Ampere's Law for H

The previous article built a field whose curl is pure free current: $\nabla\times\mathbf{H} = \mathbf{J}_f$ . This article turns that differential statement into the working tool of applied magnetism — $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_{f,\rm enc}$ , Ampère's law for $\mathbf{H}$ — and then spends it the way the dielectric chapter spent Gauss's law for $\mathbf{D}$ . The kinship is exact and worth announcing up front: Gauss's Law in Dielectrics was the worked catalog where $\mathbf{D}$ flowed from free charge before the material was even named; this is the worked catalog where $\mathbf{H}$ flows from free current with the same magnificent indifference. Wind a coil around a core and $H$ is set by your ammeter — the same $nI$ whether the core is air, aluminium, or transformer iron. All the material's drama is postponed to a single labeled conversion step. The catalog has two flagships. The filled solenoid and toroid show the law at full power: $H$ material-blind, $B$ amplified by the core, the bound currents harvested afterward like a dividend. The gapped electromagnet — iron ring, coil, and a 2 mm slice of air — shows the law running a budget: the drive $NI$ is divided between iron path and air gap, and the arithmetic delivers the sobering, load-bearing fact of electromagnet design: a gap of two millimetres can consume ninety-five percent of everything the coil supplies. And because this chapter inherited the honesty doctrine of the displacement field, the sermon comes with the tool. The law is always true — and usually insufficient. A bar magnet threads zero free current through every loop, so the law says only $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = 0$ ; it cannot see the pole-sourced $\mathbf{H}$ that fills that magnet from end to end. Symmetry must pin $\mathbf{H}$ — including the magnetization's geometry, not just the current's — before the two-line ritual is legal. Knowing exactly when that is, and what to do when it is not, is the skill this article installs.

Key takeaways

Integral law: $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_{f,\rm enc}$ — the circulation of $\mathbf{H}$ counts only the free current threading the loop, by Stokes's theorem applied to $\nabla\times\mathbf{H} = \mathbf{J}_f$ .
The ritual (twin of the $\mathbf{D}$ catalog's): symmetry → $\mathbf{H}$ from $I_f$ → convert $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$ or $\mu\mathbf{H}$ → harvest $\mathbf{M}$ , $\mathbf{J}_b$ , $\mathbf{K}_b$ → audit against Ampère's law for $\mathbf{B}$ with total currents.
Flagship one: solenoid/toroid with a linear core — $H = nI$ (or $NI/2\pi s$ ) material-blind; the core multiplies $B$ by $(1+\chi_m)$ and wears a bound surface current $K_b = \chi_m nI$ , a phantom solenoid wound on top of the real one.
Flagship two: the gapped electromagnet — $NI = H_{\rm iron}\ell_{\rm iron} + H_{\rm gap}\,g$ with $B$ continuous across the gap; for $\mu_r = 5000$ , $\ell = 0.5$ m, $g = 2$ mm, the gap eats ~95% of the budget and $B \approx 0.60$ T.
The sermon: the law is always true and usually insufficient — $I_f = 0$ gives $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = 0$ , not $\mathbf{H} = 0$ (bar magnet). Symmetry must pin $\mathbf{H}$ including the magnetization's geometry.
Interface fine print ( $H^\parallel$ jumps by $K_f$ , $H^\perp$ by the magnetization mismatch) is owned by Magnetic Boundary Conditions, next in the chapter; here only $B^\perp$ -continuity is spent, at the gap.
KEY IDEA: $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_{f,\rm enc}$ turns free current into $\mathbf{H}$ in two lines wherever symmetry pins the field — material-blind first, constitutive conversion second, bound-current audit third — and it never, by itself, determines $\mathbf{H}$ without that symmetry.

The core insight

Ampère's law for the auxiliary field — $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_{f,\rm enc}$ , from Stokes on $\nabla\times\mathbf{H} = \mathbf{J}_f$ — computes $\mathbf{H}$ from controllable free current alone in symmetric geometries, postponing all material response to one constitutive conversion; but it constrains only the circulation, so without symmetry (the bar magnet) it is true, insufficient, and silent.

Where this lives today

Every transformer, motor, relay, and inductor on Earth is designed by iterating this article's two flagships. The engineering formalism — magnetic circuits, with $NI$ as "magnetomotive force" and $\ell/\mu A$ as "reluctance" — is Ampère's law for $\mathbf{H}$ dressed in Ohm's-law clothing, and modern finite-element magnetics packages are the same bookkeeping run on meshes of millions of elements: assemble $\nabla\times\mathbf{H} = \mathbf{J}_f$ per element, close with each material's $B(H)$ curve, iterate through saturation. The 95%-to-the-gap budget of the flagship reappears at every scale, from the sub-millimetre gaps of hard-drive write heads (where the gap field must switch in nanoseconds) to the multi-tesla dipoles of accelerator rings. The law also quietly defines how magnetic materials are characterized. A standards lab measuring a new electrical steel winds it into a toroid precisely because this article says so: closed flux path, poles impossible, $H$ known from the ammeter to metrological accuracy — then $B$ read from a secondary winding gives the $B$ – $H$ curve with nothing to correct for. When the datasheet for your transformer laminations quotes " $B = 1.5$ T at $H = 800$ A/m," it is reporting a measurement whose entire epistemology is $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = NI$ on a ring.

Origins · 1 minJoseph Henry and how this was discovered

Joseph Henry

1797–1878 · Albany, Princeton, Washington

Henry built the machine this article analyzes. A schoolteacher at the Albany Academy with no formal physics training, he took Sturgeon's feeble 1825 electromagnet — bare wire, a few loose turns — and re-engineered it with the one insight the gapped-ring budget makes obvious: amp-turns are the currency. Insulating the wire (with silk from his wife's petticoats, the story goes) let him wind hundreds of tight layers; by 1831 his Albany magnet lifted 750 pounds, over a thousand times its own weight, and a successor at Yale exceeded a ton — the strongest artificial magnetic devices then in existence. The same year he rang a bell through a mile of wire, the germ of the telegraph Morse would commercialize, and his relay — a small gapped electromagnet switching a larger circuit — became the amplifying element of nineteenth-century telecommunication. Henry also discovered self-induction independently of Faraday (the SI unit of inductance is his), directed the new Smithsonian Institution for three decades, and died the acknowledged dean of American science. Every flagship in this article — the tightly wound core, the budgeted gap, the iron highway for flux — is Albany engineering, formalized.

History

1820

— Ørsted and then Ampère tie magnetism to current; the circulation law's ancestry begins

1825

— Sturgeon bends soft iron into a horseshoe and winds it: the first electromagnet

1831

— Henry's insulated multi-layer windings lift 750 pounds at Albany — amp-turns understood in practice

1873

— Maxwell's Treatise gives the H-circulation law its modern macroscopic form

1886

— The Hopkinsons publish the magnetic-circuit law, engineering's Ohm's-law reading of this article

1930

s — Toroidal permeameters standardize B-H measurement on closed rings, poles engineered away

The physical picture

Picture the workflow as a relay with four labeled batons, run in strict order. First symmetry: argue from the free-current geometry — and from the material's, which must respect it — that $\mathbf{H}$ has a known direction and depends on one coordinate. Second circulation: choose an Amperian loop along which that known $\mathbf{H}$ is constant, and read $H$ off $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_{f,\rm enc}$ — two lines, no material constants anywhere. Third conversion: now, and only now, the material enters, through $\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})$ — which for the linear media of the coming articles collapses to one multiplication, $\mathbf{B} = \mu\mathbf{H}$ (the constants $\chi_m$ and $\mu$ are owned by Magnetic Susceptibility and Magnetic Permeability; here they are rented with a citation). Fourth audit: harvest the bound currents $\mathbf{M}$ implies and check that Ampère's law for $\mathbf{B}$ , run with total current, reproduces the same $\mathbf{B}$ — the books must balance both ways.

Figure

The free-current ritual — four batons in strict order, material enters only at baton three

Why does baton two work at all? Because $\mathbf{H}$ was built so that the material's response currents cancel out of its circulation. When you wrap a toroid in $N$ turns and drive current $I$ , the iron inside erupts in bound currents thousands of times stronger than what your supply delivers — and the circulation of $\mathbf{H}$ around the ring registers none of it. $H = NI/2\pi s$ , the same formula as for an empty ring. This is the precise magnetostatic echo of the insulated wire in Gauss's Law in Dielectrics, where $\mathbf{D} = \lambda/2\pi s$ held serenely through unknown rubber: the auxiliary field reads the sources you control and postpones the rest. The lab consequence is everything: materials scientists set $H$ (amp-turns are adjustable), measure $B$ (flux through a secondary winding), and plot one against the other — every $B$ – $H$ curve ever published is this ritual performed with instruments. But the ritual's first baton hides the sermon, and the bar magnet of the previous article is its eternal text. No free current anywhere: the law dutifully reports zero circulation around every conceivable loop — and stops. It does not report that $\mathbf{H} = -\mathbf{M}/3$ fills the magnetized sphere, because that field is curl-free: sourced by $\nabla\cdot\mathbf{H} = -\nabla\cdot\mathbf{M}$ , invisible to any circulation measurement, exactly as a point charge's electrostatic field is invisible to $\oint\mathbf{E}\cdot\mathrm{d}\boldsymbol{\ell} = 0$ . The symmetry demanded in baton one is therefore a double demand: the free currents must have it, and the magnetization must not break it. A uniform toroid qualifies (its $\mathbf{M}$ runs in closed azimuthal loops, divergence-free); a horseshoe magnet never does (its $\mathbf{M}$ ends at two faces, spraying poles). The worked catalog below is a tour of the geometries that pass.

The law

\oint_{\mathcal{C}}\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_{f,\rm enc} \qquad (1)

where: $\mathcal{C}$ — any closed loop; $I_{f,\rm enc}$ — the net free current threading any surface bounded by $\mathcal{C}$ (conduction current in windings and wires; bound currents explicitly excluded — they were folded into $\mathbf{H}$ 's definition by the previous article). Units check: $H$ in A/m times metres gives amperes — $\mathbf{H}$ 's circulation is a current count, which is why electromagnet engineers speak of "amp-turns" as naturally as volts. For a coil of $N$ turns carrying $I$ , the enclosed free current is $NI$ — each turn threads the loop once, and the law counts it once:

\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = NI \qquad (2)

where: the product $NI$ (amp-turns) is the magnetomotive force of engineering practice — the entire drive available to push $\mathbf{H}$ around a magnetic circuit, a reading the gapped-electromagnet flagship will make quantitative. Note what is absent from both equations: $\mu_0$ , $\chi_m$ , $\mu$ , and every other material symbol. The law is constitutive-free; that is its whole charm.

Derivation AMinting the integral law — Stokes plus one honesty clause

The goal is equation (1) from the differential law of the previous article, which takes one honest step, plus the two clauses that govern its use: why the answer cannot depend on which spanning surface counts the current, and why the law nevertheless underdetermines $\mathbf{H}$ . The route is Stokes's theorem (E0, recalled), the same minting that turned $\nabla\times\mathbf{B} = \mu_0\mathbf{J}$ into the integral Ampère's law two articles' worth of magnetostatics ago — the scope clauses are where the content lives.

Step 1. Integrate $\nabla\times\mathbf{H} = \mathbf{J}_f$ over any open surface $\mathcal{S}$ spanning a closed loop $\mathcal{C}$ , and apply Stokes's theorem to the left side:

\oint_{\mathcal{C}}\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = \int_{\mathcal{S}}(\nabla\times\mathbf{H})\cdot\mathrm{d}\mathbf{a} = \int_{\mathcal{S}}\mathbf{J}_f\cdot\mathrm{d}\mathbf{a} = I_{f,\rm enc}.

Why only free current appears: the bound current $\nabla\times\mathbf{M}$ was absorbed into $\mathbf{H}$ 's definition — this line is the cash value of that construction.

Step 2. Certify surface-independence. Two surfaces spanning the same loop bound a closed volume; the difference of their current counts is $\oint\mathbf{J}_f\cdot\mathrm{d}\mathbf{a}$ over that closed volume, which vanishes because steady free currents are divergence-free, $\nabla\cdot\mathbf{J}_f = 0$ (the Continuity Equation's steady-state clause). So "enclosed current" is well defined — any spanning surface gives the same $I_{f,\rm enc}$ . (The famous failure of this clause between capacitor plates is a time-dependent story, scheduled for the displacement current in the electrodynamics chapter.)

Step 3. State the insufficiency clause. Equation (1) is one scalar statement per loop about a three-component field; it constrains only the curl-carrying part of $\mathbf{H}$ . The divergence-sourced part — nonzero wherever $\nabla\cdot\mathbf{M} \neq 0$ , per the previous article — circulates nothing and is invisible to (1). Formally: add to any solution a pole field $-\nabla W$ and every circulation audit still passes. The law therefore determines $\mathbf{H}$ only when symmetry guarantees the invisible part vanishes along the loop — which requires the magnetization's geometry, not merely the current's, to respect the symmetry. That double demand is checked explicitly, case by case, in the catalog below.

The catalog

The filled solenoid and toroid — H is material-blind

An infinite solenoid, $n$ turns per metre carrying $I$ , is filled with a uniform linear magnetic core. This is the geometry where every clause of the ritual can be watched operating, so it earns a full derivation; the toroid follows by the same logic bent into a ring.

Go deeper · Derivation B · 2 minFilled solenoid — two-line H, one-line B, and the bound-current dividend

Derivation BFilled solenoid — two-line H, one-line B, and the bound-current dividend

The goal is the complete ritual on the solenoid with a linear core: $\mathbf{H}$ from free current alone, $\mathbf{B}$ and $\mathbf{M}$ by conversion, the bound surface current harvested, and the audit against total-current Ampère closed exactly. The symmetry baton must be checked for both the winding and the magnetization — watch that step, it is where bar-magnet problems would disqualify themselves.

Step 1. Symmetry, both demands. The winding is translation- and rotation-symmetric, so any $\mathbf{H}$ it determines is axial, a function of $s$ only, and zero at infinity; the exterior $\mathbf{H}$ vanishes by the same rectangle-pair argument as the vacuum solenoid's (Ampère's Law, E4). The core's response: a uniform axial $\mathbf{M}$ has $\nabla\cdot\mathbf{M} = 0$ in the bulk and no end faces (infinite), so it sprays no poles and breaks no symmetry — the divergence-sourced part of $\mathbf{H}$ is identically zero, and the circulation law will deliver all of $\mathbf{H}$ . Baton one passes.

Step 2. Circulation. Run the standard rectangle, length $L$ , one long side inside, one outside:

\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = H\,L = I_{f,\rm enc} = nLI \qquad\Longrightarrow\qquad H = nI,

axial, uniform, and — the point of the whole construction — independent of the core. Iron, bismuth, or vacuum: $H = nI$ , set by the ammeter.

Step 3. Conversion, the material's one appearance. For a linear core, $\mathbf{M} = \chi_m\mathbf{H}$ (the constitutive law, quoted from Magnetic Susceptibility):

B = \mu_0(H + M) = \mu_0(1+\chi_m)\,nI, \qquad M = \chi_m nI.

A paramagnetic core nudges $B$ up by parts in $10^{5}$ ; transformer iron multiplies it by thousands (with the honesty flag that "linear" for iron holds only far from saturation — Linear and Nonlinear Magnetic Media owns that fine print).

Step 4. Harvest the dividend. The magnetized core is a uniformly magnetized cylinder, so it wears the surface bound current of Magnetization and Bound Currents: $K_b = M\hat{\boldsymbol{\phi}}\times\hat{\mathbf{s}}$ -wise, magnitude $K_b = M = \chi_m nI$ , circulating the same way as the winding for $\chi_m > 0$ . Read it physically: the core is a phantom solenoid of strength $\chi_m nI$ wound on top of the real one — paramagnetic cores reinforce, diamagnetic cores (tiny, opposite) buck.

Step 5. Audit with the physical field's law. Ampère for $\mathbf{B}$ counts total current per unit length, $nI + K_b$ :

B = \mu_0(nI + K_b) = \mu_0(nI + \chi_m nI) = \mu_0(1+\chi_m)\,nI

✓ — identical to Step 3. The two ledgers — free-current bookkeeping through $\mathbf{H}$ , total-current physics through $\mathbf{B}$ — close on the same field, which is the whole consistency of the chapter in one line.

The toroid transposes without new ideas, and is worth recording because it is the geometry magnetics labs actually wind (no ends, no demagnetizing poles, no stray field). For $N$ total turns on a ring core, the circular Amperian loop of radius $s$ inside the core gives

H = \frac{NI}{2\pi s}, \qquad B = \frac{\mu_0(1+\chi_m)NI}{2\pi s} \qquad (3)

where: $s$ — distance from the ring's central axis; the $1/s$ profile and the outside-is-zero verdict carry over verbatim from the vacuum toroid of the E4 catalog. The core's $\mathbf{M}$ runs in closed azimuthal circles — divergence-free everywhere, no end faces anywhere — which is precisely why the toroid passes baton one with honors and why every serious $B$ – $H$ measurement is toroidal: the sample generates no poles to contaminate the $H$ you think you applied.

The gapped electromagnet — where the amp-turn budget goes

Cut a thin slice of air, width $g$ , out of an iron ring (mean path length $\ell$ , relative permeability $\mu_r$ , cross-section uniform) and drive the winding with $NI$ amp-turns. This is the primal geometry of every electromagnet, relay, motor, and MRI yoke — and the cleanest showcase of the law running a budget.

Figure

The gapped ring — one flux all the way around, but the amp-turn budget belongs to the air

Go deeper · Derivation C · 2 minThe gapped ring — B continuous, H partitioned, the gap wins

Derivation CThe gapped ring — B continuous, H partitioned, the gap wins

The goal is the gap field $B$ as a function of $NI$ , $\ell$ , $\mu_r$ , $g$ , and the budget statement of how the drive divides between iron and air. The inputs: the circulation law around the ring's centerline, the continuity of $B^\perp$ across the iron–air faces (the one boundary rule this article borrows — from Boundary Conditions for the Magnetostatic Field, since $\nabla\cdot\mathbf{B} = 0$ makes normal $B$ continuous always; the full $\mathbf{H}$ -interface rules belong to Magnetic Boundary Conditions, next lane), and linearity of the iron. Watch the small-gap idealization: we neglect fringing, so the flux tube keeps constant cross-section and $B$ is one number all the way around.

Step 1. Circulation around the centerline. $\mathbf{H}$ runs along the path everywhere (azimuthal symmetry of the ring, gap thin enough not to break it), with one value $H_{\rm iron}$ in the metal and another $H_{\rm gap}$ in the air:

\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = H_{\rm iron}\,\ell + H_{\rm gap}\,g = NI.

Why two H values but one loop: the integrand is piecewise — the law happily sums segments; nothing requires $\mathbf{H}$ to be continuous, and indeed it will not be.

Step 2. One flux, one $B$ . The gap faces are perpendicular to the flux, and normal $B$ is continuous across any surface ( $\nabla\cdot\mathbf{B} = 0$ , pillbox): $B_{\rm iron} = B_{\rm gap} = B$ . Convert each segment's $H$ : $H_{\rm iron} = B/\mu_0\mu_r$ , $H_{\rm gap} = B/\mu_0$ .

Step 3. Solve the budget equation:

\frac{B}{\mu_0\mu_r}\,\ell + \frac{B}{\mu_0}\,g = NI \qquad\Longrightarrow\qquad B = \frac{\mu_0\,NI}{\dfrac{\ell}{\mu_r} + g}.

Read the denominator as the reluctance ledger: the iron path contributes its length demagnified by $\mu_r$ — half a metre of $\mu_r = 5000$ iron "costs" the same as $0.1$ mm of air — while the gap contributes at face value. Air is five thousand times more expensive per metre.

Step 4. Run the numbers. $N = 500$ , $I = 2.0\,\mathrm{A}$ , $\ell = 0.50\,\mathrm{m}$ , $\mu_r = 5000$ , $g = 2.0\,\mathrm{mm}$ :

B = \frac{(1.257\times10^{-6})(1000)}{1.0\times10^{-4} + 2.0\times10^{-3}} = 0.60\,\mathrm{T}.

Budget audit: $H_{\rm gap}\,g = (B/\mu_0)(0.002) = 952$ amp-turns; $H_{\rm iron}\,\ell = (B/\mu_0\mu_r)(0.5) = 48$ amp-turns; total $1000$ ✓ exactly. The 2 mm of air consumes 95% of the drive; the half-metre of iron takes 5%. The iron is not where the effort goes — it is the cheap highway that delivers the flux to the expensive crossing.

Step 5. The honesty flags. (i) Remove the gap and the formula predicts $B = \mu_0\mu_r NI/\ell = 12.6\,\mathrm{T}$ — nonsense, because real iron saturates near $1.5$ – $2$ T: the linear $\mu_r$ is a small-field tangent, not a law of nature (Ferromagnetism owns saturation; Linear and Nonlinear Magnetic Media owns the design consequences). The gapped formula survives precisely because the gap keeps $B$ below saturation. (ii) Fringing: flux bulges outward at the gap, effectively widening the tube; for $g$ much smaller than the core's transverse size the correction is small, and precision designs add the standard fringing allowances.

Limiting cases

Case	Result	Meaning
vacuum core	$\chi_m = 0$ : $B = \mu_0 nI$	the E4 solenoid recovered; ledgers coincide
linear core	$H$ unchanged, $B \times(1+\chi_m)$	material enters only at conversion
iron core, small field	$B$ amplified $\sim10^3$ – $10^4$	linear approximation of the virgin curve
iron core, large field	saturation: $M \to M_s$ , amplification dies	linearity honesty — Ferromagnetism
gap $g \to 0$	$B \to \mu_0\mu_r NI/\ell$ (until saturation)	the closed ring; check against $M_s$
gap $g \gg \ell/\mu_r$	$B \approx \mu_0 NI/g$	the gap owns the budget; iron nearly free
$I_f = 0$ (bar magnet)	law says $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = 0$ only	$\mathbf{H} \neq 0$ ; use the pole fiction instead
no symmetry	law true, silent	solve honestly, audit with (1) afterwards

Common trapThe law is always true — and usually insufficient

Every wrong answer this law has ever produced came from skipping baton one. The equation $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = I_{f,\rm enc}$ holds for every loop in every magnetostatic problem — but it determines $\mathbf{H}$ only when symmetry forces $\mathbf{H}$ to be constant-and-parallel along a loop you can draw, and that requires the magnetization's geometry to cooperate, not just the winding's. The bar magnet is the canonical prosecution: $I_f = 0$ , so the law reports zero circulation — and a student concludes $\mathbf{H} = 0$ while the magnet stands there full of pole-sourced $\mathbf{H}$ , invisible to every circulation audit because it is curl-free. Same trap, subtler dress: a straight wire through a magnetized block whose $\mathbf{M}$ ends on faces — the wire's symmetry is perfect, the block's poles break it, and $NI/2\pi s$ silently stops being the whole of $\mathbf{H}$ . The safe envelope: closed flux paths (toroids), infinite translation symmetry (solenoids, cylinders with axial or azimuthal $\mathbf{M}$ ), or explicitly divergence-free magnetization. Outside it: pole fiction or honest matching (previous article), with the circulation law demoted to a consistency check it will always, uselessly, pass.

Worked examples

1 · A ferrite toroid, the full ritual (confidence builder)

A toroidal core of linear ferrite ( $\mu_r = 400$ ) has mean radius $s = 12\,\mathrm{cm}$ and carries $N = 800$ turns at $I = 0.50\,\mathrm{A}$ . Run all four batons at the mean radius.

Step 1. Symmetry. Winding: azimuthal loops of free current, rotationally symmetric ⇒ $\mathbf{H}$ azimuthal, function of $s$ . Core response: $\mathbf{M}$ azimuthal, closed loops, $\nabla\cdot\mathbf{M} = 0$ , no faces ⇒ no poles, symmetry intact. Ritual licensed.

Step 2. Circulation, loop of radius $s$ : $H = NI/2\pi s = (800)(0.50)/(2\pi\times0.12) = 5.3\times10^{2}\,\mathrm{A/m}$ .

Step 3. Conversion: $B = \mu_0\mu_r H = (1.257\times10^{-6})(400)(531) = 0.27\,\mathrm{T}$ — comfortably below ferrite saturation ( $\approx 0.4$ T), so the linear rental is honest. Magnetization: $M = \chi_m H = (399)(531) = 2.1\times10^{5}\,\mathrm{A/m}$ .

Step 4. Harvest and audit. Bound surface current on the core: $K_b = M = 2.1\times10^{5}\,\mathrm{A/m}$ , wound poloidally like the copper — the ferrite is a phantom winding carrying $\chi_m = 399$ times the free amp-turns. Audit: total-current Ampère gives $B = \mu_0(1+\chi_m)NI/2\pi s = \mu_0(400)(531)$ — the same $0.27$ T ✓. $H = 531\,\mathrm{A/m}$ , $B = 0.27\,\mathrm{T}$ , $M = 2.1\times10^{5}\,\mathrm{A/m}$ — and the supply only ever knew about the $400$ amp-turns.

2 · Designing the gap field (exam level)

The electromagnet of the flagship derivation ( $N = 500$ , $\ell = 0.50\,\mathrm{m}$ , $\mu_r = 5000$ , $g = 2.0\,\mathrm{mm}$ ) must produce $B = 1.0\,\mathrm{T}$ in its gap for a beamline steering test. Find the required current, re-audit the budget, and check the design's honesty.

Step 1. Invert the flagship formula for amp-turns:

NI = \frac{B\left(\ell/\mu_r + g\right)}{\mu_0} = \frac{(1.0)(1.0\times10^{-4} + 2.0\times10^{-3})}{1.257\times10^{-6}} = 1.67\times10^{3}\ \text{amp-turns},

so $I = 1671/500 = \mathbf{3.3\,A}$ .

Step 2. Budget at the new operating point: $H_{\rm gap}\,g = (1.0/\mu_0)(0.002) = 1592$ amp-turns (95.2%); $H_{\rm iron}\,\ell = (1.0/\mu_0\times5000)(0.5) = 80$ amp-turns (4.8%); sum $1671$ ✓. The proportions are unchanged from the flagship — they depend only on the geometry ratio $g \,/\,(\ell/\mu_r)$ , not on the drive.

Step 3. Honesty checks. Iron flux density equals the gap's $1.0$ T (continuity), safely under saturation ( $\approx 2$ T for silicon steel): the $\mu_r = 5000$ rental remains valid ✓. Fringing at a 2 mm gap in a centimetre-scale pole: percent-level, acceptable for a steering test.

Step 4. Read the design lesson. Doubling the gap to 4 mm would nearly double the required current ( $NI \to 3263$ ); halving it to 1 mm nearly halves it ( $NI \to 875$ ). In a gapped magnet, the air is the load — which is why motor and relay designers fight for every tenth of a millimetre of clearance, and why an MRI magnet's bore (a gap measured in tens of centimetres) drove the field into superconducting coils where amp-turns are, at last, cheap.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.3 · Jackson, Classical Electrodynamics, 3rd ed., §5.8 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, §13.3. Key constants and worked results: $\mu_0=4\pi\times10^{-7}\,$ T m/A; the Stokes-theorem step and the limiting reductions ( $\mathbf{M}\to0$ and $\chi_m\to0$ ) all hold. Differential-law construction and boundary conditions are treated in The Auxiliary Field H and Magnetic Boundary Conditions respectively. Worked numbers: ferrite toroid ( $N=800$ , $I=0.5$ A, $s=0.12$ m, $\mu_r=400$ ): $H = 531$ A/m, $B = 0.267$ T, $M = K_b = 2.12\times10^{5}$ A/m, audit closes; gapped ring ( $N=500$ , $I=2$ A, $\ell=0.5$ m, $\mu_r=5000$ , $g=2$ mm): $B = 0.598$ T, budget $952 + 48 = 1000$ amp-turns (95.2% to the gap), no-gap linear value $12.57$ T vs saturation $\approx 2$ T; design inverse $NI = 1671$ ⇒ $I = 3.34$ A for $B = 1.0$ T.

Builds on: Magnetization · Bound Currents · The Auxiliary Field H · Magnetic Dipole Moment of Atoms

Leads to: Magnetic Permeability · Magnetic Susceptibility · Magnetic Boundary Conditions · Linear and Nonlinear Magnetic Media

Why does the iron core not change H, when it changes B by a factor of thousands?

Because $\mathbf{H}$ was engineered to be deaf to the material. The core's contribution to the real field is carried entirely by bound currents, and those were folded into $\mathbf{H}$ 's definition — its circulation counts only what your supply drives. In the solenoid the geometry is kind enough that this circulation statement pins all of $\mathbf{H}$ , so $H = nI$ stands whatever you insert. The physics has not disappeared: the iron's thousands-fold response is sitting in $\mathbf{M} = \chi_m\mathbf{H}$ and reappears the moment you convert to $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$ . One warning: this material-blindness is a property of symmetric problems, not of $\mathbf{H}$ in general — insert a core with end faces (poles!) and $\mathbf{H}$ itself shifts, because the divergence-sourced part wakes up.

What exactly goes wrong if I use the loop trick on a horseshoe electromagnet without a formula for the gap?

The horseshoe's iron ends at two pole faces, where $\mathbf{M}$ terminates and sprays effective magnetic charge. That pole field is curl-free, so it adds to $\mathbf{H}$ without registering in any circulation — your loop equation $H_{\rm iron}\ell + H_{\rm gap}g = NI$ remains true but no longer involves just two unknowns with a clean geometry: the field fans out between the poles, fringes heavily, and varies along the path. The gapped-ring flagship dodges this because its "poles" are the two gap faces held millimetres apart with flux crossing straight over — the pole fields are exactly what the $B$ -continuity step accounts for. Rule of thumb: the smaller the gap relative to the pole-face size, the more your magnet is a "ring with a slice" and the better the budget formula; the wider the gap, the more it is two poles in open air, and you graduate to the pole fiction or a field solver.

Is magnetomotive force a real force? The magnetic-circuit analogy feels like it is cheating.

It is a line integral wearing a costume, and the costume is honest as long as you remember what is underneath. The analogy maps $NI \leftrightarrow$ EMF, flux $\Phi \leftrightarrow$ current, and reluctance $\ell/\mu A \leftrightarrow$ resistance — and it works because $\oint\mathbf{H}\cdot\mathrm{d}\boldsymbol{\ell} = NI$ around a flux loop has the same structure as Kirchhoff's voltage law around a circuit, with $\nabla\cdot\mathbf{B} = 0$ playing current conservation. Nothing is being pushed by a "force"; no energy is dissipated in a reluctance. The analogy's real failure modes are physical: leakage flux (field lines that skip the iron path), fringing at gaps, and saturation (reluctance becomes flux-dependent) — all handled in practice by correction factors or by abandoning the circuit picture for a field solver.

In the gap, H jumps by a factor of 5000 while B stays the same. What is physically different between the two regions?

The magnetization is. $B$ is one honest flux density crossing the boundary ( $\nabla\cdot\mathbf{B} = 0$ tolerates no jump in the normal component), but that flux is composed differently on the two sides: in the iron, almost all of $B/\mu_0$ is $M$ — massed atomic moments — with a whisper of $H$ ; in the air there is no matter, $M = 0$ , and the whole of $B/\mu_0$ must be carried by $H$ alone. The jump in $H$ is exactly the jump in $-M$ across the face — which is also why the gap dominates the amp-turn budget: your winding pays for $H$ , and only the air demands $H$ in bulk.

Why do measurement labs insist on toroidal samples instead of the much easier bar shape?

Poles. A bar sample magnetized along its length ends at two faces; the resulting demagnetizing field (previous article) subtracts from whatever $H$ your solenoid applies, so the field inside the sample — the one the material actually responds to — is neither known nor uniform: $H_{\rm in} = H_{\rm applied} - NM$ with a shape factor $N$ you can only estimate. Toroids have no ends: $\mathbf{M}$ closes on itself, $\nabla\cdot\mathbf{M} = 0$ , zero poles, and $H = NI/2\pi s$ inside the material to the accuracy of your ammeter and tape measure. Every point of a published $B$ – $H$ curve implicitly says "measured on a closed ring, poles engineered away." Bar-sample measurements exist, but they arrive with demagnetization corrections — and the corrections are the previous article's shape trio doing penance.

Does the surface I choose to count the enclosed free current ever matter?

For magnetostatics, never — that is Step 2 of the minting derivation. Two spanning surfaces differ by a closed volume, and steady currents have $\nabla\cdot\mathbf{J}_f = 0$ , so what flows in flows out and both surfaces count the same $I_{f,\rm enc}$ . The classic apparent paradox is a wiggly surface that dodges between a solenoid's turns versus a flat one that slices the winding $N$ times: count carefully and each turn pierces any spanning surface a net once. The escape clause is time dependence: charge a capacitor through your loop and the two surfaces genuinely disagree — that scandal, and the displacement-current term that resolves it, is the electrodynamics chapter's opening act.

The ferrite example rents χ_m = 399 — where does that number actually come from?

From the next two articles, and ultimately from the microscopic physics of the chapter's opening. Magnetic Susceptibility defines $\chi_m$ as the constitutive dial $M = \chi_m H$ and catalogs measured values with their signs and temperature fingerprints; Magnetic Permeability packages it as $\mu = \mu_0(1+\chi_m)$ ; and the mechanisms that set the magnitude — induced orbital response for diamagnets, thermal alignment of permanent moments for paramagnets, cooperative ordering for ferro- and ferrimagnets like your ferrite — are owned by Diamagnetism, Paramagnetism, and Ferromagnetism. Within this article, $\chi_m$ is honestly a rented constant: the ritual works for any value, which is exactly the point of postponing the material to baton three.