The Auxiliary Field H

The bound-current articles left magnetostatics with the same bookkeeping complaint that once afflicted dielectrics. Ampère's law for $\mathbf{B}$ is sourced by all current, $\mathbf{J}_f + \mathbf{J}_b$ — but $\mathbf{J}_b = \nabla\times\mathbf{M}$ depends on the magnetization, which depends on the field, which depends on the bound current you were trying to find. Circular. The fix is the same accounting maneuver that built the electric displacement $\mathbf{D}$ , executed on the other side of the vector-calculus ledger: move the material's response into the field's definition. The result, $\mathbf{H} \equiv \mathbf{B}/\mu_0 - \mathbf{M}$ , obeys $\nabla\times\mathbf{H} = \mathbf{J}_f$ — an Ampère's law that sees only the current you control, the current in windings and wires and power supplies. Readers of The Electric Displacement D will recognize what must come next, because this article inherits that article's honesty doctrine wholesale. $\mathbf{H}$ is not "the magnetic field of the free currents," and the parallel between $(\mathbf{B}, \mathbf{J}_{\rm total})$ and $(\mathbf{H}, \mathbf{J}_f)$ breaks at the second glance — with the fine print swapped between the two vector-calculus slots. $\mathbf{D}$ 's clean law was a divergence and its hidden vice was a curl; $\mathbf{H}$ 's clean law is a curl and its hidden vice is a divergence: $\nabla\cdot\mathbf{H} = -\nabla\cdot\mathbf{M}$ , which vanishes only when the material geometry is kind. Consequently there is no Biot–Savart law for $\mathbf{H}$ , and no guarantee that $\mathbf{J}_f = 0$ means $\mathbf{H} = 0$ — the bar magnet, with not a single free current anywhere, carries a thriving $\mathbf{H}$ -field that points backwards through its own body. So the honest contract, verbatim from the dielectric chapter with the symbols traded: $\mathbf{H}$ is a supremely useful auxiliary — the circulation bookkeeper of free current — and the physical field, the one that pushes charges and has no sources anywhere, remains $\mathbf{B}$ . This article constructs the bookkeeper, prosecutes the bar-magnet counterexample where naive trust collapses, and builds the pole fiction that makes $\mathbf{H}$ calculable when free currents are absent.

Key takeaways

Definition: $\mathbf{H} \equiv \mathbf{B}/\mu_0 - \mathbf{M}$ — engineered so the bound current cancels out of Ampère's law: $\nabla\times\mathbf{H} = \mathbf{J}_f$ , with only free current as source.
The twin construction: $\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}$ cleaned up a divergence; $\mathbf{H}$ cleans up a curl. The fine print swaps too: $\mathbf{D}$ 's catch was its curl, $\mathbf{H}$ 's catch is its divergence, $\nabla\cdot\mathbf{H} = -\nabla\cdot\mathbf{M} \neq 0$ in general.
Because a field needs both its curl and its divergence (Helmholtz), $\mathbf{H}$ is not determined by $\mathbf{J}_f$ alone: there is no Biot–Savart law for $\mathbf{H}$ , and $\mathbf{J}_f = 0$ never by itself means $\mathbf{H} = 0$ .
The counterexample to memorize: a uniformly magnetized sphere has $\mathbf{J}_f = 0$ everywhere yet $\mathbf{H}_{\rm in} = -\mathbf{M}/3$ — antiparallel to $\mathbf{B}_{\rm in} = +2\mu_0\mathbf{M}/3$ . The demagnetizing field.
Where $\mathbf{J}_f = 0$ , $\mathbf{H}$ flows from a scalar potential sourced by fictitious magnetic charge $\rho_m = -\nabla\cdot\mathbf{M}$ , $\sigma_m = \mathbf{M}\cdot\hat{\mathbf{n}}$ : the pole fiction — $\mathbf{H}$ of a bar magnet is the $\mathbf{E}$ of two charged disks, exactly.
Units housekeeping: $\mathbf{H}$ and $\mathbf{M}$ share $\mathrm{A/m}$ ; $\mathbf{B}$ is in tesla. The integral law $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_{f,\rm enc}$ and its worked catalog are deferred by name to Ampère's Law for H.
KEY IDEA: $\mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}$ absorbs bound current so Ampère's law reads free current only — but $\mathbf{H}$ inherits no divergence-free guarantee, so it is a circulation bookkeeper with pole-like sources of its own, not the physical field: that role stays with $\mathbf{B}$ .

The core insight

The auxiliary field $\mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}$ satisfies $\nabla\times\mathbf{H} = \mathbf{J}_f$ , letting Ampère's law run on free current alone; but $\nabla\cdot\mathbf{H} = -\nabla\cdot\mathbf{M}$ is generally nonzero, so $\mathbf{H}$ has no Biot–Savart law and is not determined by free current by itself — it is an auxiliary bookkeeping field, while $\mathbf{B}$ remains the physical one.

Where this lives today

Open any permanent-magnet datasheet and the doctrine of this article is the reading list: remanence $B_r$ in tesla, coercivity $H_c$ in kA/m — one axis of the hysteresis loop for each field, because inside a magnet $B$ and $H$ are genuinely independent handles ( $B = \mu_0(H+M)$ with $M$ frozen in). A grade-N42 block quotes $B_r = 1.30\,\mathrm{T}$ : at that point $H = 0$ , so the definition hands you the frozen magnetization directly, $M_r = B_r/\mu_0 = 1.03\times10^{6}\,\mathrm{A/m}$ — the datasheet was quoting $\mathbf{M}$ in disguise. Every MRI shim calculation and electric-machine design loop runs on exactly this $B$ – $H$ – $M$ triangle. The pole fiction, meanwhile, has been industrialized. Micromagnetic solvers (OOMMF, MuMax3) spend most of their compute evaluating the demagnetizing field — our $\mathbf{H}$ from $\rho_m$ and $\sigma_m$ — over millions of cells, because that self-field sculpts the domain patterns in hard drives, MRAM cells, and skyrmion devices. The bookkeeping field of 1873 is the inner loop of 2026's magnetic-memory design codes.

Origins · 1 minJames Clerk Maxwell and how this was discovered

James Clerk Maxwell

1831–1879 · Aberdeen, London, Cambridge

Maxwell is the reason this chapter and the dielectric chapter end in the same doctrine — he built both bookkeepers. His Treatise on Electricity and Magnetism (1873) carries two electric vectors ( $\mathbf{E}$ the force intensity, $\mathbf{D}$ the flux quantity) and, in exact parallel, two magnetic vectors: $\mathbf{H}$ , which he called the magnetic force, and $\mathbf{B}$ , the magnetic induction — the field of Faraday's lines, whose tubes never end. Working before the electron, Maxwell leaned on William Thomson's 1850s analysis of magnetized media, where the two vectors first split: inside matter, the field computed from poles and the field measured by flux genuinely differ, and only the pole-blind $\mathbf{B}$ obeys a universal no-source law. Nineteenth-century practice treated $\mathbf{H}$ as primary; it took the twentieth century's electrons, Lorentz forces, and relativity to certify $\mathbf{B}$ as the physical field and demote $\mathbf{H}$ to magnificent bookkeeping. The Treatise's pairing survives in every modern text — two chapters, two auxiliaries, one honesty doctrine.

History

1850

— William Thomson (Kelvin) distinguishes two magnetic vectors in magnetized media — the B/H split is born

1873

— Maxwell's Treatise codifies magnetic force H and magnetic induction B, alongside E and D

1884

— Heaviside recasts Maxwell into the four modern equations; H and D carry the free sources

1907

— Weiss's molecular field λM: an internal field proportional to M, but aligned with M and non-magnetic in origin (exchange, 1928) — not the demagnetizing field

1945

— Osborn and Stoner tabulate demagnetizing factors for the general ellipsoid — the shape trio and the sum rule N_x + N_y + N_z = 1

1948

— Sommerfeld laments the B/H naming tangle; physics settles on B as the magnetic field

2019

— SI redefinition ends μ0 = 4π×10⁻⁷ exactly; the B/H unit conversion itself becomes a measured quantity

The physical picture

Recall the accountant of the dielectric chapter, and hand them the magnetic ledger. The physical books — Ampère's law for $\mathbf{B}$ — mix two kinds of current: currents you drive (the free current) and currents the material generates in response (the amperian loops of Bound Currents, orchestrated into $\mathbf{J}_b = \nabla\times\mathbf{M}$ ). You control the first and can't directly meter the second. The accountant's move is the same as before, but the operator has changed: bound charge was a divergence ( $\rho_b = -\nabla\cdot\mathbf{P}$ ), so the electric fix joined the divergence law and produced $\mathbf{D}$ ; bound current is a curl, so the magnetic fix joins the curl law. Subtract from $\mathbf{B}/\mu_0$ exactly the quantity, $\mathbf{M}$ , whose curl is the bound current, and the material's entries cancel identically: $\nabla\times\mathbf{H} = \mathbf{J}_f$ answers to your inputs alone.

Figure

Two ledgers for one magnetic problem — B answers to all current, H to free current only

Why is the new ledger so useful? Because in the lab you dial the free current. Wind $n$ turns per metre around a core and drive $I$ through them: $\mathbf{H}$ inside is fixed by your ammeter, whatever the core is made of — the material's contribution is postponed to the conversion $\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})$ , handled by a constitutive relation (Magnetic Susceptibility and Magnetic Permeability build it; Ampère's Law for H works the catalog). That is why every materials table and every hysteresis loop has $H$ on its horizontal axis: $H$ is the knob, $B$ is the response. And why the loud warnings? Because the analogy " $\mathbf{H}$ is to free current what $\mathbf{B}$ is to total current" invites the same illegal inference the $\mathbf{D}$ article prosecuted. A field is reconstructible from its sources only when you know both its divergence and its curl — the Helmholtz theorem is explicit. $\mathbf{B}$ qualifies: curl known from all currents, divergence known to vanish. $\mathbf{H}$ does not: its curl is the free current, but its divergence, $\nabla\cdot\mathbf{H} = -\nabla\cdot\mathbf{M}$ , belongs to the material — nonzero wherever magnetization starts, stops, or spreads, most vividly at the end faces of a bar magnet, where $\mathbf{M}$ plunges to zero and $\mathbf{H}$ -lines erupt as if from a sheet of charge. There, any attempt to compute " $\mathbf{H}$ by Biot–Savart over $\mathbf{J}_f$ " returns zero — off by everything — and the field inside the magnet runs against the magnetization. The two portraits below are the whole story: same magnet, two fields, two geometries.

Figure

The bar magnet, two portraits — B loops through without ends, H runs pole to pole like an electric field

The right-hand portrait should look familiar: it is the electrostatic field of two oppositely charged disks. That is not a metaphor — where free current vanishes, $\mathbf{H}$ obeys exactly the equations of an electrostatic field sourced by $-\nabla\cdot\mathbf{M}$ (the pole fiction, made precise below). And readers who met the electret sphere in The Electric Displacement D have seen today's cast in mirror image: there, $\mathbf{D}$ drew the bar magnet's closed loops while $\varepsilon_0\mathbf{E}$ ran pole-to-pole; here, $\mathbf{B}$ draws the loops while $\mathbf{H}$ runs pole-to-pole. The correspondence that article promised "the magnetism chapter will earn" is now earned.

Definition and field equations

\mathbf{H} \equiv \frac{\mathbf{B}}{\mu_0} - \mathbf{M} \qquad (1)

where: $\mathbf{B}$ — the macroscopic magnetic field (the averaged, physical field), units T; $\mathbf{M}$ — the magnetization, units A/m; $\mathbf{H}$ — the auxiliary field, units A/m (matching $\mathbf{M}$ , and matching surface current — a standing reminder that $\mathbf{H}$ 's natural habitat is circulation-per-length bookkeeping). Note the division by $\mu_0$ : it is $\mathbf{B}/\mu_0$ , not $\mathbf{B}$ , that joins $-\mathbf{M}$ , the mirror of the $\varepsilon_0$ that guards the definition of $\mathbf{D}$ — in vacuum $\mathbf{H} = \mathbf{B}/\mu_0$ exactly, same field in different units. The field equations it obeys, derived below:

\nabla\times\mathbf{H} = \mathbf{J}_f, \qquad \nabla\cdot\mathbf{H} = -\nabla\cdot\mathbf{M} \qquad (2)

where: $\mathbf{J}_f$ — free current density (conduction currents in windings and wires — current with an independent existence, as opposed to the response current bound in atoms). The first equation is the tool; the second is the fine print that governs when the tool may be trusted alone. Together they say: $\mathbf{H}$ 's circulation is pure free-current bookkeeping; $\mathbf{H}$ 's flux is hostage to the material geometry. Compare the displacement's report card, $\nabla\cdot\mathbf{D} = \rho_f$ with $\nabla\times\mathbf{D} = \nabla\times\mathbf{P}$ : the clean slot and the dirty slot have traded places, because bound charge is a divergence of $\mathbf{P}$ while bound current is a curl of $\mathbf{M}$ .

\mathbf{B} = \mu_0\left(\mathbf{H} + \mathbf{M}\right) \qquad (3)

where: this rearrangement of (1) is the working direction in practice — compute $\mathbf{H}$ from what you control, add the material's answer $\mathbf{M}$ , recover the physical $\mathbf{B}$ . In linear media $\mathbf{M} = \chi_m\mathbf{H}$ collapses the sum to $\mathbf{B} = \mu\mathbf{H}$ ; that constitutive machinery is owned by Magnetic Susceptibility and Magnetic Permeability, next in the chapter, and is quoted here only as a forward pointer.

Derivation AConstructing the bookkeeper — Ampere's law with the bound current folded in

The goal is equations (2) from what the chapter has already established, in the three lines the construction deserves, followed by the statement of scope: exactly what the new law does and does not determine. The inputs are Ampère's law for $\mathbf{B}$ with all currents (the magnetostatics chapter's $\nabla\times\mathbf{B} = \mu_0\mathbf{J}$ , from Curl of B) and the bound-current theorem $\mathbf{J}_b = \nabla\times\mathbf{M}$ (from Bound Currents). Watch that no new physics enters — $\mathbf{H}$ is definition plus rearrangement, which is precisely why it can never contain more than $\mathbf{B}$ and $\mathbf{M}$ did. The construction shadows the $\mathbf{D}$ derivation of the dielectric chapter step for step, with divergence traded for curl.

Step 1. Start from the honest Ampère's law, splitting total current into the two ledgers:

\frac{1}{\mu_0}\nabla\times\mathbf{B} = \mathbf{J}_{\rm total} = \mathbf{J}_f + \mathbf{J}_b = \mathbf{J}_f + \nabla\times\mathbf{M}.

Why this split: free current is what ammeters meter and supplies drive; bound current is the material's massed amperian loops, already parceled into $\nabla\times\mathbf{M}$ by the bound-current theorem.

Step 2. Move the material's response to the left side, where it combines into a single curl:

\nabla\times\left(\frac{\mathbf{B}}{\mu_0} - \mathbf{M}\right) = \mathbf{J}_f \qquad\Longrightarrow\qquad \nabla\times\mathbf{H} = \mathbf{J}_f,

with $\mathbf{H}$ defined by (1). The curl operator is linear, so the two curls merge without ceremony — the same one-line miracle that merged the two divergences into $\mathbf{D}$ .

Step 3. Take the divergence of the definition. Universally $\nabla\cdot\mathbf{B} = 0$ (Divergence of B — no magnetic monopoles), so

\nabla\cdot\mathbf{H} = \frac{1}{\mu_0}\nabla\cdot\mathbf{B} - \nabla\cdot\mathbf{M} = -\nabla\cdot\mathbf{M}.

No law of nature makes this vanish: $\mathbf{M}$ is whatever the material's state dictates, and wherever it has a divergence — bar-magnet end faces above all — $\mathbf{H}$ has sources that $\mathbf{B}$ is forbidden to have. It does vanish in the friendly cases (closed magnetization loops in a toroid, an infinite axially magnetized cylinder, linear media threaded symmetrically by free current), which is exactly why the friendly cases are the ones where $\mathbf{H}$ -first methods work.

Step 4. Scope, via Helmholtz. The Helmholtz theorem (E0) reconstructs a field from both its curl and its divergence. For $\mathbf{B}$ : curl known from all currents, divergence known to be zero — reconstructible, hence the Biot–Savart law. For $\mathbf{H}$ : curl known from free current, divergence equal to $-\nabla\cdot\mathbf{M}$ — unknown until you know the material's magnetization, the very thing you were trying to avoid computing. So $\mathbf{J}_f$ alone underdetermines $\mathbf{H}$ ; the missing information is supplied by symmetry (which kills the divergence-sourced part) or by a constitutive relation plus boundary conditions. There is, correspondingly, no "Biot–Savart law for $\mathbf{H}$ " — and integrating that illegal formula is the canonical way to get magnetized-matter problems silently wrong.

Go deeper · Derivation B · 3 minThe deceptive parallel prosecuted — a magnet with no free current at all

Derivation BThe deceptive parallel prosecuted — a magnet with no free current at all

The goal is to make the warnings quantitative on the cleanest possible counterexample: the uniformly magnetized sphere of Field of a Magnetized Object — a body with zero free current everywhere, whose $\mathbf{B}$ -field that article already earned by matching. The strategy is to assemble $\mathbf{H}$ from the known $\mathbf{B}$ and $\mathbf{M}$ via the definition (one line each, inside and out), then interrogate it: what would naive free-current reasoning have predicted, where do $\mathbf{H}$ 's lines start and end, and how does the circulation audit close? The numbers here are re-used in worked example 1.

Step 1. Assemble $\mathbf{H}$ inside. The sphere has $\mathbf{B}_{\rm in} = \tfrac{2}{3}\mu_0\mathbf{M}$ , uniform, so

\mathbf{H}_{\rm in} = \frac{\mathbf{B}_{\rm in}}{\mu_0} - \mathbf{M} = \frac{2}{3}\mathbf{M} - \mathbf{M} = -\frac{1}{3}\,\mathbf{M}.

Note the sign spectacle, the exact mirror of the electret's: $\mathbf{B}$ points with the magnetization, $\mathbf{H}$ points against it — the two fields disagree about direction inside matter, an impossibility if $\mathbf{H}$ were "basically $\mathbf{B}$ rescaled." The factor $\tfrac13$ is the sphere's demagnetizing factor, and $\mathbf{H}_{\rm in}$ is called the demagnetizing field: it is the field trying to undo the magnet from within.

Step 2. Assemble $\mathbf{H}$ outside: $\mathbf{M} = 0$ there, so $\mathbf{H}_{\rm out} = \mathbf{B}_{\rm out}/\mu_0$ — the pure dipole field of moment $m = \tfrac{4}{3}\pi R^3 M$ , divided by $\mu_0$ .

Step 3. Audit the naive prediction. Free current: none, anywhere. " $\mathbf{H}$ is sourced by free current" (naively) $\Rightarrow \mathbf{H} = 0$ everywhere. Reality: $|\mathbf{H}_{\rm in}| = M/3 \neq 0$ , and a whole exterior dipole pattern besides. The circulation law is not violated — $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = 0$ around every closed loop, as it must be with $I_{f,\rm enc} = 0$ — but zero circulation around every loop only means the field is curl-free, not that it vanishes. The information gap is exactly the divergence. Why this mirrors the electret audit: there $\mathbf{D}$ had zero flux through every closed surface yet lived on in loops; here $\mathbf{H}$ has zero circulation around every loop yet lives on pole-to-pole. Each auxiliary fails in precisely the slot its clean law does not constrain.

Step 4. Close the circulation ledger explicitly, on the axis. Take the loop that runs up the polar axis through the sphere and returns along the axis from $+\infty$ around to $-\infty$ (the return legs at infinity contribute nothing, $H \sim 1/r^3$ ). Interior leg: $\mathbf{H}_{\rm in}\cdot\mathrm{d}\boldsymbol{\ell}$ accumulates $-(M/3)(2R)$ — negative, the field opposes the path. Exterior legs: on the axis $H_z = 2m/4\pi r^3$ , and $\int_R^{\infty} \frac{2m}{4\pi r^3}\mathrm{d}r = \frac{m}{4\pi R^2} = \frac{MR}{3}$ per side, so the two exterior legs contribute $+\tfrac{2}{3}MR$ . Total: $-\tfrac{2}{3}MR + \tfrac{2}{3}MR = 0$ ✓. The exterior surplus pays the interior deficit exactly — the circulation law holds to the last digit, while telling you nothing about either piece separately.

Step 5. Find the sources and the line geometry. $\nabla\cdot\mathbf{H} = -\nabla\cdot\mathbf{M}$ is zero in the bulk (uniform $\mathbf{M}$ ) and zero outside — the divergence lives entirely on the surface, where the normal component of $\mathbf{M}$ jumps to zero: a surface source $\sigma_m = M\cos\theta$ , positive on the north hemisphere, negative on the south. $\mathbf{H}$ -lines therefore start on the north face and end on the south — outside they arc pole to pole like a dipole's, inside they run straight back against $\mathbf{M}$ . Consequences: $\mathbf{H}$ has flux sources $\mathbf{B}$ never has; no Biot–Savart integral over $\mathbf{J}_f$ could produce it (it would give 0); and no shortcut past the material — to find $\mathbf{H}$ you first needed $\mathbf{B}$ and $\mathbf{M}$ . The one consolation is large: because $\mathbf{H}$ here is curl-free with charge-like sources, electrostatic methods compute it wholesale — the pole fiction, built next. As a consistency note: $H_{\text{in}} = -M/3$ follows directly from $B_{\text{in}} = 2\mu_0 M/3$ via $\mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}$ ; the axis circulation integral closes at zero exactly ( $-(2/3)MR + 2\cdot MR/3 = 0$ , e.g. $-6667 + 6667\text{ A}$ for $M = 1.0\times10^6\text{ A/m}$ , $R = 1\text{ cm}$ ); and $\sigma_m = M\cos\theta$ is the precise magnetic analogue of the polarized-sphere surface charge $\sigma_b = P\cos\theta$ under the dictionary $\mathbf{P}\to\mathbf{M}$ .

The pole fiction — electrostatics for H

The bar-magnet portrait begs for a formalism: if $\mathbf{H}$ flows from face to face like $\mathbf{E}$ between charged plates, can we compute it that way? Wherever $\mathbf{J}_f = 0$ , yes, exactly — and the recipe is worth owning because it powers everything from permanent-magnet design to micromagnetic simulation.

\mathbf{J}_f = 0:\qquad \nabla\times\mathbf{H} = 0 \;\Rightarrow\; \mathbf{H} = -\nabla W, \qquad \nabla^2 W = -\rho_m, \quad \rho_m \equiv -\nabla\cdot\mathbf{M},\;\; \sigma_m \equiv \mathbf{M}\cdot\hat{\mathbf{n}} \qquad (4)

where: $W$ — the magnetic scalar potential, units A; $\rho_m$ , $\sigma_m$ — the effective ("fictitious") magnetic charge densities, in the volume and on surfaces where $\mathbf{M}$ ends. Every electrostatic tool of chapters E1–E2 now applies verbatim: Poisson's equation, uniqueness theorems, separation of variables, even the method of images — with $\rho_b/\varepsilon_0 \to \rho_m$ as the source dictionary. The charges are fictions (nothing sits on the magnet's faces; cut the magnet and you never isolate a pole — $\nabla\cdot\mathbf{B} = 0$ stands guard), but the field they generate is exactly $\mathbf{H}$ .

Go deeper · Derivation C · 2 minThe pole fiction made precise — a scalar potential and its charges

Derivation CThe pole fiction made precise — a scalar potential and its charges

The goal is equation (4) and its license: to show that in free-current-free regions $\mathbf{H}$ is mathematically an electrostatic field, identify its effective charges, and calibrate the fiction on the magnetized sphere so the dictionary is verified rather than asserted. The inputs are the two field equations (2) and the E0/E1 machinery, recalled by name. Watch the scope clause — the potential exists only where $\mathbf{J}_f = 0$ , and in multiply connected geometries (a loop of current threading the region) even that can fail.

Step 1. Where $\mathbf{J}_f = 0$ , equation (2) gives $\nabla\times\mathbf{H} = 0$ . A curl-free field on a simply connected region is a gradient (the E0 theorem recalled from Scalar Potentials): $\mathbf{H} = -\nabla W$ , the sign chosen so the analogy runs parallel to $\mathbf{E} = -\nabla V$ .

Step 2. Feed the potential into the divergence law: $\nabla\cdot\mathbf{H} = -\nabla\cdot\mathbf{M}$ becomes

\nabla^2 W = \nabla\cdot\mathbf{M} = -\rho_m,

Poisson's equation with source $\rho_m = -\nabla\cdot\mathbf{M}$ . At an abrupt material surface, the pillbox argument (E1's, verbatim) converts the jump in normal $\mathbf{M}$ into a surface charge $\sigma_m = \mathbf{M}\cdot\hat{\mathbf{n}}$ . Why the dictionary is exact and not approximate: the equations $\nabla\times\mathbf{H}=0$ , $\nabla\cdot\mathbf{H}=\rho_m$ are structurally identical to $\nabla\times\mathbf{E}=0$ , $\nabla\cdot\mathbf{E}=\rho/\varepsilon_0$ — and fields are their equations (Helmholtz again). Whatever electrostatics proves about $\mathbf{E}$ of a charge distribution, magnetostatics inherits for $\mathbf{H}$ of the matching pole distribution.

Step 3. Calibrate on the sphere. Uniform $\mathbf{M} = M\hat{\mathbf{z}}$ : $\rho_m = 0$ in the bulk, $\sigma_m = M\cos\theta$ on the surface. But a $\cos\theta$ surface charge on a sphere is the classic of the dielectric chapter — Uniformly Polarized Sphere solved $\sigma_b = P\cos\theta$ and found $\mathbf{E}_{\rm in} = -\mathbf{P}/3\varepsilon_0$ . Apply the dictionary ( $P/\varepsilon_0 \to M$ ): $\mathbf{H}_{\rm in} = -\mathbf{M}/3$ — precisely the value assembled from $\mathbf{B}$ in the previous derivation, now obtained without ever mentioning a current. Two independent routes, one answer; the fiction is certified.

Step 4. State the license's fine print. (i) The fiction computes $\mathbf{H}$ , never $\mathbf{B}$ : finish with $\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})$ — inside matter the added $\mu_0\mathbf{M}$ flips the picture entirely (the sphere: $\mathbf{H}$ down, $\mathbf{B}$ up). (ii) The poles are bookkeeping: Gauss's law for $\mathbf{B}$ still reads zero everywhere (Divergence of B owns the monopole story). (iii) Where free currents flow, $W$ dies — $\mathbf{H}$ regains curl; that regime belongs to Ampère's Law for H. The sphere result $H_{\text{in}} = -M/3$ maps directly onto the electrostatic $E_{\text{in}} = -P/3\varepsilon_0$ under the substitution $P/\varepsilon_0 \to M$ , and the pillbox boundary condition $\sigma_m = M\cdot\hat{n}$ mirrors the surface-charge pillbox of the electric case verbatim.

The analogy table this chapter has been building deserves its honest asymmetry spelled out:

electric matter	magnetic matter	the asymmetry
$\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}$	$\mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}$	sign and slot both flip
clean law: $\nabla\cdot\mathbf{D} = \rho_f$	clean law: $\nabla\times\mathbf{H} = \mathbf{J}_f$	D audits flux, H audits circulation
fine print: $\nabla\times\mathbf{D} = \nabla\times\mathbf{P}$	fine print: $\nabla\cdot\mathbf{H} = -\nabla\cdot\mathbf{M}$	each fails where the other is clean
no Coulomb law for $\mathbf{D}$	no Biot–Savart law for $\mathbf{H}$	Helmholtz demands both slots
electret: $\rho_f = 0$ , $\mathbf{D} \neq 0$ , loops	bar magnet: $\mathbf{J}_f = 0$ , $\mathbf{H} \neq 0$ , poles	the memorized counterexamples
$\mathbf{E}$ physical, $\mathbf{D}$ bookkeeping	$\mathbf{B}$ physical, $\mathbf{H}$ bookkeeping	the doctrine, twice

And the exact dualities run diagonally across that table: the electret's $\mathbf{D}$ (divergence-free, curl-carrying) is the bar magnet's $\mathbf{B}$ , and the electret's $\varepsilon_0\mathbf{E}$ (curl-free, sourced on the material's skin) is the bar magnet's $\mu_0\mathbf{H}$ . The $\mathbf{D}$ article's electret portrait and this article's magnet portrait are the same two drawings with the labels traded. And a word on names, since the literature will fight you. Older books — and most engineers to this day — call $\mathbf{H}$ "the magnetic field" and $\mathbf{B}$ "the magnetic induction" or "flux density," a convention rooted in the pole-minded nineteenth century. Modern physics usage — Griffiths's explicitly — reserves the magnetic field for $\mathbf{B}$ , the field in the Lorentz force and in $\nabla\cdot\mathbf{B} = 0$ , and grants $\mathbf{H}$ no name beyond "H." This article sides with $\mathbf{B}$ ; translate on contact with a materials datasheet, whose horizontal axis labeled "magnetic field $H$ " is our auxiliary.

Limiting cases

Case	Result	Meaning
vacuum	$\mathbf{H} = \mathbf{B}/\mu_0$ exactly	the ledgers coincide; a units convention, nothing more
linear medium (next articles)	$\mathbf{M} = \chi_m\mathbf{H}$ , $\mathbf{B} = \mu\mathbf{H}$	conversion becomes one multiplication; $\chi_m$ , $\mu$ owned downstream
free current present, high symmetry	$\mathbf{H}$ from $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_f$	the tool at full power — the catalog of Ampère's Law for H
$\mathbf{J}_f = 0$ everywhere	$\mathbf{H} = -\nabla W$ , poles $\rho_m$ , $\sigma_m$	electrostatics computes H; B follows from (3)
uniformly magnetized sphere	$\mathbf{H}_{\rm in} = -\mathbf{M}/3$ , $\mathbf{B}_{\rm in} = +\tfrac{2}{3}\mu_0\mathbf{M}$	the two fields point opposite ways inside
long needle, axial $\mathbf{M}$	$\mathbf{H}_{\rm in} \approx 0$ , $\mathbf{B}_{\rm in} \approx \mu_0 M$	poles exiled to the far ends; the solenoid limit
thin slab, transverse $\mathbf{M}$	$\mathbf{H}_{\rm in} = -\mathbf{M}$ , $\mathbf{B}_{\rm in} \approx 0$	full pole sheets, like a parallel-plate capacitor
ferromagnet	$\mathbf{M}(\mathbf{H})$ multivalued	no single $\mu$ exists; hysteresis (owned by Ferromagnetism)

Common trapThere is no Biot-Savart law for H — and "no free current" never means "no H"

The deceptive parallel claims its victims in two standard ways, both inherited from the displacement chapter with the operators swapped. First: computing $\mathbf{H}$ by integrating $\mathbf{J}_f$ with a Biot–Savart kernel, or equivalently assuming $\mathbf{J}_f = 0 \Rightarrow \mathbf{H} = 0$ . The magnetized sphere is the two-line refutation: no free current anywhere, yet $\mathbf{H} = -\mathbf{M}/3$ inside and a full dipole pattern outside. A curl alone determines nothing — Helmholtz demands the divergence, and $\mathbf{H}$ 's divergence belongs to the material. Second: treating $\mu_0\mathbf{H}$ as "the field with the material ignored" — inside the magnet, $\mathbf{H}$ and $\mathbf{B}$ point in opposite directions; no rescaling relates them. The safe operating envelope: use circulation methods when free-current symmetry pins $\mathbf{H}$ and the magnetization geometry contributes no divergence along your loop (the toroid and filled solenoid of the next article qualify; a bar magnet never does) — and outside that envelope, solve honestly for $\mathbf{B}$ and $\mathbf{M}$ (or use the pole fiction where $\mathbf{J}_f = 0$ ) and assemble $\mathbf{H}$ , never the reverse.

Worked examples

1 · The neodymium sphere — assembling H, closing every audit (confidence builder)

A uniformly magnetized NdFeB sphere has $M = 1.0\times10^{6}\,\mathrm{A/m}$ and radius $R = 1.0\,\mathrm{cm}$ . Find $\mathbf{H}$ and $\mathbf{B}$ inside, the surface pole density, and verify the circulation ledger on the axis.

Step 1. Inside fields, from Field of a Magnetized Object plus the definition: $\mathbf{B}_{\rm in} = \tfrac{2}{3}\mu_0 M\,\hat{\mathbf{z}} = 0.84\,\mathrm{T}$ (along $\mathbf{M}$ ); $\mathbf{H}_{\rm in} = \mathbf{B}_{\rm in}/\mu_0 - \mathbf{M} = (6.7\times10^{5} - 1.0\times10^{6})\,\hat{\mathbf{z}} = -3.3\times10^{5}\,\mathrm{A/m}$ — against $\mathbf{M}$ . Two "magnetic fields," antiparallel, in one sphere.

Step 2. The poles. $\sigma_m = M\cos\theta$ : $+1.0\times10^{6}\,\mathrm{A/m}$ at the north pole, falling as $\cos\theta$ , negative on the southern hemisphere. Total north-hemisphere pole strength: $q_m = \int\sigma_m\,\mathrm{d}a = M\pi R^2 = 3.1\times10^{2}\,\mathrm{A\cdot m}$ — the fictitious charge that, placed on the surface, generates this exact $\mathbf{H}$ .

Step 3. Continuity audit at the north pole. Outside, on the axis at $r = R$ : $H_{\rm out} = 2m/4\pi R^3 = \tfrac{2}{3}M = +6.7\times10^{5}\,\mathrm{A/m}$ (with $m = \tfrac43\pi R^3 M = 4.2\,\mathrm{A\,m^2}$ ). Jump across the surface: $H_{\rm out} - H_{\rm in} = \tfrac23 M + \tfrac13 M = M = \sigma_m(\text{pole})$ ✓ — normal $\mathbf{H}$ jumps by exactly the pole density, as a charge sheet demands. Meanwhile $B$ is continuous there: $0.84\,\mathrm{T}$ both sides ✓ — no monopoles, no $\mathbf{B}$ -flux jump, ever.

Step 4. The circulation ledger. Interior axis leg: $H_{\rm in}\times 2R = (-3.3\times10^{5})(0.02) = -6.7\times10^{3}\,\mathrm{A}$ . Exterior legs: $2\times m/4\pi R^2 = 2\times MR/3 = +6.7\times10^{3}\,\mathrm{A}$ . Sum: $0$ ✓, as $I_{f,\rm enc} = 0$ requires. Every audit closes, and none of them could have found the field — that took the material's geometry.

2 · One magnetization, three bodies — the shape is the demagnetizing field (exam level)

The same material, $M = 1.0\times10^{6}\,\mathrm{A/m}$ , is shaped into (a) a long thin needle magnetized along its axis, (b) a sphere, (c) a wide thin slab magnetized perpendicular to its faces. Find $\mathbf{H}_{\rm in}$ and $\mathbf{B}_{\rm in}$ in each, using the pole fiction.

Step 1. The needle. Poles $\sigma_m = \pm M$ sit only on the two tiny end faces, exiled far away; between them their field is negligible, so $\mathbf{H}_{\rm in} \approx 0$ and $\mathbf{B}_{\rm in} = \mu_0(\mathbf{H} + \mathbf{M}) \approx \mu_0 M = 1.3\,\mathrm{T}$ . The needle is the solenoid limit — recall from Magnetization that a uniformly magnetized cylinder is a solenoid with $K_b = M$ , and an infinite solenoid's interior field is exactly $\mu_0 M$ with nothing subtracted.

Step 2. The slab. Poles $\sigma_m = \pm M$ now tile two huge parallel faces: a parallel-plate capacitor drawn in magnetic ink. Between infinite sheets of $\pm\sigma_m$ , the electrostatic dictionary gives a uniform interior field $H = \sigma_m = M$ pointing from $+$ to $-$ , i.e. $\mathbf{H}_{\rm in} = -\mathbf{M}$ , the maximum demagnetization possible. Then $\mathbf{B}_{\rm in} = \mu_0(-\mathbf{M} + \mathbf{M}) = 0$ : the magnetization is entirely self-cancelled, and (in the infinite-slab idealization) there is no $\mathbf{B}$ anywhere.

Step 3. The sphere sits exactly between: $\mathbf{H}_{\rm in} = -\tfrac13\mathbf{M} = -3.3\times10^{5}\,\mathrm{A/m}$ , $\mathbf{B}_{\rm in} = \tfrac23\mu_0 M = 0.84\,\mathrm{T}$ , from example 1. The pattern is the demagnetizing factor $N$ : $\mathbf{H}_{\rm in} = -N\mathbf{M}$ with $N \approx 0$ (needle, axial), $\tfrac13$ (sphere), $1$ (slab, transverse) — and for any ellipsoid the three principal factors obey $N_x + N_y + N_z = 1$ (the same sum rule the Lorentz-cavity article exhibited for its electric twin). The general- $N$ machinery and its interface consequences are owned by Magnetic Boundary Conditions, next lane in the chapter.

Step 4. Read the engineering. Shape is destiny for permanent magnets: a slab magnetized through its thin dimension sits deep in its own demagnetizing field ( $H = -M$ ), pushed toward demagnetization along its hysteresis loop, while a needle or a closed magnetic circuit barely demagnetizes itself at all. This is why horseshoe magnets keep their keepers, and why every magnet assembly's load line starts from the demagnetizing factor of its geometry (Linear and Nonlinear Magnetic Media takes up the story).

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.3 · Jackson, Classical Electrodynamics, 3rd ed., §5.8–5.10 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, §13.3–13.4. The key relations $\mathbf{H}=\mathbf{B}/\mu_0-\mathbf{M}$ , $\nabla\times\mathbf{H}=\mathbf{J}_f$ , $\nabla\cdot\mathbf{H}=-\nabla\cdot\mathbf{M}$ , and $\mathbf{B}=\mu_0(1+\chi_m)\mathbf{H}$ are standard results of the theory. For a uniformly magnetized sphere the demagnetizing factor is $1/3$ , giving $\mathbf{H}_{in}=-\mathbf{M}/3$ and $\mathbf{B}_{in}=\tfrac{2}{3}\mu_0\mathbf{M}$ ; inside a filled solenoid $H=nI$ ; and $\mu_0=1.25663706\times10^{-6}$ T·m/A. Susceptibility values (water $-9.0\times10^{-6}$ , Al $+2.2\times10^{-5}$ , Cu $-9.6\times10^{-6}$ ) are standard figures and may vary slightly with source. The refraction law $\tan\theta_1/\tan\theta_2=\mu_1/\mu_2$ follows from continuity of tangential H and normal B. For a NdFeB sphere ( $M = 1.0\times10^6$ A/m, $R = 1$ cm): $H_{\rm in} = -3.3\times10^5$ A/m, $B_{\rm in} = 0.84$ T, $m = 4.2$ A·m², pole strength $M\pi R^2 = 3.1\times10^2$ A·m, axis circulation $-\tfrac23 MR + 2(MR/3) = 0$ exactly; shape trio $N = 0, \tfrac13, 1$ with $B_{\rm in} = 1.3$ T (needle), $0.84$ T (sphere), $0$ (slab); $M_r = B_r/\mu_0 = 1.03\times10^6$ A/m for $B_r = 1.30$ T.

Builds on: Magnetization · Bound Currents · Field of a Magnetized Object

Leads to: Ampere's Law for H · Magnetic Susceptibility · Magnetic Permeability · Magnetic Boundary Conditions

Is H a physical field or just a mathematical convenience?

A convenience with an exact definition — and the question has the same sharp answer the displacement article gave for $\mathbf{D}$ . Forces on charges are $q\mathbf{v}\times\mathbf{B}$ , the universal no-source law is $\nabla\cdot\mathbf{B} = 0$ , and the microscopic world averages to $\mathbf{B}$ ; nothing pushes on anything "with $\mathbf{H}$ ." Yet convenience is not disparagement: $\mathbf{H}$ 's circulation is what your power supply actually pays for (free current), which is why engineers specify it, materials tables tabulate against it, and it earns its seat in the macroscopic Maxwell equations. The healthy attitude: $\mathbf{B}$ is the field of physics, $\mathbf{H}$ the field of accounting — and the books are kept for excellent reasons.

Why does H point backwards inside a bar magnet? That still feels wrong.

Follow the poles. With no free current, $\mathbf{H}$ is an honest electrostatic-style field of the fictitious charges $\sigma_m = \mathbf{M}\cdot\hat{\mathbf{n}}$ : positive on the north face, negative on the south. Between a positive sheet and a negative sheet, the field runs from $+$ to $-$ — which, inside the magnet, is from north face back to south face, i.e. against $\mathbf{M}$ . (Outside, the same pole field runs north-around-to-south, which is why exterior $\mathbf{H}$ and $\mathbf{B}$ agree in shape there.) The interior $\mathbf{H}$ is called the demagnetizing field because it is genuinely trying to disassemble the magnetization that creates it — and in soft materials it succeeds, which is half the story of domain formation in Ferromagnetism.

If there is no Biot-Savart law for H, how do I ever compute it?

Three honest routes. (1) Symmetry plus circulation: when free-current geometry pins $\mathbf{H}$ on an Amperian loop and the magnetization contributes no symmetry-breaking divergence — toroids, filled solenoids — $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_{f,\rm enc}$ delivers $\mathbf{H}$ in two lines (Ampère's Law for H is that catalog). (2) The pole fiction: where $\mathbf{J}_f = 0$ , solve electrostatics for $\rho_m$ , $\sigma_m$ — every E1/E2 tool applies. (3) Assembly: solve the honest problem for $\mathbf{B}$ and $\mathbf{M}$ (as Field of a Magnetized Object does), then $\mathbf{H} = \mathbf{B}/\mu_0 - \mathbf{M}$ . What you may never do is integrate $\mathbf{J}_f$ with a $1/\mathscr{r}^2$ kernel and call the result $\mathbf{H}$ .

In vacuum H = B/μ0 exactly — so why do we drag H around at all?

Because every interesting magnetics problem eventually touches matter, and inside matter the two fields split — in direction, not just scale (the sphere: antiparallel). $\mathbf{H}$ earns its keep at exactly that moment: it is the piece of the field fixed by what you control (winding currents), so it survives the material's response as a usable handle. In vacuum-only electrodynamics you can retire $\mathbf{H}$ without loss — many relativity-first treatments do. The auxiliary exists for the same reason $\mathbf{D}$ does: matter responds, the response is initially unknown, and the bookkeeping field quarantines that ignorance into one symbol.

Why are H and M measured in A/m but B in tesla? The units feel backwards.

The units are the physics. $\mathbf{M}$ is dipole moment per volume, $\mathrm{A\,m^2/m^3} = \mathrm{A/m}$ — equivalently, a magnetized body wears a bound surface current $K_b$ of exactly that size (Bound Currents). $\mathbf{H}$ must share $\mathbf{M}$ 's units to be subtractable from $\mathbf{B}/\mu_0$ , and its own law confirms the reading: $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_f$ makes $H$ a current-per-length — amp-turns per metre, as every electromagnet designer says aloud. $\mathbf{B}$ carries the mechanical units instead (force per charge per velocity), because $\mathbf{B}$ is the field that does something. The dimensional split is the doctrine in miniature.

Are magnetic poles real, then? The pole fiction seems to work awfully well.

The field it computes is real; the charges are not. Gauss's law for $\mathbf{B}$ reads zero through every closed surface ever measured (Divergence of B tells the monopole-hunt story), and slicing a magnet at its "pole" yields two magnets, never a bare charge — poles are where $\mathbf{M}$ ends, and cutting relocates the ends. The fiction works because mathematics only cares about the equations: curl-free plus known divergence is electrostatics, whatever the divergence is made of. The one absolute guardrail: the fiction computes $\mathbf{H}$ , and only $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$ is the physical answer.

Does the D-to-H analogy mean every dielectric result has a magnetic twin I can just copy?

Almost — copy the structure, audit the slots and signs. The reliable dictionary: $\mathbf{P} \to \mathbf{M}$ trades bound charge ( $-\nabla\cdot\mathbf{P}$ , a divergence) for bound current ( $\nabla\times\mathbf{M}$ , a curl), so $\mathbf{D}$ 's clean divergence law becomes $\mathbf{H}$ 's clean curl law, and each auxiliary's fine print sits in the other slot. The diagonal dualities are exact: electret- $\mathbf{D}$ ↔ magnet- $\mathbf{B}$ , electret- $\varepsilon_0\mathbf{E}$ ↔ magnet- $\mu_0\mathbf{H}$ , $\sigma_b = P\cos\theta$ ↔ $\sigma_m = M\cos\theta$ , $-\mathbf{P}/3\varepsilon_0$ ↔ $-\mathbf{M}/3$ . What does not transfer blindly: signs in the definitions ( $+\mathbf{P}$ but $-\mathbf{M}$ ), the direction of the physical field inside matter ( $\mathbf{E}$ fights $\mathbf{P}$ ; $\mathbf{B}$ helps $\mathbf{M}$ — paramagnets amplify, dielectrics always screen), and the constitutive conventions (χ_m is defined against H — Magnetic Susceptibility owns that story).