Magnetic Boundary Conditions

The magnetostatics of vacuum ended with a complete set of crossing rules: at a current sheet, $B^\perp$ glides through untouched while $\vec B^\parallel$ jumps by $\mu_0 K$ . Then the chapter filled space with matter, and the rules need renegotiating — not because the old ones fail (they hold verbatim, bound currents included), but because at a material surface the bound current $\vec K_b=\vec M\times\hat n$ is precisely the thing you don't know yet. Writing the tangential rule with $\vec B$ and total current is true and useless; the chicken-and-egg escape is the same one that motivated $\vec H$ in the first place. This article derives the media crossing rules: $B^\perp$ continuous always, $\vec H^\parallel$ jumping only by the free surface current — usually zero — and, as the fine print, $H^\perp$ jumping by the magnetization mismatch, the boundary-value voice of $\nabla\cdot\vec H=-\nabla\cdot\vec M$ . Two dividends follow. For linear media the rules combine into a refraction law, $\tan\theta_1/\tan\theta_2=\mu_1/\mu_2$ , with a consequence worth staring at: field lines meeting iron at almost any angle are bent nearly parallel to the surface inside it, and exit the far side essentially perpendicular. Iron does not merely respond to flux — it collects and pipes it, which is why transformer cores, recording heads, and mu-metal shields work at all. Second, the rules resurrect the demagnetizing field quantitatively: the shape factors $N$ (slab 1, sphere $\tfrac13$ , needle 0) that decide how much of a material's susceptibility you actually measure — for soft iron shaped into a sphere, a $\chi$ of a thousand masquerades as three. The time-dependent upgrade of these rules — where induced fields join the ledger — waits for the electrodynamic boundary conditions at this chapter's far end. Here: static interfaces, magnetized matter, and the exact geometry of the jump.

Key takeaways

$B^\perp$ is continuous across every interface — vacuum, magnet, iron, superconductor — because $\nabla\cdot\vec B=0$ admits no magnetic surface charge. This rule crossed unchanged from the vacuum chapter.
The workhorse: $\vec H^\parallel_{1}-\vec H^\parallel_{2}=\vec K_f\times\hat n$ — the tangential jump of $\vec H$ is set by the free surface current alone. At an ordinary material interface $\vec K_f=0$ and $\vec H^\parallel$ is continuous even though $\vec B^\parallel$ jumps by the (unknown-in-advance) bound current.
The fine print: $H^\perp_1-H^\perp_2=-(M^\perp_1-M^\perp_2)$ — normal $\vec H$ jumps wherever magnetization ends, the boundary form of $\nabla\cdot\vec H=-\nabla\cdot\vec M$ .
Linear media: $\tan\theta_1/\tan\theta_2=\mu_1/\mu_2$ . Into iron ( $\mu_r\sim5000$ ), a line at 1° from the normal in air bends to 89.3° inside; leaving iron, lines exit almost exactly normal. High- $\mu$ material gathers flux and pipes it.
Shape rules the measurement: a uniformly magnetized body generates $\vec H_{\rm in}=-N\vec M$ (demagnetizing field), and a measured susceptibility is $\chi_{\rm meas}=\chi/(1+N\chi)$ — negligible correction for para/diamagnets, everything for soft ferromagnets.
KEY IDEA: $B^\perp$ continuous (no monopoles, no exceptions); $\vec H^\parallel$ jumps only by free surface current — so at bare material interfaces both are continuous, and those two continuities are the glue of every magnetostatic boundary-value problem.

The core insight

$B^\perp$ is continuous always; $\vec H^\parallel$ jumps by $\vec K_f\times\hat n$ — free current only, because $\vec H$ was built to hide the bound currents that make $\vec B^\parallel$ 's jump incomputable in advance.

Where this lives today

Magnetic recording lives and dies by these rules. A hard-drive write head is a flux pipe engineered to microns: a high- $\mu$ yoke gathers the write coil's flux (refraction law), pipes it to a nanoscale gap, and the fringing field at the gap — shaped entirely by the crossing rules between yoke, gap, and medium — flips bits in a granular film chosen for its own boundary-condition properties. Modern perpendicular recording added a soft underlayer beneath the medium precisely to close the magnetic circuit through the bit being written: a boundary-conditions redesign, worth an order of magnitude in areal density when it shipped in the mid-2000s. At the other end of the scale, magnetically shielded rooms for MEG (magnetoencephalography — femtotesla brain fields) and for atomic-clock and fundamental-physics laboratories are nested mu-metal shells, sometimes seven layers deep, each layer a working instance of the flux-gathering limit; the record facilities suppress ambient fields by factors beyond $10^6$ . Their design formula — the shell suppression factor with its $\mu t/r$ scaling — belongs to the linear-media article's sphere machinery, but every qualitative decision (why walls must overlap at seams, why holes for cables are piped through high- $\mu$ tubes, why layers are separated by air gaps) is read directly off the refraction law and the continuity pair. And in every finite-element magnetics code — the daily tool of motor, transformer, and MRI-magnet designers — equations (1) and (3) are literally the interface conditions the solver imposes between mesh regions; the crossing rules are not used by the software so much as they are the software's contract between elements.

Origins · 1 minArnold Sommerfeld and how this was discovered

Arnold Sommerfeld

1868–1951 · Munich — the theorist who made boundary conditions a discipline

Sommerfeld ran the most productive school of theoretical physics of the twentieth century — Heisenberg, Pauli, Debye, Bethe, and Pauling trained in his Munich seminar, and he was nominated for the Nobel Prize eighty-four times without winning. His own fingerprints are on every corner of this article's subject. His 1896 solution of diffraction by a half-plane was the first rigorous boundary-value solution of Maxwell's equations, an existence proof that interfaces could be treated exactly rather than by hand-waving; his radiation condition disciplined what "boundary condition at infinity" means; and his Lectures on Theoretical Physics — six volumes distilled from four decades of teaching, written out in his seventies — fixed the pillbox-and-loop presentation of interface conditions that this article and every modern textbook run verbatim. Sommerfeld's motto for the trade was that the special problems, worked honestly, teach the general theory: the magnetized sphere, the refracting field line, and the shielded cavity were all seminar staples. He also gave physics the fine-structure constant and the first quantum corrections to the Bohr atom — but it is in the grammar of boundary-value electromagnetism that working physicists quote him daily, usually without knowing it.

History

1824

— Poisson's magnetism memoir introduces surface distributions of magnetic "matter" — pole sheets as the first interface bookkeeping.

1873

— Maxwell's Treatise states the continuity of normal induction and tangential force at magnetized surfaces; the modern pair in Victorian dress.

1885

— Heaviside's vector recasting makes the pillbox/loop jump conditions explicit working tools.

1896

— Sommerfeld solves half-plane diffraction: the first exact boundary-value solution of the field equations, boundary conditions elevated to rigorous mathematics.

1933

— Meissner and Ochsenfeld find superconductors expelling flux: a material that enforces its own boundary condition, B-perpendicular zero at the surface.

2005

— Perpendicular magnetic recording ships: a soft underlayer redesigns the write head's magnetic circuit, boundary conditions buying an order of magnitude in storage density.

The physical picture

Start from what is already banked. The vacuum crossing rules — derived by pillbox and thin loop where this track met its first current sheet — say $B^\perp$ continuous, $\vec B^\parallel$ jumping by $\mu_0\vec K\times\hat n$ with $\vec K$ the total surface current. Those statements are exact at a magnetized surface too: a bar magnet's skin carries a genuine bound sheet $\vec K_b=\vec M\times\hat n$ , and $\vec B$ jumps across it exactly as the vacuum rule demands. So why a new article? Because at a material boundary the total-current rule has the logical structure of a trap. You want the field; the field's jump needs $\vec K_b$ ; $\vec K_b$ needs $\vec M$ ; and in any material that responds to fields, $\vec M$ needs the field. The tangential- $\vec B$ condition is true, and circular. The way out is the same bookkeeping swap that built $\vec H$ . Since $\nabla\times\vec H=\vec J_f$ involves free current only — the current you dial, the current you know — the thin-loop argument run on $\vec H$ produces a jump condition with only knowns on the right-hand side. At the vast majority of interfaces (magnet–air, core–gap, iron–iron) no free sheet flows, and the condition collapses to a clean continuity: $\vec H^\parallel$ matches. Pair it with $B^\perp$ 's unconditional continuity and you have two computable glue conditions, one from each field — exactly the pattern the dielectric chapter established with $D^\perp$ and $\vec E^\parallel$ , with the normal/tangential roles swapped. The swap is not decoration; it is the deep grammar of the two theories. Electric fields begin on charges, so the free-source condition lives in the perpendicular component ( $D^\perp$ jumps by $\sigma_f$ ); magnetic fields circulate around currents, so the free-source condition lives in the tangential one ( $\vec H^\parallel$ jumps by $K_f$ ).

Figure

The media crossing kit — one pillbox and one loop per field, four rules, two of them computable in advance

One more piece of vacuum kit crosses the door intact: the instruments themselves. The pillbox and the thin loop are local probes; they shrink to zero size, every smooth surface flattens in that limit, and so every rule below holds pointwise on curved interfaces — magnet poles, toroidal cores, shield shells — with $\hat n$ and the fields read locally. What changes is only which law we feed the instruments.

The crossing rules for magnetized media

Conventions for everything that follows: $\hat n$ points from medium 2 into medium 1; superscripts $\perp$ and $\parallel$ are components along and transverse to $\hat n$ ; $\vec K_f$ is the free surface current (A/m), and each medium carries its own magnetization $\vec M_i$ just inside the surface.

Derivation AThe media rules — one pillbox and one loop, with H hiding the bound currents

The goal is the complete set of magnetostatic crossing rules at a material interface: $B^\perp$ continuous, $\vec H^\parallel$ jumping by the free surface current only, and the less-advertised normal- $\vec H$ condition that follows from $\vec H$ 's nonzero divergence. The strategy deliberately reuses the vacuum instruments — this derivation's interest is not the geometry (that was settled at the current-sheet article) but which law goes into which instrument, because that choice is what turns a true-but-circular statement into a computable one. Watch the right-hand sides: everything hangs on what survives the thin-limit squeeze.

Step 1. Pillbox on the divergence laws. Straddle the surface with a thin cylinder, faces of area $a$ , wall height $\epsilon\to0$ . Feeding it $\nabla\cdot\vec B=0$ repeats the vacuum result verbatim — no magnetic surface charge exists to be enclosed, in matter or out of it:

B^\perp_1=B^\perp_2 . \qquad (1)

where: both components are taken along $\hat n$ at the same surface point. This rule is unconditional — it survived the passage into matter untouched because matter contains currents, not monopoles. But the same pillbox fed the auxiliary field's divergence law, $\nabla\cdot\vec H=-\nabla\cdot\vec M$ , encloses the "magnetic surface charge" of the pole fiction, $\sigma_m=-\,\Delta(\vec M\cdot\hat n)$ across the interface, and returns

H^\perp_1-H^\perp_2=-\left(M^\perp_1-M^\perp_2\right) . \qquad (2)

where: $M^\perp_i=\vec M_i\cdot\hat n$ . Normal $\vec H$ jumps wherever magnetization terminates — at a magnet's flat pole face ( $M^\perp$ drops from $M$ to $0$ ), $H^\perp$ steps by $M$ . Equation (2) is the boundary-value voice of the fine print the auxiliary-field article prosecuted: $\vec H$ is not divergence-free, and its divergence is fed by the ends of $\vec M$ .

Step 2. Loop on the curl law — the computable side. Now the thin rectangle: long sides $\ell$ parallel to the surface, short sides $\epsilon\to0$ . Fed $\nabla\times\vec H=\vec J_f$ via Stokes, the circulation reads the tangential mismatch while the spanning ribbon catches only free sheet current, $I_{f,\rm enc}=K_{f}\,\ell$ (bound currents are absent from this law by construction — that was the entire point of minting $\vec H$ ). Running the loop in the two independent tangential orientations, exactly as the vacuum derivation did, and assembling:

\vec H^\parallel_1-\vec H^\parallel_2=\vec K_f\times\hat n \qquad\Longleftrightarrow\qquad \hat n\times\left(\vec H_1-\vec H_2\right)=\vec K_f . \qquad (3)

where: the jump is tangential, perpendicular to $\vec K_f$ , with the component of $\vec H^\parallel$ along the free current continuous. At the overwhelmingly common interface — magnet–air, core–gap, iron against iron — nobody has installed a free sheet, $\vec K_f=0$ , and (3) collapses to the working continuity $\vec H^\parallel_1=\vec H^\parallel_2$ .

Step 3. Audit against the total-current rule. The vacuum rule $\vec B_1-\vec B_2=\mu_0\vec K_{\rm tot}\times\hat n$ must be — and is — the same physics. Write $\vec B_i=\mu_0(\vec H_i+\vec M_i)$ and subtract: the tangential part gives $\mu_0(\Delta\vec H^\parallel+\Delta\vec M^\parallel)=\mu_0(\vec K_f+\vec K_b)\times\hat n$ , and the magnetization mismatch supplies precisely the bound sheet, $\Delta\vec M^\parallel$ across the surface being $(\vec K_{b,\rm net})\times\hat n$ with $\vec K_{b,\rm net}=(\vec M_1-\vec M_2)\times\hat n\cdot(-1)$ — the two descriptions are one, cell for cell. Nothing was gained in content; everything was gained in usability, because (1) and (3) contain no unknown sources. Why the electric swap is exact and not a mnemonic: $D^\perp$ jumps by $\sigma_f$ because free charge is a divergence-side source; $\vec H^\parallel$ jumps by $K_f$ because free current is a curl-side source. Each auxiliary field carries the free-source condition in the component its own law governs, and each fundamental field ( $\vec E^\parallel$ , $B^\perp$ ) carries an unconditional continuity where nature supplies no source at all.

The pair to memorize is the computable pair: $B^\perp$ and $\vec H^\parallel$ continuous (bare interface). The pair that jumps — $\vec B^\parallel$ and $H^\perp$ — jumps by amounts the material decides after the problem is solved. Every magnetostatic boundary-value method in this chapter is a scheme for imposing the first pair; in scalar-potential language ( $\vec H=-\nabla W$ where $\vec J_f=0$ ) they read: $W$ continuous, and $\mu\,\partial W/\partial n$ continuous for linear media — the exact grammar of the dielectric matching problems, ported.

The refraction law — how iron pipes flux

For linear media the two continuities talk to each other through $\vec B=\mu\vec H$ , and their conversation is a geometry statement. Let a field line cross from medium 1 into medium 2 with no free sheet on the interface, making angles $\theta_1$ , $\theta_2$ with the normal.

Go deeper · Derivation B · 2 minField-line refraction and the flux-gathering limit

Derivation BField-line refraction and the flux-gathering limit

The goal is the refraction law $\tan\theta_1/\tan\theta_2=\mu_1/\mu_2$ and its high-contrast reading — the single most consequential picture in applied magnetostatics: high- $\mu$ material swallows field lines and runs them nearly parallel to its own surface. The strategy is two lines of algebra on the two continuities; the payoff is all in the limits, so watch what happens as $\mu_2/\mu_1$ reaches the thousands that soft iron actually delivers.

Step 1. Impose the pair. Normal components: $B_1\cos\theta_1=B_2\cos\theta_2$ , from (1). Tangential components: $H^\parallel$ continuous means $(B_1/\mu_1)\sin\theta_1=(B_2/\mu_2)\sin\theta_2$ , from (3) with $\vec K_f=0$ and the linear constitutive law spent per medium.

Step 2. Divide. The unknown magnitudes cancel:

\frac{\tan\theta_1}{\tan\theta_2}=\frac{\mu_1}{\mu_2} . \qquad (4)

where: angles are measured from the normal, so $\theta\to90^\circ$ means grazing along the surface. Field lines bend away from the normal in the higher- $\mu$ medium — Snell's law's cousin, with two structural differences worth registering: it is the tangents (not sines) that scale, and the bending is typically enormous, because $\mu_2/\mu_1$ for iron against air is $10^3$ – $10^4$ where optical index ratios struggle to reach $2$ .

Step 3. The high-contrast limits, both directions. Entering iron ( $\mu_2=\mu_r\mu_0$ , $\mu_r=5000$ ): a line arriving just $1^\circ$ off the normal in air continues at $\tan\theta_{\rm iron}=5000\tan1^\circ\approx87$ , i.e. $\theta_{\rm iron}\approx89.3^\circ$ — inside the iron the line runs essentially parallel to the surface, whatever it was doing outside. Leaving iron: a line at $45^\circ$ inside exits into air at $\tan\theta_{\rm air}=\tan45^\circ/5000$ , $\theta_{\rm air}\approx0.011^\circ$ — B emerges from a high- $\mu$ surface almost exactly along the normal, the magnetic twin of "E leaves a conductor perpendicular." Iron surfaces are to $\vec B$ nearly what equipotential surfaces are to $\vec E$ .

Step 4. Read it as engineering. Combine the two limits and a high- $\mu$ bar in an ambient field becomes a flux pipe: lines dive in normal at one region, run along the material's interior (the tangential- $B$ inside is enhanced — $H^\parallel$ matching means $B^\parallel_{\rm in}=\mu_r B^\parallel_{\rm out}$ ), and exit normal elsewhere. Wrap the pipe around a cavity and the ambient flux is guided around the protected volume instead of through it: magnetic shielding. How much the interior field is suppressed — the famous $\sim\mu t/\mu_0 r$ shell factor of mu-metal design — is the linear-media article's business, delivered by its sphere-and-shell machinery; this derivation owns the mechanism, and the mechanism is (4). Why the tangents and not the sines: light refracts by matching wave phase along the interface — a kinematic condition on wavefronts. Field lines refract by conserving flux ( $B\cos\theta$ ) while matching circulation ( $H\sin\theta$ ) — a statics condition on sources. Same geometry vocabulary, entirely different physics; the tangent form is the fingerprint of the static origin. *The computed limits are $\tan^{-1}(5000\tan1^\circ)=89.34^\circ$ and $\tan^{-1}(\tan45^\circ/5000)=0.0115^\circ$ ; the conductor analogy is exact in the $\mu_r\to\infty$ limit, where the iron surface becomes a surface of constant scalar potential $W$ .*

Figure

Flux gathering — a high-permeability slab in an ambient field collects lines, pipes them, and releases them normal to its far face

Shape strikes back — demagnetizing factors

The crossing rules have one more piece of unfinished chapter business: the demagnetizing field, met twice already as a bare fact — the magnetized sphere's interior $\vec H_{\rm in}=-\vec M/3$ , and the susceptibility article's warning that a measured $\chi$ can be a property of the shape rather than the material. Both are boundary-condition phenomena, and the rules just derived let us own them.

Go deeper · Derivation C · 3 minDemagnetizing factors and the shape-corrected susceptibility

Derivation CDemagnetizing factors and the shape-corrected susceptibility

The goal is the working demagnetization kit: the factor $N$ in $\vec H_{\rm in}=-N\vec M$ for the three canonical shapes, its trace identity, and the correction formula $\chi_{\rm meas}=\chi/(1+N\chi)$ that decides when a magnetometer reads the material and when it reads the geometry. The strategy is to work the extreme shapes directly from the crossing rules — that is the derivation's point, that $N$ is nothing but equations (1)–(3) expressed per geometry — then assemble the correction by self-consistency, the same feedback bookkeeping the dielectric chapter ran for its cavity fields.

Step 1. The slab, magnetized through its faces. Take an infinite slab, uniform $\vec M$ perpendicular to its faces, no free current anywhere. Outside: far above and below an infinite slab all sources look like two opposite, uniform pole sheets ( $\sigma_m=\pm M$ where $\vec M$ terminates, per equation (2)), the exact magnetic image of the charged-capacitor geometry — their fields cancel outside, so $\vec H_{\rm out}=0$ and $\vec B_{\rm out}=0$ . Inside, apply the rules across a face: $B^\perp$ continuous gives $B_{\rm in}=B_{\rm out}=0$ , hence

\vec B_{\rm in}=0, \qquad \vec H_{\rm in}=\frac{\vec B_{\rm in}}{\mu_0}-\vec M=-\vec M \qquad\Rightarrow\qquad N_{\rm slab}^{\perp}=1 .

where: the entire magnetization is spent fighting itself — a flat plate magnetized through its thin dimension generates the maximum possible demagnetizing field. (This is the precise magnetic twin of the polarized slab, whose $\vec E_{\rm in}=-\vec P/\varepsilon_0$ and $\vec D=0$ the dielectric chapter exhibited; the roles of the vanishing field swap, $D\leftrightarrow B$ .)

Step 2. The needle and the sphere. A long needle magnetized along its axis: the pole sheets are exiled to the far ends, their $\vec H$ at the waist falls off as end-effects, and the crossing rule that matters is the tangential one along the flank — $\vec H^\parallel$ continuous with $\vec H_{\rm out}\to0$ gives $\vec H_{\rm in}\to0$ : $N_{\rm needle}^{\rm axial}=0$ , $\vec B_{\rm in}=\mu_0\vec M$ — none of the magnetization is wasted. The sphere sits exactly between the extremes: the field-of-a-magnetized-object article solved it outright, $\vec H_{\rm in}=-\vec M/3$ , so $N_{\rm sphere}=\tfrac13$ — and the crossing rules audit it: at the pole, $H^\perp$ jumps by $-\Delta M^\perp=+M$ (from $-M/3$ inside to $+2M/3$ outside ✓, the dipole's polar value); at the equator, $\vec H^\parallel$ is continuous ( $-M/3$ both sides ✓). For a general ellipsoid, uniform $\vec M$ gives uniform $\vec H_{\rm in}=-\mathsf N\vec M$ with shape factors along the principal axes obeying the trace identity

N_x+N_y+N_z=1 ,

where: the slab ( $0,0,1$ ), needle ( $\tfrac12,\tfrac12,0$ ), and sphere ( $\tfrac13,\tfrac13,\tfrac13$ ) are the identity's three famous rows. The identity itself is the pole-fiction twin of the depolarization sum rule the Lorentz-cavity article exhibited for dielectrics — same potential theory, relabeled.

Step 3. The feedback loop — measured versus intrinsic susceptibility. Now put a linear body ( $\vec M=\chi\vec H_{\rm in}$ , demagnetizing factor $N$ along the applied direction) into an applied field $\vec H_0$ . The field inside is the applied field minus the self-generated demagnetizing field:

\vec H_{\rm in}=\vec H_0-N\vec M =\vec H_0-N\chi\vec H_{\rm in} \qquad\Rightarrow\qquad \vec H_{\rm in}=\frac{\vec H_0}{1+N\chi} ,

and the magnetometer, which knows only what it applied, reports

\chi_{\rm meas}\equiv\frac{M}{H_0}=\frac{\chi}{1+N\chi} . \qquad (5)

where: the correction term $N\chi$ is the whole story. For diamagnets and paramagnets, $|\chi|\lesssim10^{-3}$ : the correction is parts in $10^4$ or less and shape is irrelevant — the Gouy-balance numbers of the susceptibility article needed no shape correction, and now we know why. For a soft ferromagnet with $\chi\sim10^3$ measured on a sphere: $\chi_{\rm meas}=1000/(1+333)\approx3.0$ — the reading is entirely geometry, within half a percent of the shape-saturated limit $1/N=3$ . Measuring the material demands the needle geometry (toroids and long strips in practice, $N\to0$ ), which is precisely how honest permeability tables are produced. Why the shape-saturated limit $\chi_{\rm meas}\to1/N$ is a feature, not a failure: when $N\chi\gg1$ the body arranges its magnetization so that $\vec H_{\rm in}\to0$ — the magnetic mirror of a conductor screening $\vec E$ . The sample has become a flux-gathering object whose response is capped by how much demagnetizing field its own shape can generate; the material's greediness beyond that cap is invisible from outside. *The sphere pole/equator audits close against the magnetized-sphere solution ( $\vec B_{\rm in}=2\mu_0\vec M/3$ , exterior dipole); equation (5) at $\chi=10^3$ , $N=\tfrac13$ gives $2.99$ , and at $\chi=2.2\times10^{-5}$ (aluminum) the correction is $7\times10^{-6}$ relative — negligible.*

Limiting cases

case	result	meaning
bare interface, $\vec K_f=0$	$B^\perp$ and $\vec H^\parallel$ both continuous	the computable glue pair — the default for every material boundary
free sheet present (wound coil surface)	$\vec H^\parallel$ jumps by $\vec K_f\times\hat n$	the driven case: windings on a core, sheet conductors
$\vec M_1=\vec M_2=0$	rules collapse to the vacuum current-sheet set	consistency with the magnetostatic-field boundary article ✓
$\mu_2\gg\mu_1$	lines inside 2 run at $\theta_2\to90^\circ$ ; $\vec B$ in medium 1 meets the surface normally	flux piping; iron surface ≈ magnetic equipotential
slab ⊥-magnetized	$N=1$ : $\vec B_{\rm in}=0$ , $\vec H_{\rm in}=-\vec M$	maximal self-defeat; twin of the polarized slab with $D\leftrightarrow B$
needle ∥-magnetized	$N=0$ : $\vec H_{\rm in}=0$ , $\vec B_{\rm in}=\mu_0\vec M$	zero demagnetization — the honest-measurement geometry
$N\chi\gg1$	$\chi_{\rm meas}\to1/N$	reading saturates at pure geometry; material greediness hidden
time-varying fields	tangential rules gain induced-field terms	deferred by name to the electrodynamic boundary conditions (E6)

Common trapThe swap is exact, and so is the temptation to invert it — B-parallel and H-perpendicular are the components that jump

The single most reliable exam error at magnetic interfaces is importing the electric pattern unswapped. Continuous here: $B^\perp$ (always) and $\vec H^\parallel$ (bare interfaces). Jumping here: $\vec B^\parallel$ (by the bound sheet you don't know in advance) and $H^\perp$ (by the magnetization mismatch, equation (2)). Writing " $\vec B^\parallel$ continuous" or " $H^\perp$ continuous" imports a monopole or deletes a pole sheet, and the resulting boundary-value solution fails its own audit. Second trap, subtler: the $\vec K$ in the $\vec H$ jump condition is the free current only — at a bar magnet's surface there is a real, physical bound sheet $\vec K_b=\vec M\times\hat n$ carrying the full tangential- $\vec B$ jump, yet $\vec H^\parallel$ sails through continuously, because $\vec H$ was built blind to exactly that current. Use $\vec B$ -rules with total currents, or $\vec H$ -rules with free currents; mixing the pairings double-counts the magnetization.

Worked examples

1 · A field line's journey through an iron plate (confidence builder)

An ambient field in air meets a large soft-iron plate ( $\mu_r=5000$ ) at $\theta_{\rm air}=1.0^\circ$ from the normal. Follow the line in, along, and out, and find the tangential amplification inside.

Step 1. Entering. Refraction law (4), air $\to$ iron: $\tan\theta_{\rm iron}=\mu_r\tan\theta_{\rm air}=5000\times0.01746=87.3$ , so $\theta_{\rm iron}=89.34^\circ$ . One degree off normal outside becomes two-thirds of a degree off the surface inside: the line has been captured into the plate.

Step 2. What amplifies and what doesn't. Across the entry face, $B^\perp$ is continuous — the normal flux density is whatever it was, $B\cos\theta_{\rm air}$ . The tangential component obeys $H^\parallel$ continuity: $B^\parallel_{\rm iron}=\mu_r B^\parallel_{\rm air}$ — a factor 5000 amplification of the along-surface flux density. The iron is not creating flux (no rule allows that); it is concentrating lines sideways into itself from a huge catchment of the surrounding air, which is why the picture shows neighboring lines converging on the plate.

Step 3. Exiting. At the far face, iron $\to$ air with the line now at $89.34^\circ$ : $\tan\theta_{\rm air}'=\tan(89.34^\circ)/5000=87.3/5000=0.0175$ — it leaves at $1.0^\circ$ , restored to nearly normal exit. Enter normal, travel parallel, exit normal: the plate is a flux pipe, and a closed loop of such plate — a transformer core — pipes essentially all of its flux around the magnetic circuit, which is why the gapped-ring analysis of Ampère's law for H could assume the same flux threads iron and gap.

2 · One material, two thermometers — the sphere that lied about its susceptibility (exam level)

A student measures a small sphere of annealed soft iron in a weak applied field and reports $\chi_{\rm meas}=2.99$ . A handbook lists the material's low-field susceptibility as $\chi\approx1.0\times10^3$ . Reconcile, and state what measurement would have recovered the handbook number.

Step 1. Apply the correction formula. Sphere: $N=\tfrac13$ . Equation (5) predicts $\chi_{\rm meas}=\chi/(1+N\chi)=1000/(1+333.3)=\mathbf{2.99}$ — the student's 2.99 is the correct measurement of a true material $\chi$ of $10^3$ ; nothing is broken but the geometry.

Step 2. Diagnose the regime. $N\chi=333\gg1$ : the reading sits within $0.3\%$ of the shape-saturated ceiling $1/N=3$ . Check the interior field: $H_{\rm in}=H_0/(1+N\chi)=H_0/334$ — the sphere's own pole sheets have annihilated 99.7% of the applied field before the material gets to respond to it. In this regime the experiment measures $1/N$ to three digits and the material to zero digits: doubling $\chi$ to $2000$ would move the reading only to $2.996$ .

Step 3. Fix the experiment. Shrink $N$ . A needle-shaped sample along the field, or better the standard laboratory geometry — a toroid, which is a needle bent to bite its own tail, $N=0$ exactly, no poles anywhere — removes the feedback: $\chi_{\rm meas}=\chi$ . This is why permeability standards are measured on ring cores with a winding, never on spheres or plates, and why the susceptibility article's warning about iron's " $\chi\sim10^3$ – $10^5$ " tables carries the silent footnote toroidal geometry, stated field point.

Step 4. The contrast case. Run the same sphere logic on aluminum, $\chi=+2.2\times10^{-5}$ : the correction factor is $1+N\chi=1.0000073$ — a 7-parts-per-million effect, far below any balance's resolution. Shape corrections are everything for ferromagnets and nothing for para/diamagnets; equation (5) contains both verdicts.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.3.3 (Eqs. 6.25–6.27) and Problem 6.24 · Jackson, Classical Electrodynamics, 3rd ed., §5.8–5.12 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, §13.3 Constants used: $\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}$ (CODATA classical value); $\mu_r=5000$ (typical soft iron, illustrative). Ellipsoid $N$ values for slab, needle, and sphere are worked directly from the crossing rules; the mu-metal shell suppression factor is derived in the Linear and Nonlinear Magnetic Media article, while only its mechanism is treated here. Worked numbers: $89.34^\circ$ / $0.0115^\circ$ refraction pair, $\chi_{\rm meas}=2.99$ at $\chi=10^3$ , aluminum correction $7\times10^{-6}$ .

Builds on: Ampere's Law for H · The Auxiliary Field H · Bound Currents · Magnetic Permeability

Leads to: Linear and Nonlinear Magnetic Media · Field of a Magnetized Object · Magnetic Susceptibility

Why do we need boundary conditions at all when the differential equations hold everywhere?

Because at an idealized sharp interface the fields are not differentiable — $\vec M$ steps from one value to another across zero thickness, and derivatives of steps are not functions the differential equations can digest pointwise. The integral forms of the same laws, applied to instruments (pillbox, loop) that straddle the step, remain perfectly well defined and encode the differential laws' content as jump conditions. Nothing new is being postulated: equations (1)–(3) are $\nabla\cdot\vec B=0$ , $\nabla\cdot\vec H=-\nabla\cdot\vec M$ , and $\nabla\times\vec H=\vec J_f$ read at point-blank range. In a real material the "step" is a few atomic layers, the fields ramp smoothly through it, and the boundary conditions are the exact statement of what survives when you observe from any distance large compared to those layers.

If the B-field rules from the vacuum chapter are still exactly true, why not just use them and skip H entirely?

You can — in principle. $\vec B$ 's rules with total current are exact at every material surface. The obstruction is practical: the bound sheet $\vec K_b=\vec M\times\hat n$ appearing in $\vec B$ 's tangential jump is determined by the magnetization, which in any responsive material is determined by the field you are trying to find. The $\vec B$ -only formulation is a fixed-point problem with the unknown on both sides. The $\vec H$ formulation moves every unknown source into the field's own definition and leaves only free current — experimentally dialed, known in advance — on the right-hand side. For hard permanent magnets (where $\vec M$ is genuinely fixed) the $\vec B$ -route is workable and sometimes used; for everything soft or linear, the $\vec H$ -route is the only one that terminates.

Is the refraction law really analogous to Snell's law? The formula looks different.

The analogy is honest but strictly geometric: a directed line crosses a surface and changes direction by a law involving a material ratio. The physics differs at the root. Snell's law matches the phase of a propagating wave along the interface — a kinematic statement, sines involved, refractive indices of order one. The field-line law matches flux ( $B\cos\theta$ continuous) against circulation ( $H\sin\theta$ continuous) for a static field — tangents involved, and material ratios of order thousands. Two consequences of the difference: static refraction is far more violent than optical refraction ever gets (a 1° line bends to 89.3°), and there is no total-internal-reflection analog — the tangent law has a solution for every incident angle, so no flux is ever "trapped" at the interface; it is merely redirected.

What physically creates the demagnetizing field — the material is not doing anything against itself on purpose?

The pole sheets. Wherever magnetization terminates against a surface (or diverges in the bulk), equation (2) says $H^\perp$ steps — equivalently, the pole fiction places effective surface charge $\sigma_m=\vec M\cdot\hat n$ there, and those sheets source an $\vec H$ that runs from north-face charge to south-face charge — through the material's inside, hence against $\vec M$ . Shape decides how much: a slab magnetized through its faces puts two big opposing sheets right next to each other (maximal internal field, $N=1$ ); a long needle exiles its little end-sheets to infinity ( $N\to0$ ); a sphere is exactly intermediate ( $N=\tfrac13$ ). Nothing is purposeful — the demagnetizing field is just the magnetostatics of the configuration the magnetization itself created, which is also why it is the term that makes the self-consistency loop in $\chi_{\rm meas}$ close.

Why is a toroid the gold-standard geometry for measuring magnetic materials?

Because it is the one convenient shape with no poles at all. Magnetize a ring along its circumference and $\vec M$ never terminates — it runs head-to-tail forever, $\vec M\cdot\hat n=0$ on every surface and $\nabla\cdot\vec M=0$ inside — so no pole sheets, no demagnetizing field, $N=0$ exactly, not approximately. The applied $H$ from a uniform winding (Ampère's law for H: $H=NI/2\pi s$ , material-blind) is then the true internal field, and the measured response is the material's own $\chi$ or $\mu$ , uncorrected. Every serious B–H curve in a materials handbook was traced on a ring core for this reason; the needle is merely the best open geometry, with small end corrections, for samples that cannot be made into rings.

Do these conditions apply to superconductors?

Yes, and instructively — a superconductor in the Meissner state is the boundary conditions' most extreme client. Flux expulsion means $\vec B=0$ inside; $B^\perp$ continuity then forces $B^\perp=0$ just outside the surface: external field lines must meet a superconductor tangentially, the exact opposite of the soft-iron case where they arrive normally. The tangential condition runs in its driven form: a free surface current (the screening supercurrent, flowing in a penetration-depth skin) takes whatever value $\hat n\times\vec H_1=\vec K_f$ requires to hold the interior at zero. In the linear-media dialect a superconductor behaves as the $\mu_r\to0$ endpoint — perfect flux expeller, mirror image of iron's perfect flux collector — a limit the linear-media article's sphere formulas contain as $\chi\to-1$ ; the quantum mechanism behind it is honestly deferred to superconductivity itself.

At a corner or edge of a magnetic body, which normal do I use?

None — the rules as stated need a tangent plane, and a corner has two. The pillbox/loop instruments assume the surface flattens as they shrink, which fails exactly at edges; there the honest statement is that each smooth face carries its own conditions up to the edge, and the fields near the edge develop the same kind of local singular behavior that electrostatics found at conductor corners (field enhancement at exterior edges, suppression in reentrant ones), governed by the opening angle and the permeability ratio. In practice this is why gapped cores and pole pieces are radiused or chamfered — sharp iron corners concentrate flux and saturate first — and why finite-element solvers refine their meshes at every material edge. Away from the singular line, at distances small compared to the body but large compared to zero, the flat-interface rules resume verbatim.