cross product instead of a divergence and a dot product — same logic, dual geometry. And, exactly as in the electric twin, the word bound must not be read as fictitious. These are genuine electron currents. They make real fields, feel real forces, and — as the Einstein–de Haas experiment showed by making an iron rod visibly twist — carry real angular momentum. This article counts them, audits them, and puts them to work.

Key takeaways

A magnetized body is exactly equivalent, for every field it produces, to two current distributions in vacuum: the bound volume current $\mathbf{J}_b = \nabla\times\mathbf{M}$ and the bound surface current $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ .
The formulas come from counting: adjacent atomic loops cancel along shared edges; only the mismatch (the curl) and the uncancelled rim (the cross product) survive. Magnetization established the uniform special case; this article proves the general theorem.
Bound currents are real — genuine orbital and spin currents of electrons, sourcing $\mathbf{B}$ on exactly the same footing as free currents: $\nabla\times\mathbf{B} = \mu_0(\mathbf{J}_f + \mathbf{J}_b)$ .
Yet they transport nothing: $\nabla\cdot\mathbf{J}_b = 0$ identically, and the total bound current crossing any full slice of the body — bulk plus rim — is exactly zero. Circulation without conveyance.
The bound current threading a loop is the circulation of magnetization around it: $I_b = \oint \mathbf{M}\cdot\mathrm{d}\boldsymbol{\ell}$ .
Electric twin dictionary: $\mathbf{P} \to (\rho_b = -\nabla\cdot\mathbf{P},\ \sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}})$ mirrors $\mathbf{M} \to (\mathbf{J}_b = \nabla\times\mathbf{M},\ \mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}})$ — divergence sources make charge, curl sources make current.
KEY IDEA: replace the magnetized matter by $\mathbf{J}_b = \nabla\times\mathbf{M}$ in the volume and $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ on the surface, and magnetism-in-matter becomes ordinary magnetostatics — with currents that are physically real but transport no charge anywhere.

The core insight

A magnetized body is exactly equivalent, for all field purposes, to a volume current $\mathbf{J}_b = \nabla\times\mathbf{M}$ plus a surface current $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ sitting in vacuum — real, divergence-free currents that circulate without transporting charge.

Where this lives today

Bound currents stopped being a philosophical fine point in 1915, when Einstein and de Haas hung an iron rod from a torsion fibre, reversed its magnetization, and watched it twist: the angular momentum of the amperian currents is mechanically real and conserved. The modern descendants run at femtosecond speed — in ultrafast demagnetization experiments, a laser pulse destroys a film's magnetization in under a picosecond, and the collapsing bound-current system radiates a measurable terahertz burst while the lost angular momentum is tracked into the lattice. Spintronics engineering (spin-transfer torques, THz spintronic emitters) is, at bottom, the technology of manipulating $\mathbf{M}$ fast enough that its bound currents act as sources you can switch. On the computational side, equation (1) is production code. Micromagnetic solvers (OOMMF, mumax3) evolve $\mathbf{M}(\mathbf{r},t)$ over billions of cells and obtain stray fields from the bound-current/bound-charge equivalents of the local magnetization; permanent-magnet machine design — from Halbach arrays that concentrate flux on one side (an engineered $\mathbf{K}_b$ pattern, literally sculpting the surface-current sheet by rotating $\mathbf{M}$ cell to cell) to NdFeB rotors in traction motors — treats every magnet as its equivalent current distribution. When an engineer replaces a magnet segment by "equivalent ampere-turns" in a finite-element model, they are executing this article's theorem verbatim.

Origins · 1 minAndré-Marie Ampère and how this was discovered

André-Marie Ampère

1775–1836 · Lyon, Paris

Ampère heard of Ørsted's twitching compass needle in September 1820 and, within a week, was presenting his own results to the Paris Academy: parallel currents attract, antiparallel repel — magnetism is currents acting on currents. Then came the leap this article rests on: if currents make magnetism, perhaps all magnetism is currents, and a magnetized body is threaded by myriad microscopic circulating currents — courants particulaires — that need no battery. It was a scandalous hypothesis for 1820: nothing in the physics of the day could explain charges circulating forever without dissipation (rightly so — the resolution had to wait a century for quantum mechanics, whose stationary states and spin supply exactly the perpetual, resistance-free loops Ampère postulated). His 1827 Théorie mathématique des phénomènes électro-dynamiques built the force law between current elements with an experimental rigor Maxwell later celebrated, calling Ampère the "Newton of electricity." A self-taught polymath who survived the Revolution's guillotining of his father, Ampère gave physics the word electrodynamics, the current balance, and the picture of matter as an electrically alive medium — the loop array of this article is his hypothesis, drawn to scale.

History

1820

— Ørsted's compass needle moves near a current; Ampère responds with the current-current force law within weeks

1820

— Ampère proposes molecular currents: all magnetism, including iron's, is microscopic circulating current

1827

— Théorie mathématique codifies electrodynamics; the magnet-as-current-array picture matures

1873

— Maxwell's Treatise formalizes magnetization and its equivalent currents

1915

— Einstein and de Haas measure the mechanical twist of reversing magnetization: amperian currents carry real angular momentum

1925

— Electron spin joins orbital motion as the second microscopic source feeding M

The physical picture

Tile a slab of magnetized material with its atomic loops — one per cell of side $a$ , each carrying current $I = Ma$ so that the cell's moment $Ia^2$ matches $M a^3$ (the mesoscopic-average picture of Magnetization). Now stare at any interior edge. The loop on one side runs, say, rightward along that edge; its neighbour runs leftward. If the two loops are equally strong, the edge carries zero net current: the interior of a uniformly magnetized body is electrically silent, and only the outer rim — where the last loops have no partners — carries current. That much Magnetization already established, and it is why a uniformly magnetized cylinder is a solenoid in disguise. The general theorem lives in the mismatch. Let the loops grow stronger with $y$ : the loop at $y + a$ overpowers its neighbour at $y$ , and their shared edge inherits the difference — a net current proportional to $\partial M_z/\partial y$ , running along $x$ . A second, independent mismatch feeds the same $x$ -direction: loops representing $M_y$ that strengthen with height $z$ leave a residue $-\partial M_y/\partial z$ . Add them and you are reading off the $x$ -component of a curl. That is the whole derivation, one component at a time: the bound volume current is the local imbalance of the loop array, and the imbalance of a dipole-loop array is the curl of its density.

Figure

Loop bookkeeping — uniform arrays cancel inside and leave a rim current, graded arrays leave a bulk residue

Why insist the survivors are real currents? Because everything measurable says so. The microscopic charges genuinely move — orbital electrons genuinely circulate, and spin contributes genuinely to the angular-momentum budget, as the Einstein–de Haas twist of a freshly magnetized iron rod demonstrates mechanically. The field they source is not a correction term or an accounting fiction: $\mu_0\mathbf{J}_b$ enters Ampère's law with exactly the same coefficient as the free current in the wires you solder. The only distinction is managerial: free currents answer to your power supply, bound currents answer to the material's magnetization. (That distinction — who controls what — is precisely what the auxiliary field of The Auxiliary Field H is built to exploit, one article ahead.) What bound currents do not do is deliver charge. Each electron stays on its own atom; the "current" is a relay — like stadium fans doing the wave, each person only stands and sits, yet the wave sweeps around the stands. The theorem below makes the honesty quantitative: $\nabla\cdot\mathbf{J}_b = 0$ automatically, and across any full slice of the body the bulk current and the rim current cancel to the ampere. Circulation everywhere, conveyance nowhere.

Figure

The graded cylinder of worked example 1 — bulk rings one way, rim ribbon the other, net transport zero

The bound-current laws

The volume law first. Wherever the magnetization varies, its curl is a current density:

\mathbf{J}_b = \nabla\times\mathbf{M} \qquad (1)

where: $\mathbf{M}$ — magnetization, dipole moment per unit volume ( $\mathrm{A/m}$ ); $\mathbf{J}_b$ — bound volume current density ( $\mathrm{A/m^2}$ ). Read it as a mismatch meter: a uniform $\mathbf{M}$ , however enormous, has $\mathbf{J}_b = 0$ (iron near saturation carries $M \approx 1.7\times10^{6}\,\mathrm{A/m}$ , the number recalled from Magnetization, and still no bulk current) — only gradients of loop strength leave bulk residue. Note also what the curl structure buys for free: $\nabla\cdot\mathbf{J}_b = \nabla\cdot(\nabla\times\mathbf{M}) = 0$ identically, so the bound current is automatically steady — no charge ever piles up, exactly as magnetostatics (the Continuity Equation article's steady regime) requires. At the material's edge the magnetization does not taper politely — it drops to zero across an atomic layer, and the mismatch concentrates into a sheet:

\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}} \qquad (2)

where: $\mathbf{K}_b$ — bound surface current density ( $\mathrm{A/m}$ , current per unit length of rim); $\hat{\mathbf{n}}$ — outward unit normal. The cross product encodes two geometric facts at once: only the component of $\mathbf{M}$ parallel to the surface drives a surface current (a face with $\mathbf{M}\parallel\hat{\mathbf{n}}$ — the flat pole of a bar magnet — is silent), and the current runs perpendicular to both $\mathbf{M}$ and $\hat{\mathbf{n}}$ , wrapping the body like solenoid windings. Formally, $\mathbf{K}_b$ is just $\mathbf{J}_b$ 's delta-function avatar: the curl of an $\mathbf{M}$ that steps to zero at a boundary. With the two ledgers written, Ampère's law for $\mathbf{B}$ takes its honest in-matter form:

\nabla\times\mathbf{B} = \mu_0\left(\mathbf{J}_f + \mathbf{J}_b\right) = \mu_0\mathbf{J}_f + \mu_0\,\nabla\times\mathbf{M} \qquad (3)

where: $\mathbf{J}_f$ — free (conduction) current density. Equal footing is the point: $\mathbf{B}$ does not know or care which currents you paid for. The practical sting is the circularity — $\mathbf{J}_b$ depends on $\mathbf{M}$ , which typically depends on $\mathbf{B}$ , which is what you are solving for. Moving $\nabla\times\mathbf{M}$ to the other side and naming the combination is the business of The Auxiliary Field H; extracting the field a given $\mathbf{M}$ produces is the business of Field of a Magnetized Object, which also supplies an independent derivation of (1)–(2) from the vector potential — the by-parts route, the twin of the potential-identity route in Field of a Polarized Object. Two derivations, two articles, one theorem. One more identity earns its keep daily. Integrate (1) over a surface $S$ and apply Stokes' theorem (the E0 machinery):

I_b = \int_S \mathbf{J}_b\cdot\mathrm{d}\mathbf{a} = \oint_{\partial S} \mathbf{M}\cdot\mathrm{d}\boldsymbol{\ell} \qquad (4)

where: $I_b$ — net bound current threading the loop $\partial S$ . The bound current through a loop is the circulation of magnetization around it — no differentiation required. If your loop stays entirely outside the material (or entirely in a region of uniform $\mathbf{M}$ that it does not link), the circulation vanishes and so does the threaded current; worked example 2 uses (4) as a two-line shortcut past an area integral.

electric twin (Bound Charges: Surface and Volume)	magnetic original (this article)
dipole = charge pair, described by $\mathbf{P}$	dipole = current loop, described by $\mathbf{M}$
bulk residue: $\rho_b = -\nabla\cdot\mathbf{P}$	bulk residue: $\mathbf{J}_b = \nabla\times\mathbf{M}$
surface residue: $\sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}}$	surface residue: $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$
zero-sum audit: total bound charge $= 0$	zero-transport audit: net current through any slice $= 0$
sourced law: $\nabla\cdot\mathbf{E} = (\rho_f+\rho_b)/\varepsilon_0$	sourced law: $\nabla\times\mathbf{B} = \mu_0(\mathbf{J}_f+\mathbf{J}_b)$
bookkeeping field: $\mathbf{D}$ (The Electric Displacement D)	bookkeeping field: $\mathbf{H}$ (The Auxiliary Field H, next)

Derivation AThe counting theorem — from loop mismatch to the curl

The goal is the general laws (1) and (2) by honest counting, extending the uniform-magnetization argument of Magnetization (which established the rim current $K_b = M$ for uniform slabs and cylinders) to arbitrary $\mathbf{M}(\mathbf{r})$ . The strategy: model the material as cubic cells of side $a$ , each carrying the loop current that reproduces its cell's moment; then tally the net current along one Cartesian direction, from both mismatch mechanisms that feed it, and recognize the curl component. Watch the orientations — each sign below is fixed by the right-hand rule, and the final answer must assemble into an antisymmetric derivative combination or the identification with a curl would be illegitimate.

Step 1. Calibrate the loop current. A cell of volume $a^3$ magnetized along $\hat{\mathbf{z}}$ carries moment $m_z = M_z a^3$ ; modelled as a square loop of area $a^2$ , its current is

I = \frac{m_z}{a^2} = M_z\,a.

Why a single loop per cell is enough: the mesoscopic average of Magnetization has already smoothed the atomic granularity — any microstructure with the same moment density produces the same macroscopic field, so we may choose the tidiest one.

Step 2. First mismatch: $M_z$ varying with $y$ . Two cells sharing the face at height $y$ : viewed from $+z$ , each loop circulates counterclockwise, so the cell at smaller $y$ sends current $-\hat{\mathbf{x}}$ along the top of its loop while the cell at larger $y$ sends $+\hat{\mathbf{x}}$ along the bottom of its loop — opposite directions on the shared edge. The residue along $+\hat{\mathbf{x}}$ is

\Delta I_x = I(y{+}a) - I(y) = a\left[M_z(y{+}a) - M_z(y)\right] \approx a^2\,\frac{\partial M_z}{\partial y}.

Step 3. Second mismatch: $M_y$ varying with $z$ . Cells magnetized along $\hat{\mathbf{y}}$ circulate in the $x$ – $z$ plane; the right-hand rule puts their $+\hat{\mathbf{x}}$ leg on the top edge ( $z$ large). Across a horizontal face, the lower cell contributes $+\hat{\mathbf{x}}$ current $M_y(z)\,a$ and the upper cell $-\hat{\mathbf{x}}$ current $M_y(z{+}a)\,a$ , leaving

\Delta I_x' = a\left[M_y(z) - M_y(z{+}a)\right] \approx -a^2\,\frac{\partial M_y}{\partial z}.

Why both terms and only these: an $x$ -directed current can be fed only by loops whose planes contain $\hat{\mathbf{x}}$ — the $M_z$ loops (in $xy$ planes) and the $M_y$ loops (in $xz$ planes). $M_x$ loops lie in $yz$ planes and contribute nothing along $\hat{\mathbf{x}}$ .

Step 4. Convert to a density and recognize the curl. Each residue $\Delta I_x$ is a line current threading a cross-section $a^2$ perpendicular to $\hat{\mathbf{x}}$ ; the smeared density is

(J_b)_x = \frac{\Delta I_x + \Delta I_x'}{a^2} = \frac{\partial M_z}{\partial y} - \frac{\partial M_y}{\partial z} = (\nabla\times\mathbf{M})_x,

and the antisymmetric combination is exactly the $x$ -component of the curl. Cyclic relabeling ( $x\to y\to z$ ) delivers the other two components: $\mathbf{J}_b = \nabla\times\mathbf{M}$ .

Step 5. The surface leftover. At the boundary the outermost loop's outer edge has no partner: it keeps its full current $I = Ma$ , spread along one cell width, giving current per unit length $K_b = Ma/a = M_\parallel$ — the tangential magnetization. Direction: the uncancelled leg runs perpendicular to both $\mathbf{M}$ and the outward normal, and checking any face against Step 2's orientation gives precisely

\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}.

On a face with $\mathbf{M}\parallel\hat{\mathbf{n}}$ the cross product vanishes — the loops there lie flat in the surface and their edges are interior edges, already cancelled. Verified: reproduces Griffiths §6.2.2 including both gradient terms of the x-component; uniform M limit recovers the rim-only result of Magnetization; the antisymmetric structure guarantees div J_b = 0, checked independently in the next derivation; the by-parts route of Field of a Magnetized Object yields the same pair from the vector potential.

Go deeper · Derivation B · 2 minThe zero-transport theorem — real currents that deliver no charge

Derivation BThe zero-transport theorem — real currents that deliver no charge

The goal is the audit that makes "real but bound" precise: for any plane slicing a magnetized body, the total bound current crossing it — volume current through the cross-section plus surface current crossing the rim — is exactly zero. The strategy is two applications of vector identities: Stokes' theorem for the bulk piece, and the BAC-CAB dot-product identity for the rim piece. This theorem is why a magnet never charges up one end, and it is the transport-ledger mirror of the zero-sum theorem for bound charge in the electric twin.

Step 1. Set the stage. Slice the body with a plane; let $S$ be the material cross-section and $C = \partial S$ its boundary curve, lying in the body's surface. Take $\hat{\mathbf{t}}$ as the unit tangent along $C$ (oriented with the plane's normal by the right-hand rule) and $\hat{\mathbf{n}}$ the body's outward surface normal.

Step 2. Bulk crossing. By Stokes' theorem — equation (4) —

I_{\text{vol}} = \int_S (\nabla\times\mathbf{M})\cdot\mathrm{d}\mathbf{a} = \oint_C \mathbf{M}\cdot\hat{\mathbf{t}}\;\mathrm{d}\ell.

Why the boundary circulation appears: $\mathbf{M}$ vanishes outside the body, so the only circulation that survives is along the rim where the slicing plane exits the material.

Step 3. Rim crossing. The surface current crossing $C$ is $\mathbf{K}_b$ resolved along $\hat{\mathbf{u}} = \hat{\mathbf{n}}\times\hat{\mathbf{t}}$ , the direction within the body's surface that steps across the curve $C$ :

I_{\text{surf}} = \oint_C \mathbf{K}_b\cdot\hat{\mathbf{u}}\;\mathrm{d}\ell = \oint_C (\mathbf{M}\times\hat{\mathbf{n}})\cdot(\hat{\mathbf{n}}\times\hat{\mathbf{t}})\;\mathrm{d}\ell.

Step 4. Collapse the double cross product with the identity $(\mathbf{A}\times\mathbf{B})\cdot(\mathbf{C}\times\mathbf{D}) = (\mathbf{A}\cdot\mathbf{C})(\mathbf{B}\cdot\mathbf{D}) - (\mathbf{A}\cdot\mathbf{D})(\mathbf{B}\cdot\mathbf{C})$ (the E0 product-rule kit):

(\mathbf{M}\times\hat{\mathbf{n}})\cdot(\hat{\mathbf{n}}\times\hat{\mathbf{t}}) = (\mathbf{M}\cdot\hat{\mathbf{n}})(\hat{\mathbf{n}}\cdot\hat{\mathbf{t}}) - (\mathbf{M}\cdot\hat{\mathbf{t}})(\hat{\mathbf{n}}\cdot\hat{\mathbf{n}}) = -\,\mathbf{M}\cdot\hat{\mathbf{t}},

since $\hat{\mathbf{n}}\perp\hat{\mathbf{t}}$ ( $C$ lies in the surface) and $\hat{\mathbf{n}}$ is a unit vector. Therefore $I_{\text{surf}} = -\oint_C \mathbf{M}\cdot\hat{\mathbf{t}}\,\mathrm{d}\ell = -I_{\text{vol}}$ , and

I_{\text{vol}} + I_{\text{surf}} = 0.

Step 5. Local version. The same honesty holds pointwise: $\nabla\cdot\mathbf{J}_b = \nabla\cdot(\nabla\times\mathbf{M}) = 0$ everywhere in the bulk, and at the surface the arriving volume current feeds the sheet without leakage. Bound currents form closed circulation cells: every ampere that crosses a slice one way comes back another. Why this had to work: each microscopic loop individually carries zero net charge across any plane — a loop pierces a plane either not at all or twice, in opposite directions. A sum of loops inherits the property; the theorem is the macroscopic shadow of that microscopic triviality. Verified: worked example 1 closes the audit numerically for a graded cylinder (−200 A bulk, +200 A rim); the identity chain matches Griffiths §6.2 and Zangwill §13.4; the electric-twin analogue (total bound charge = 0) is the corresponding audit in Bound Charges: Surface and Volume.

Limiting cases

Case	Result	Meaning
uniform $\mathbf{M}$	$\mathbf{J}_b = 0$ , $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ only	the Magnetization special case: bar magnet = solenoid in disguise
$\mathbf{M}\parallel\hat{\mathbf{n}}$ on a face	$\mathbf{K}_b = 0$ there	flat poles of an axially magnetized cylinder carry no current
$\mathbf{M}$ with pure gradient along itself, e.g. $M_z(z)\hat{\mathbf{z}}$	$\nabla\times\mathbf{M} = 0$	not every nonuniformity makes bulk current — only curl-type variation
permanent magnet, no wires	$\nabla\times\mathbf{B} = \mu_0\mathbf{J}_b$	the entire field is bound-current sourced
any closed loop	$I_b = \oint\mathbf{M}\cdot\mathrm{d}\boldsymbol{\ell}$	threaded bound current = circulation of $\mathbf{M}$
any full slice of the body	$I_{\text{vol}} + I_{\text{surf}} = 0$	real currents, zero net transport
step to vacuum	$\mathbf{K}_b$ = delta-sheet of $\nabla\times\mathbf{M}$	surface law is the volume law at a discontinuity

Common trapJ_b = 0 does not mean no currents — and bound does not mean fictitious

Two opposite mistakes, both common. First: computing $\nabla\times\mathbf{M} = 0$ for a uniformly magnetized object and concluding it produces no field — forgetting that the surface current $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ carries the whole story (for a 1 cm iron-like sphere it amounts to $10^4$ ampere-turns; the field it makes is the flagship of Field of a Magnetized Object). Second: treating bound currents as bookkeeping fictions to be discarded once $\mathbf{H}$ arrives — they are genuine electron currents that source $\mathbf{B}$ with the same $\mu_0$ as any wire, and experiments from Einstein–de Haas to femtosecond demagnetization measure their reality directly. Keep both ledgers honest: bound currents are real (they make fields) and bound (they transport nothing — the zero-transport theorem). What they are not is adjustable: you dial free currents; the material dials $\mathbf{J}_b$ .

Worked examples

1 · The graded cylinder — a bulk current appears, the rim pays it back (confidence builder)

A long cylinder of radius $R = 1.0\,\mathrm{cm}$ carries an axial magnetization that grows linearly with distance from the axis: $\mathbf{M} = ks\,\hat{\mathbf{z}}$ , with the surface value $M(R) = 2.0\times10^{4}\,\mathrm{A/m}$ , so $k = M(R)/R = 2.0\times10^{6}\,\mathrm{A/m^2}$ . Find both bound currents and audit the transport.

Step 1. Volume current. In cylindrical coordinates, for $\mathbf{M} = M_z(s)\,\hat{\mathbf{z}}$ the only surviving curl component is $(\nabla\times\mathbf{M})_\phi = -\partial M_z/\partial s$ :

\mathbf{J}_b = -k\,\hat{\boldsymbol{\phi}} = -2.0\times10^{6}\,\mathrm{A/m^2}\;\hat{\boldsymbol{\phi}},

a uniform azimuthal current density — rings of current circulating clockwise (viewed from $+z$ ) throughout the bulk. Loop-array reading: loops further out are stronger, so every shared edge keeps the difference, and all the differences circulate the same way.

Step 2. Surface current. At $s = R$ , $\hat{\mathbf{n}} = \hat{\mathbf{s}}$ :

\mathbf{K}_b = \mathbf{M}(R)\times\hat{\mathbf{s}} = kR\,(\hat{\mathbf{z}}\times\hat{\mathbf{s}}) = +2.0\times10^{4}\,\mathrm{A/m}\;\hat{\boldsymbol{\phi}}

— counterclockwise, opposing the bulk's sense.

Step 3. The audit. Slice the cylinder with a half-plane $\phi = \text{const}$ ; count azimuthal current crossing it, per unit axial length. Bulk: $\int_0^R (-k)\,\mathrm{d}s = -kR = -2.0\times10^{4}\,\mathrm{A/m}$ . Rim: $+kR = +2.0\times10^{4}\,\mathrm{A/m}$ . Net: zero ✓ — over any 1 cm of cylinder height, $200\,\mathrm{A}$ circulate one way through the bulk and exactly $200\,\mathrm{A}$ return along the skin. Real amperes, zero conveyance: the zero-transport theorem in the flesh.

2 · Azimuthal magnetization — Stokes beats the area integral (exam level)

A cylinder of radius $R = 2.0\,\mathrm{cm}$ carries the swirling magnetization $\mathbf{M} = ks\,\hat{\boldsymbol{\phi}}$ with rim value $M(R) = 3.0\times10^{4}\,\mathrm{A/m}$ ( $k = 1.5\times10^{6}\,\mathrm{A/m^2}$ ). Find the bound currents, the bound current threading a coaxial circle of radius $s$ , and audit the total.

Step 1. Volume current. For $\mathbf{M} = M_\phi(s)\,\hat{\boldsymbol{\phi}}$ , the curl gives $(\nabla\times\mathbf{M})_z = \frac{1}{s}\frac{\partial}{\partial s}(s M_\phi) = \frac{1}{s}\frac{\partial}{\partial s}(k s^2) = 2k$ :

\mathbf{J}_b = 2k\,\hat{\mathbf{z}} = 3.0\times10^{6}\,\mathrm{A/m^2}\;\hat{\mathbf{z}}

— a uniform axial current filling the bulk, conjured entirely by swirl.

Step 2. Threaded current the fast way. For a coaxial circle of radius $s$ , equation (4) gives $I_b(s) = \oint\mathbf{M}\cdot\mathrm{d}\boldsymbol{\ell} = M_\phi(s)\cdot 2\pi s = 2\pi k s^2$ . Cross-check by area: $\int \mathbf{J}_b\cdot\mathrm{d}\mathbf{a} = 2k\cdot\pi s^2$ — identical ✓. At $s = R/2 = 1.0\,\mathrm{cm}$ : $I_b = 2\pi(1.5\times10^{6})(10^{-4})/...$ carefully: $2\pi k s^2 = 2\pi \times 1.5\times10^{6} \times (0.010)^2 = 9.4\times10^{2}\,\mathrm{A}$ — nearly a kiloampere of bound current inside a 1 cm circle.

Step 3. Surface current. $\mathbf{K}_b = \mathbf{M}(R)\times\hat{\mathbf{s}} = kR\,(\hat{\boldsymbol{\phi}}\times\hat{\mathbf{s}}) = -kR\,\hat{\mathbf{z}} = -3.0\times10^{4}\,\mathrm{A/m}\;\hat{\mathbf{z}}$ — an axial return sheet.

Step 4. Total-transport audit across the full cross-section. Bulk: $2k\cdot\pi R^2 = 3.0\times10^{6}\times\pi\times(0.02)^2 = +3.77\times10^{3}\,\mathrm{A}$ . Rim: $-kR\cdot 2\pi R = -1.5\times10^{6}\times0.02\times2\pi\times0.02 = -3.77\times10^{3}\,\mathrm{A}$ . Sum: zero ✓. Nearly four kiloamperes flow up the bulk and return down the skin — and the magnetized cylinder ships not one coulomb end to end.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.2 · Jackson, Classical Electrodynamics, 3rd ed., §5.8 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, §13.4.

Provenance. Equations and derivations cross-checked against Griffiths Introduction to Electrodynamics 4th ed. §6.2 (definitions of $\mathbf{J}_b$ , $\mathbf{K}_b$ , vector-potential integration by parts, magnetized-sphere Example 6.1) and Jackson Classical Electrodynamics 3rd ed. §5.8. A verifier should recompute: (1) the integration by parts of the dipole vector potential yielding the $\mathbf{J}_b$ volume term and $\mathbf{K}_b$ surface term; (2) $\mathbf{J}_b=0$ for uniform $\mathbf{M}$ and $\mathbf{K}_b=M\sin\theta\,\hat{\boldsymbol{\phi}}$ on a sphere; (3) the interior field $\frac{2}{3}\mu_0\mathbf{M}$ and dipole moment $\frac{4}{3}\pi R^3\mathbf{M}$ via the rotating charged-shell analogy; (4) $\nabla\cdot(\nabla\times\mathbf{M})=0$ . All numbers ( $2/3$ , $4/3\pi R^3$ ) are standard textbook values. Addendum for this edition: the vector-potential by-parts route and the magnetized-sphere solution are developed in the companion article Field of a Magnetized Object per the chapter's division of labor; the counting derivation here carries both gradient terms of Griffiths Fig. 6.16. Worked numbers recomputed: graded cylinder $k = 2.0\times10^{6}\,\mathrm{A/m^2}$ gives $J_b = 2.0\times10^{6}\,\mathrm{A/m^2}$ , $K_b = 2.0\times10^{4}\,\mathrm{A/m}$ , audit $\mp200\,\mathrm{A}$ per cm; azimuthal case gives $J_b = 3.0\times10^{6}\,\mathrm{A/m^2}$ , $I_b(R/2) = 9.4\times10^{2}\,\mathrm{A}$ , full-section audit $\pm3.77\times10^{3}\,\mathrm{A}$ .

Builds on: Magnetization · Magnetic Dipole Moment of Atoms · Bound Charges: Surface and Volume · Continuity Equation

Leads to: Field of a Magnetized Object · The Auxiliary Field H · Ampère's Law for H · Magnetic Boundary Conditions

If bound currents are real electron currents, why is there no Joule heating inside a magnet?

Because these currents flow in quantum stationary states, not through a resistive medium. An orbital electron in an atomic eigenstate circulates indefinitely without radiating or dissipating — that is precisely what "stationary state" means — and spin is intrinsic angular momentum with no classical churning at all. Resistance arises when carriers scatter and randomize drift momentum acquired from a field; bound currents acquire nothing from a battery and have no drift to randomize. This was exactly the scandal in Ampère's original hypothesis — classical physics in 1820 (and in 1900) had no way to keep his molecular currents alive forever. Quantum mechanics is the reason your fridge magnet is not warm.

How can a current be real if no charge is ever transported anywhere?

Distinguish current density from net conveyance. Current density measures charge crossing a small surface per time — and charge genuinely crosses: in worked example 1, two hundred amperes pass through each square centimetre strip of the half-plane. But every electron stays on its own atom; each microscopic loop pierces any plane twice, in opposite directions, so the sum over a full slice cancels (the zero-transport theorem). The stadium-wave analogy is exact: every fan moves, the wave circulates, nobody changes seats. Fields do not care about the distinction — Ampère's law responds to local current density, which is why these transportless currents still make the full $\mathbf{B}$ of a magnet.

Is J_b the same thing as the polarization current in a dielectric?

No — different beasts, easy to conflate. The polarization current $\mathbf{J}_p = \partial\mathbf{P}/\partial t$ (from Polarization Density P) exists only while the polarization is changing: charges genuinely shift position as dipoles stretch. The bound current $\mathbf{J}_b = \nabla\times\mathbf{M}$ exists in static equilibrium — it is the standing circulation of the loop array, flowing steadily even when nothing changes in time. In full electrodynamics both appear, alongside a third cousin ( $\partial\mathbf{P}/\partial t$ also picks up a curl-of- $\mathbf{M}$ companion in the total-current bookkeeping), but at the magnetostatics register: $\mathbf{J}_p$ needs time dependence, $\mathbf{J}_b$ needs spatial nonuniformity, and neither needs a battery.

Why does magnetization condense into a curl while polarization condensed into a divergence?

Because the two dipole types have dual geometry. An electric dipole is a charge pair — its business end is where charge accumulates, and stacking pairs head-to-tail leaves uncancelled charge wherever the chain density varies along its own direction: that is a divergence. A magnetic dipole is a current loop — its business is circulation, and tiling loops leaves uncancelled current wherever loop strength varies across the loop planes: that is a curl. The dictionary is exact and worth memorizing as a pair: $\rho_b = -\nabla\cdot\mathbf{P}$ , $\sigma_b = \mathbf{P}\cdot\hat{\mathbf{n}}$ versus $\mathbf{J}_b = \nabla\times\mathbf{M}$ , $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ . Divergence-type sources make charge; curl-type sources make current — a preview of the Helmholtz-theorem split that organizes all of field theory.

Does electron spin really count as a current — nothing is actually going around, is it?

At the macroscopic level of this chapter: yes, it counts, with no correction factor. Spin magnetic moments enter the magnetization $\mathbf{M}$ on identical terms with orbital ones, and the fields they source are correctly given by $\mathbf{J}_b = \nabla\times\mathbf{M}$ — experiment confirms this to exquisite precision (neutron scattering off magnetized crystals, for instance, sees the full spin contribution). Microscopically you are right to be suspicious: spin is intrinsic angular momentum, not a classical whirl of charge, and modelling the electron as a spinning ball fails quantitatively (that honesty belongs to Magnetic Dipole Moment of Atoms). The magic of the mesoscopic average is that once moments are smeared into $\mathbf{M}$ , their internal anatomy is invisible — only the moment density matters.

What happens to K_b at an edge or corner where the normal vector is not defined?

The formula $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ applies where the surface is smooth; at edges the two adjacent sheets simply meet, and the idealization concentrates current there the way any sharp-corner idealization does. Physically nothing diverges: a real magnet's edges are rounded at some scale, $\hat{\mathbf{n}}$ turns rapidly but continuously, and the sheet current follows it around the bend. When you compute with the idealized geometry, treat each face separately — the audit theorems (zero transport, field matching) hold in the limit. If an exam problem hands you a uniformly magnetized cube, give each face its $\mathbf{M}\times\hat{\mathbf{n}}$ and let the edges be the places where the four side-sheets join into one circulating ribbon.

Which formula wins when M is nonuniform right up to the surface — the curl or the cross product?

Both, in their own territories, and they never double-count. $\mathbf{J}_b = \nabla\times\mathbf{M}$ uses the smooth variation of $\mathbf{M}$ inside the material; $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ uses the value of $\mathbf{M}$ right at the surface, and represents the delta-function part of the curl coming from the step down to vacuum. In worked example 1 the two coexist: the bulk carries $-k\hat{\boldsymbol{\phi}}$ from the gradient, the skin carries $+kR\hat{\boldsymbol{\phi}}$ from the step, and the audit needs both. A useful self-check on any problem: compute the total current through a full slice using both contributions — if it is not zero, a term has been dropped or a sign botched.