Linear and Nonlinear Magnetic Media

This is the article where the magnetism-in-matter chapter cashes out. The constitutive pair is minted — $\mathbf{M} = \chi_m\mathbf{H}$ from Magnetic Susceptibility, $\mathbf{B} = \mu\mathbf{H}$ from Magnetic Permeability — and the question left standing is the working physicist's: when may I use it, and what does it buy me? The first answer is the linearity doctrine: for diamagnets and paramagnets, essentially always at laboratory fields; for ferromagnets, never honestly. The second is a pair of structural prizes. First, a proportionality theorem — in a homogeneous linear medium the bound current is locked to the free current, $\mathbf{J}_b = \chi_m\mathbf{J}_f$ , so where no free current threads the material there is no volume bound current at all: every linear-media problem is secretly a surface-current problem. Second, the flagship geometry the chapter has been training for: a linear sphere in a uniform field, solved by the magnetic-scalar-potential machinery of The Auxiliary Field H riding the one-rung Legendre ladder of Dielectric Sphere in a Uniform Field — ending in twin verdicts that are the chapter's best plot twist. Crank $\mu_r\to\infty$ and the sphere gathers flux, $\mathbf{B}_{\rm in}\to3\mathbf{B}_0$ , where its dielectric twin screened its interior to zero; crank $\mu_r\to0$ — the superconductor's $\chi_m=-1$ — and the field is expelled entirely: the Meissner endpoint. The nonlinear half is taxonomy rather than theorem: how to read a $B$ – $H$ curve, where saturation and hysteresis live, and the working rule for permanent magnets ( $\mathbf{M}$ frozen — forget $\chi$ entirely). And because this article closes the constitutive arc, it ends by naming the road ahead: boundary conditions, Curie's law, domains, hysteresis, energy, exchange.

Key takeaways

Linearity doctrine: $\mathbf{M} = \chi_m\mathbf{H}$ holds for dia- and paramagnets at essentially all laboratory fields (alignment parameter $\mu B/k_BT \sim 10^{-3}$ at 1 T, 300 K); for ferromagnets it never honestly holds.
Proportionality theorem: in a homogeneous linear medium $\mathbf{J}_b = \chi_m\mathbf{J}_f$ — no free current through the material, no volume bound current. Linear-media fields are surface-current stories; core problems are solenoid-like.
Flagship: a linear sphere in uniform $\mathbf{B}_0$ , by scalar-potential matching, has uniform interior fields $\mathbf{H}_{\rm in} = 3\mathbf{H}_0/(\mu_r+2)$ , $\mathbf{B}_{\rm in} = 3\mu_r\mathbf{B}_0/(\mu_r+2)$ .
The twin verdicts: $\mu_r\to\infty$ gives $\mathbf{B}_{\rm in}\to3\mathbf{B}_0$ — flux gathering — where the dielectric sphere's $\varepsilon_r\to\infty$ gave $\mathbf{E}_{\rm in}\to0$ , screening. And $\mu_r\to0$ ( $\chi_m=-1$ ) gives $\mathbf{B}_{\rm in}\to0$ : perfect expulsion, the Meissner effect (mechanism deferred to superconductivity).
Shielding payoff: a high- $\mu_r$ shell shields its cavity by $S \approx 1+\tfrac{2}{9}\mu_r\left(1-a^3/b^3\right) \approx \tfrac{2}{3}\mu_r t/R$ — mu-metal at $\mu_r \approx 8\times10^4$ , $t = 2$ mm, $R = 0.5$ m gives $S \approx 213$ : Earth's field drops to a quarter-microtesla. MEG and MRI rooms are this formula, nested.
Nonlinear taxonomy: saturation (the tanh ceiling of Paramagnetism), hysteresis (Ferromagnetism), and the permanent-magnet rule — treat $\mathbf{M}$ as frozen, compute from $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$ , never from a $\chi$ .
KEY IDEA: linearity turns magnetized matter into two numbers and one boundary-value method — surface currents only, scalar-potential matching for geometry — delivering flux gathering at $\mu_r\to\infty$ and Meissner expulsion at $\mu_r\to0$ ; nonlinearity (iron) revokes the whole license and hands you a curve instead.

The core insight

In homogeneous linear media the constitutive law makes bound current proportional to free current ( $\mathbf{J}_b = \chi_m\mathbf{J}_f$ ) and reduces every source-free geometry problem to scalar-potential matching — the linear sphere yields uniform $\mathbf{B}_{\rm in} = 3\mu_r\mathbf{B}_0/(\mu_r+2)$ , gathering flux as $\mu_r\to\infty$ and expelling it (Meissner, $\chi_m = -1$ ) as $\mu_r\to0$ — while nonlinear media (ferromagnets) void the license and must be read from the $B$ – $H$ curve.

Where this lives today

The linear-with-large- $\chi$ frontier is commercial reality. Ferrofluids — nanometre magnetite grains in oil, superparamagnetic and honestly linear at modest fields with $\chi \sim 1$ – $5$ — seal hard-drive spindles, dampen loudspeaker voice coils, and make the spiky demonstration sculptures every physics department owns. Their cousins, magnetorheological (MR) fluids, stiffen under field in milliseconds and sit in the semi-active dampers of production vehicles and seismic mounts: a constitutive law you can switch, the linear/nonlinear boundary of this article turned into a product. At the ladder's endpoints, both flagship verdicts are industrial: flux gathering runs every transformer core and magnetic shield (the shielded-room market for MEG and quantum-sensor labs runs on nested mu-metal), while Meissner expulsion — $\chi_m = -1$ as a working material constant — levitates superconducting bearings and demonstration trains, and defines the field-free bores that dilution refrigerators guard with, fittingly, more mu-metal.

Origins · 1 minHeike Kamerlingh Onnes and how this was discovered

Heike Kamerlingh Onnes

1853–1926 · Groningen, Leiden

Onnes built the coldest place on Earth and found this article's endpoint waiting there. Appointed at Leiden at 29, he ran his laboratory under the motto door meten tot weten — "through measurement to knowledge" — and spent two decades industrializing cryogenics until, in 1908, he liquefied helium itself. On 8 April 1911, measuring the resistance of frozen mercury at $4.2\,\mathrm{K}$ , his team watched it vanish — not diminish, vanish — and Onnes named the new state "supraconductivity." He won the 1913 Nobel Prize and pushed on to find that superconductivity dies above a critical field and current — the first hint that magnetism and the new state were locked in opposition. The full magnetic verdict came after his death: Meissner and Ochsenfeld's 1933 discovery that superconductors expel flux rather than merely trap it, certifying $\chi_m = -1$ as an equilibrium property — the perfect-diamagnet endpoint, $\mu_r = 0$ , that completes this article's ladder and this chapter's story.

History

1824

— Poisson's magnetism memoir builds the scalar-potential machinery this article's flagship spends

1873

— Maxwell's Treatise formalizes linear magnetic media alongside dielectrics — the twin ladders

1908

— Kamerlingh Onnes liquefies helium at Leiden: the road to 4 K opens

1911

— Onnes discovers superconductivity in mercury — the chi equals minus one endpoint, unrecognized as such

1933

— Meissner and Ochsenfeld show flux expulsion: perfect diamagnetism is equilibrium physics, not induction

1963

— Papell at NASA files the ferrofluid patent — linear magnetic media become a product

2026

— Nested mu-metal rooms shield MEG and quantum-sensor labs to nanotesla; MR dampers ship in production vehicles

The physical picture

Line up three spheres in the same uniform field. A high- $\mu_r$ sphere (soft iron) invites the field: lines bend toward it, crowd through it, and the interior field nearly triples. A vacuum-adjacent sphere (water, aluminium) leaves the field almost untouched. A superconducting sphere refuses it: lines part around it and the interior is swept clean. One formula, $\mathbf{B}_{\rm in} = 3\mu_r\mathbf{B}_0/(\mu_r+2)$ , spans all three.

Figure

Three spheres in the same uniform field — gathering, indifference, expulsion, one formula

Why does linear response permit such clean answers? Because of a structural theorem proved below: in a homogeneous linear medium, volume bound current exists only where free current threads the material. No windings through the body means all response condenses onto the surface — a bound sheet $\mathbf{K}_b$ — so the problem has the anatomy of a solenoid or a shell of current, never a tangle of volume sources. That is why the chapter's worked catalog (cored solenoids and toroids in Ampere's Law for H, the sphere here) looks the way it does — the magnetic mirror of the dielectric chapter's discovery that homogeneous linear dielectrics hide all their bound charge on their surfaces. And when geometry defeats symmetry, the chapter's bookkeeping pays its dividend: with no free current anywhere, $\nabla\times\mathbf{H} = 0$ everywhere, so the pole-fiction machinery of The Auxiliary Field H applies globally: $\mathbf{H} = -\nabla W$ , with $W$ obeying Laplace's equation wherever the medium is uniform. Every electrostatic tool transfers. The flagship below is Dielectric Sphere in a Uniform Field re-fought with $W$ in place of $V$ — same ladder, same one surviving rung, different winner.

The linearity doctrine — who gets the license

The license to write $\mathbf{M} = \chi_m\mathbf{H}$ is issued by the microscopics, and the chapter has done the inspections. For diamagnets, the induced-orbit response of Diamagnetism is linear with no saturation mechanism at any laboratory strength — the license is unconditional. For paramagnets, linearity requires the alignment energy small against thermal disorder: the parameter is $\mu B/k_BT$ , and at $1\,\mathrm{T}$ , $300\,\mathrm{K}$ it is $2.2\times10^{-3}$ (recalled from Paramagnetism) — three orders of headroom; the response bends toward its tanh ceiling only in millikelvin refrigerators or hundred-tesla pulses. For ferromagnets, there is no regime: the cooperative order of Ferromagnetism makes the response enormous, saturating, and history-dependent from the outset. The doctrine in one line: dia/para — always, in practice; ferro — never, honestly.

medium	license	why
diamagnet	unconditional	induced response, no saturation mechanism (Diamagnetism)
paramagnet	$\mu B/k_BT \ll 1$ — lab-always	$2.2\times10^{-3}$ at 1 T, 300 K; ceiling far away (Paramagnetism)
ferromagnet	none	cooperative, saturating, hysteretic (Ferromagnetism)
superconductor	$\chi_m = -1$ , exactly, below critical field	perfect diamagnetism — linear with the most extreme value allowed

The proportionality theorem — linear media are surface-current stories

The first structural prize costs three lines and reorganizes every problem you will meet.

Derivation ABound current locked to free current — and the surface-only corollary

The goal is $\mathbf{J}_b = \chi_m\mathbf{J}_f$ for a homogeneous linear medium, and its corollary: absent free current threading the material, the only bound current is the surface sheet $\mathbf{K}_b$ — every source-free linear-media problem is a surface-current problem. The inputs are the bound-current theorem $\mathbf{J}_b = \nabla\times\mathbf{M}$ (owned by Bound Currents), the constitutive law, and Ampère's law for $\mathbf{H}$ . Watch the homogeneity assumption: it does real work, and Step 1 shows where an inhomogeneous $\chi_m$ smuggles volume currents back in.

Step 1. Take the curl of the constitutive law in a region of constant $\chi_m$ :

\mathbf{J}_b = \nabla\times\mathbf{M} = \nabla\times(\chi_m\mathbf{H}) = \chi_m\,(\nabla\times\mathbf{H}) = \chi_m\,\mathbf{J}_f. \qquad (1)

Why the constant matters: for varying susceptibility the product rule adds $\nabla\chi_m\times\mathbf{H}$ — a genuine volume bound current on the material's gradients. Homogeneity kills it.

Step 2. Read the theorem. Bound current is not merely related to free current — it is a scaled copy, flowing alongside it point by point: a wire of homogeneous magnetic material carries $\chi_m$ times its free current as bound current in its own volume, amplifying the field by $(1+\chi_m) = \mu_r$ — the microscopic reading of $\nabla\times\mathbf{B} = \mu\mathbf{J}_f$ . For a diamagnet the copy runs antiparallel: the medium fights the wire.

Step 3. The corollary. If no free current threads the material — a core beside a coil, a sphere in an external field, a shield around a room — then $\mathbf{J}_b = 0$ throughout the volume. All response lives on the boundary as $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ (Bound Currents' surface leftover): the material behaves as an empty region wrapped in an invisible solenoid winding whose strength self-consistently follows the field. This is why cored problems are solenoid-like, and why the uniformly magnetized bodies of Field of a Magnetized Object were the right training set.

Step 4. Audit with the divergence. $\nabla\cdot\mathbf{J}_b = \chi_m\nabla\cdot\mathbf{J}_f = 0$ in steady state (Continuity Equation recalled) — the bound copy is as conserved as the original. Note: the inhomogeneous correction term $\nabla\chi_m\times\mathbf{H}$ is the volume bound current that reappears on material gradients; the $\mu_r$ amplification is consistent with the toroid result ( $K_b = \chi_m nI$ , the surface solenoid-on-solenoid reading); and $\nabla\cdot\mathbf{J}_b = 0$ follows from steady-state continuity, as expected.

The flagship — a linear sphere in a uniform field

Now the geometry masterpiece. A sphere of linear material, permeability $\mu = \mu_r\mu_0$ , sits in an applied uniform field $\mathbf{B}_0 = \mu_0 H_0\hat{\mathbf{z}}$ (vacuum outside). Find the fields everywhere. No free current exists anywhere, so by the corollary the response is a surface story, and by the pole fiction the whole problem is electrostatics in disguise.

Go deeper · Derivation B · 2 minThe linear sphere by scalar-potential matching

Derivation BThe linear sphere by scalar-potential matching

The goal is the interior solution $\mathbf{H}_{\rm in} = 3\mathbf{H}_0/(\mu_r+2)$ and $\mathbf{B}_{\rm in} = 3\mu_r\mathbf{B}_0/(\mu_r+2)$ — uniform, remarkably — plus the exterior dipole, by matching the magnetic scalar potential across the surface. The strategy spends two earlier purchases rather than re-deriving them: the pole-fiction formalism of The Auxiliary Field H ( $\mathbf{J}_f = 0$ everywhere $\Rightarrow \mathbf{H} = -\nabla W$ with $\nabla^2 W = 0$ in each uniform region), and the one-rung Legendre ladder of Separation of Variables in Spherical Coordinates as deployed in Dielectric Sphere in a Uniform Field — including that article's honest kill of every $l \neq 1$ term, quoted rather than redone: a $\cos\theta$ drive at infinity feeds only the $l = 1$ rung, and the matching conditions, diagonal in $l$ , force every other coefficient to vanish identically. Watch the boundary conditions: the magnetic pair ( $H^{\parallel}$ continuous, $B^{\perp}$ continuous), and that one swap is where $\mu_r$ enters.

Step 1. Set up the two potentials on the $l = 1$ rung. Inside (regular at the origin) and outside (uniform field plus decaying dipole):

W_{\rm in} = -C\,r\cos\theta, \qquad W_{\rm out} = -H_0\,r\cos\theta + \frac{D\,\cos\theta}{r^2},

with $C$ , $D$ to be found. Both satisfy Laplace's equation; the drive fixes the $-H_0 r\cos\theta$ asymptotics.

Step 2. Match the tangential condition. With no free surface current, $H^{\parallel}$ is continuous, which for gradients of $W$ is guaranteed by continuity of $W$ itself at $r = R$ :

C R = H_0 R - \frac{D}{R^2}.

Step 3. Match the normal condition. $\nabla\cdot\mathbf{B} = 0$ demands $B^\perp$ continuous: $\mu\,(\partial W_{\rm in}/\partial r) = \mu_0\,(\partial W_{\rm out}/\partial r)$ at $r = R$ , i.e.

\mu_r\,C = H_0 + \frac{2D}{R^3}.

Why this is the step that differs from the dielectric twin: there, $\varepsilon\,\partial V/\partial r$ matched across the surface; here $\mu\,\partial W/\partial r$ does. The algebra is isomorphic under $\varepsilon_r \to \mu_r$ , $V \to W$ — one ladder, two problems.

Step 4. Solve the two-by-two system:

C = \frac{3H_0}{\mu_r+2}, \qquad D = \frac{(\mu_r-1)}{(\mu_r+2)}\,R^3 H_0.

Hence the interior fields, both uniform:

\mathbf{H}_{\rm in} = \frac{3\,\mathbf{H}_0}{\mu_r+2}, \qquad \mathbf{B}_{\rm in} = \mu\,\mathbf{H}_{\rm in} = \frac{3\mu_r}{\mu_r+2}\,\mathbf{B}_0, \qquad (2)

and outside, the applied field plus a pure dipole of moment $m = 4\pi D$ — the magnetic Clausius–Mossotti shape, mirroring the dielectric sphere's induced moment under the dictionary.

Step 5. Audit against the chapter. The magnetization is $M = \chi_m H_{\rm in} = 3\chi_m H_0/(\chi_m+3)$ ; feed it to the demagnetizing relation of The Auxiliary Field H: $H_{\rm in} = H_0 - M/3 = 3H_0/(\mu_r+2)$ ✓ — the scalar-potential route and the demagnetizing-factor route agree exactly, the famous $\tfrac13$ visible inside the $(\mu_r+2)$ denominator. The same demagnetizing relation covers the companion case: set $H_0=0$ and impose a frozen $\mathbf{M}$ — not an induced one — and it returns Field of a Magnetized Object's sphere (a companion result, not a limit of the linear solution, which for $\chi_m>0$ always has $\mathbf{H}_{\rm in}$ parallel to $\mathbf{H}_0$ ): $\mathbf{H}_{\rm in} = -\mathbf{M}/3$ , $\mathbf{B}_{\rm in} = 2\mu_0\mathbf{M}/3$ . Note: equation (2) is equivalent to $\mathbf{B}_{\rm in} = 3\mu\mathbf{B}_0/(2\mu_0+\mu)$ after dividing numerator and denominator by $\mu_0$ ; $\mu_r = 1$ returns $\mathbf{B}_{\rm in} = \mathbf{B}_0$ exactly; the demagnetizing audit closes as shown in Step 5; and the spot-checks confirm: $\mu_r = 5000$ gives $B_{\rm in}/B_0 = 2.9988$ ; water gives $B_{\rm in}/B_0 = 1 - 6.0\times10^{-6}$ ; $\mu_r = 0$ gives $B_{\rm in} = 0$ , $H_{\rm in} = \tfrac32 H_0$ .

The twin verdicts

Hold the result against its dielectric twin — the comparison is the chapter's best twist. The dielectric sphere gave $\mathbf{E}_{\rm in} = 3\mathbf{E}_0/(\varepsilon_r+2)$ : as $\varepsilon_r\to\infty$ the interior field dies — matter screens $\mathbf{E}$ , always. The magnetic sphere's $\mathbf{H}_{\rm in} = 3\mathbf{H}_0/(\mu_r+2)$ has the identical form — but $\mathbf{H}$ is the bookkeeper. The physical field is $\mathbf{B}_{\rm in} = \mu_r \cdot 3\mathbf{B}_0/(\mu_r+2)$ , and the extra $\mu_r$ upstairs flips the verdict:

\mu_r \to \infty:\quad \mathbf{B}_{\rm in} \to 3\,\mathbf{B}_0 \qquad\text{(flux gathering, not screening)} \qquad (3)

where: the factor 3 is geometric (the sphere's demagnetizing $\tfrac13$ , inverted) — a high- $\mu$ sphere concentrates three times the ambient flux density, and no permeability, however large, buys more from this shape; iron at $\mu_r = 5000$ already delivers $2.9988\,\mathbf{B}_0$ . Matter aids $\mathbf{B}$ where it opposes $\mathbf{E}$ — the paramagnetic sign of the chapter, made geometric at last. And the opposite end of the ladder delivers the second verdict:

\mu_r \to 0 \;(\chi_m = -1):\quad \mathbf{B}_{\rm in} \to 0, \qquad \mathbf{H}_{\rm in} \to \tfrac{3}{2}\,\mathbf{H}_0 \qquad\text{(perfect expulsion)} \qquad (4)

where: $\chi_m = -1$ is the extreme diamagnet — magnetization exactly cancelling the field inside. This is the phenomenological face of the Meissner effect: a superconductor below its critical temperature behaves, for field-geometry purposes, as an exactly linear medium with $\mu_r = 0$ , expelling flux from its bulk and forcing field lines to part around it (the surface current $\mathbf{K}_b$ doing all the work — the proportionality corollary at its most dramatic). The mechanism — why a quantum condensate enforces $\chi_m = -1$ rather than merely conducting perfectly — is honestly deferred to superconductivity; here it enters as the inhabited endpoint of the ladder, occupied by the superconductivity Onnes discovered in mercury at Leiden in 1911.

The payoff — magnetic shielding

Flux gathering is not a curiosity; it is the working principle of every magnetically shielded room. Wrap a cavity in high- $\mu_r$ material and ambient field lines, preferring the easy path, funnel through the wall instead of the interior. The quantitative version is the sphere solve iterated — a shell has two boundaries, so the same $l = 1$ matching runs with four coefficients instead of two — sketched honestly here rather than re-derived in full.

Go deeper · Derivation C · 1 minThe high-permeability shell — the sphere solve iterated, sketched

Derivation CThe high-permeability shell — the sphere solve iterated, sketched

The goal is the shielding factor $S \equiv B_0/B_{\rm cavity}$ of a spherical shell (inner radius $a$ , outer $b$ , permeability $\mu_r$ ) in a uniform field, at sketch honesty: set up the three-region potential, quote the closed-form result, extract the thin-shell working formula with numbers. This is the flagship's machinery with one more surface — nothing new enters but algebra, which is why the grind is summarized rather than performed.

Step 1. Three regions, all on the $l = 1$ rung: cavity $W_1 = -\alpha r\cos\theta$ ; shell $W_2 = (-\beta r + \gamma/r^2)\cos\theta$ ; exterior $W_3 = (-H_0 r + \delta/r^2)\cos\theta$ . Four constants, four conditions: $W$ continuous and $\mu\,\partial W/\partial r$ continuous at both $r = a$ and $r = b$ .

Step 2. Solve the linear system (quoted, not ground out). The exact cavity field is

\frac{B_{\rm cavity}}{B_0} = \frac{9\mu_r}{(2\mu_r+1)(\mu_r+2) - 2\dfrac{a^3}{b^3}(\mu_r-1)^2},

which for $\mu_r \gg 1$ collapses to the working form

S \;\approx\; 1 + \frac{2}{9}\,\mu_r\!\left(1 - \frac{a^3}{b^3}\right) \;\approx\; \frac{2\,\mu_r\, t}{3\,R} \quad\text{(thin shell, } t = b-a \ll R\text{)}. \qquad (5)

Why (5) is believable without the grind: the shell dilutes the sphere's full gathering power by the fraction of volume that is actually permeable, $1 - a^3/b^3 \approx 3t/R$ — a thin skin funnels proportionally less.

Step 3. Numbers — the mu-metal room. $\mu_r = 8\times10^4$ , $t = 2\,\mathrm{mm}$ , $R = 0.5\,\mathrm{m}$ : $S \approx \tfrac{2}{3}\times 8\times10^4 \times 0.002/0.5 \approx 213$ (exact: $213.5$ ). Earth's $50\,\mathrm{\mu T}$ becomes $\approx 0.23\,\mathrm{\mu T}$ — a two-hundred-fold, single-layer cleanup.

Step 4. Audit the linear pretense. The funneled flux concentrates in the wall: $B_{\rm wall} \sim B_0 R/2t \approx 6.3\,\mathrm{mT}$ — far below mu-metal's $\sim0.7\,\mathrm{T}$ saturation, so the constant- $\mu_r$ treatment is self-consistent for Earth-strength fields. Try to shield a tesla this way and the wall saturates, $\mu_{\rm diff}$ collapses (Magnetic Permeability's honesty), and the shield quietly dies. Note: the exact three-region solution gives $B_{\rm cav}/B_0 = 4.68\times10^{-3}$ , i.e.\ $S = 213.5$ , at the quoted parameters; $\mu_r = 1$ returns $S = 1$ exactly; and the thin-shell form $\tfrac23\mu_r t/R$ is the correct high-permeability limit of the full expression.

Multiply the trick and it compounds: MEG chambers, resolving the $\sim100\,\mathrm{fT}$ fields of the human brain, nest three to six mu-metal shells (factors multiply, roughly) plus active cancellation; MRI suites line walls with high-permeability steel to keep fringe fields and disturbances apart. Every such room is equation (5), iterated and engineered.

The nonlinear half — reading the curve instead of the constant

Everything above rested on the license $\mathbf{M} = \chi_m\mathbf{H}$ . Ferromagnets revoke it; what replaces the constant is a curve, and the taxonomy is short:

Saturation. Drive any moment-bearing material hard enough and the alignment runs out of moments: $M \to M_s$ and the ceiling appears — the tanh flattening derived for the two-level paramagnet in Paramagnetism, reappearing at vastly lower fields in ferromagnets because exchange pre-aligns the moments within domains. Above saturation, $B$ grows only by the vacuum term: $\mathrm{d}B/\mathrm{d}H \to \mu_0$ .

Hysteresis. Cycle $H$ and the ferromagnetic $B(H)$ refuses to retrace itself: domain-wall pinning makes the response multivalued and history-dependent — the loop of Ferromagnetism, with remanence $B_r$ and coercivity $H_c$ . No function $B(H)$ exists, only trajectories; the loop's anatomy, its area-equals-dissipation theorem, and its engineering (soft cores versus hard magnets) are owned by Hysteresis and Magnetic Energy in Materials.

Permanent magnets — the frozen-M rule. For a hard magnet in operation, the honest constitutive statement is brutal and liberating: $\mathbf{M}$ is frozen; forget $\chi$ entirely. Take $\mathbf{M}$ as given (near-uniform, at its remanent value) and compute from the universal $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$ with the bound-current or pole machinery of Field of a Magnetized Object and The Auxiliary Field H — that is how every NdFeB assembly is actually designed. The refinement — where on its loop the operating point sits, found by intersecting the demagnetizing line $H = -NM$ with the material's demagnetization curve (the load line) — is gestured at here and owned by Hysteresis and Magnetic Energy in Materials.

Figure

The three constitutive personalities — linear line, saturating curve, hysteresis loop

Common trapA superconductor is not merely a perfect conductor — and chi equals minus one is an equilibrium property, not an induction effect

Tempting shortcut, wrong physics: "zero resistance traps whatever flux was present, so of course $B$ stays out." A perfect conductor would indeed freeze the flux threading it — including keeping field in if cooled in a field. The Meissner effect is stronger and stranger: a superconductor expels pre-existing flux on cooling through its transition, settling into the same $\mathbf{B} = 0$ bulk state regardless of history. That is why $\chi_m = -1$ deserves its seat on the equilibrium susceptibility ladder — a thermodynamic material property, not electromagnetic inertia — and why this article may legally treat the superconducting sphere as a linear medium with $\mu_r = 0$ . The quantum mechanism (condensate rigidity, penetration depth, the type-II caveat of quantized vortices) belongs to superconductivity, by name; the chapter borrows only the boundary-value behavior.

Limiting cases

Case	Result	Meaning
$\mu_r = 1$	$\mathbf{B}_{\rm in} = \mathbf{B}_0$ exactly	vacuum sphere: no response, formula sanity check
water sphere	$B_{\rm in}/B_0 = 1 - 6.0\times10^{-6}$	diamagnetic screening, six parts per million
iron sphere ( $\mu_r = 5000$ )	$B_{\rm in} = 2.9988\,B_0$	the gathering ceiling, effectively reached
$\mu_r \to \infty$	$\mathbf{B}_{\rm in} \to 3\mathbf{B}_0$ , $\mathbf{H}_{\rm in} \to 0$	flux gathering; geometric cap of 3 for a sphere
$\mu_r \to 0$ ( $\chi_m = -1$ )	$\mathbf{B}_{\rm in} = 0$ , $\mathbf{H}_{\rm in} = \tfrac32\mathbf{H}_0$	Meissner expulsion; superconducting endpoint
frozen- $M$ companion case	$\mathbf{H}_{\rm in} = -\mathbf{M}/3$ , $\mathbf{B}_{\rm in} = \tfrac23\mu_0\mathbf{M}$	recovers Field of a Magnetized Object's magnetized sphere
thin high- $\mu$ shell	$S \approx \tfrac23\mu_r t/R$	shielding: the sphere solve iterated
ferromagnet	no $\chi$ ; read the $B$ – $H$ curve	saturation + hysteresis; frozen- $M$ rule for hard magnets

Worked examples

1 · The iron sphere in a weak field (confidence builder)

A soft-iron sphere ( $\mu_r = 5000$ at this low-field operating point) sits in a uniform applied field $B_0 = 0.50\,\mathrm{mT}$ — weak enough that iron stays on its initial slope. Find $\mathbf{H}_{\rm in}$ , $\mathbf{M}$ , $\mathbf{B}_{\rm in}$ , and audit.

Step 1. The applied auxiliary field: $H_0 = B_0/\mu_0 = 5.0\times10^{-4}/1.2566\times10^{-6} = 398\,\mathrm{A/m}$ .

Step 2. Interior auxiliary field from the flagship: $H_{\rm in} = 3H_0/(\mu_r+2) = 3\times398/5002 = 0.239\,\mathrm{A/m}$ — three orders below the applied: the material's own poles nearly cancel $\mathbf{H}$ inside, demagnetizing feedback at full strength.

Step 3. Magnetization: $M = \chi_m H_{\rm in} = 4999 \times 0.239 = 1.19\times10^{3}\,\mathrm{A/m}$ — just $0.07\%$ of iron's $M_s = 1.7\times10^{6}\,\mathrm{A/m}$ : deep in the linear regime, license valid.

Step 4. The physical field: $B_{\rm in} = \mu_0(H_{\rm in} + M) = 1.2566\times10^{-6} \times 1193 = 1.50\,\mathrm{mT} = 2.9988\,B_0$ . The sphere gathers essentially the full factor of 3. Audit via the closed form: $3\mu_r/(\mu_r+2) = 15000/5002 = 2.9988$ ✓. Note the division of labor: $\mathbf{H}$ collapsed a thousandfold while $\mathbf{B}$ tripled — the two fields moving opposite ways inside matter, exactly the chapter's doctrine.

2 · Designing a mu-metal enclosure (exam level)

A magnetometer lab needs Earth's field ( $50\,\mathrm{\mu T}$ ) reduced below $0.5\,\mathrm{\mu T}$ at the bench. Available: mu-metal sheet, $\mu_r = 8\times10^{4}$ , thickness $2.0\,\mathrm{mm}$ . The workspace fits in an effective sphere of radius $0.50\,\mathrm{m}$ . Does a single shell suffice? Check the wall for saturation, and state what a second shell would buy.

Step 1. Required shielding factor: $S_{\rm req} = 50/0.5 = 100$ .

Step 2. Single-shell factor from the thin-shell formula (5): $S \approx \tfrac{2}{3}\mu_r t/R = \tfrac{2}{3} \times 8\times10^{4} \times 0.0020/0.50 \approx 213$ . (The exact three-region solution gives $213.5$ — the thin-shell form is honest here since $t/R = 0.4\%$ .) Interior field: $50/213 = 0.23\,\mathrm{\mu T}$ . A single shell meets spec with a factor-two margin.

Step 3. Saturation audit — the step that separates a design from a formula. The shell funnels the flux previously crossing its $\sim R$ -scale cross-section through walls of thickness $t$ : $B_{\rm wall} \sim B_0 R/2t = 50\,\mathrm{\mu T} \times 125 = 6.3\,\mathrm{mT}$ , versus mu-metal saturation $\sim0.7\,\mathrm{T}$ : margin of 100, linear treatment self-consistent. (This audit is why the same enclosure cannot shield a $0.1\,\mathrm{T}$ environment: the wall saturates, $\mu_{\rm diff}$ collapses toward $\mu_0$ , and $S$ with it.)

Step 4. The second shell. Nested shells with an air gap multiply their factors approximately: two shells $\Rightarrow S \sim 213^2 \approx 4.5\times10^{4}$ , taking the interior to $\sim1\,\mathrm{nT}$ — MEG territory. (The multiplication is approximate; the honest coupled solution is the four-boundary version of the same $l = 1$ matching.) One shell: $0.23\,\mathrm{\mu T}$ , spec met with margin; two shells: nanotesla.

Closing the chapter — the road ahead

The constitutive arc of E5 is complete: microscopic mechanisms (Diamagnetism, Paramagnetism, Ferromagnetism), bookkeeping machinery (Bound Currents, Field of a Magnetized Object, The Auxiliary Field H, Ampere's Law for H), and the constitutive laws with their geometry payoffs (Magnetic Susceptibility, Magnetic Permeability, this closer). What remains is the second act, each piece named as it was deferred: Magnetic Boundary Conditions systematizes the interface rules and demagnetizing factors this article used for one shape; Curie Law and Curie Temperature makes the $1/T$ fingerprint quantitative and finds where ferromagnetic order dies; Magnetic Domains and Domain Walls explains how a ferromagnet hides its saturation; Hysteresis anatomizes the loop and the load line; Magnetic Energy in Materials prices it all in joules ( $B^2/2\mu$ and what replaces it when the loop opens); and Exchange Interaction and Ferromagnetic Ordering finally pays the quantum debt behind the cooperative story.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.4 · Jackson, Classical Electrodynamics, 3rd ed., §5.8–5.12 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, Ch. 13. The sphere result $\mathbf{B}_{\mathrm{in}}=3\mu\mathbf{B}_0/(2\mu_0+\mu)$ and its $\mu\to\infty$ limit of $3\mathbf{B}_0$ follow from the scalar-potential matching above, with the demagnetizing self-consistency route retained as the Step-5 audit. The hysteresis-loss identity $W/V=\oint H\,\mathrm{d}B$ (derived via toroid EMF) and the Curie–Weiss relation are treated in Hysteresis / Magnetic Energy in Materials and Curie Law and Curie Temperature respectively. The relation $\mu_r=1+\chi_m$ is standard. Quoted susceptibilities are standard SI volume values: water $\approx-9.0\times10^{-6}$ , bismuth $\approx-1.66\times10^{-4}$ , aluminium $\approx+2.1\times10^{-5}$ . $\mu_0=4\pi\times10^{-7}$ T·m/A (exact in the pre-2019 convention; numerical value $1.25663706\times10^{-6}$ N/A²). Numbers used throughout: iron sphere $B_{\rm in}/B_0 = 2.9988$ , water $1-6.0\times10^{-6}$ , superconductor $H_{\rm in} = 1.5H_0$ ; shell shielding factor $S = 213.5$ for $\mu_r = 8\times10^4$ , $t = 2$ mm, $R = 0.5$ m (thin-shell form $\tfrac23\mu_r t/R$ ); wall-field saturation audit $\sim6.3$ mT.

Builds on: Magnetization · The Auxiliary Field H · Ampere's Law for H · Magnetic Susceptibility

Leads to: Magnetic Permeability · Ferromagnetism · Magnetic Boundary Conditions · Paramagnetism

The formulas for H_in and E_in look identical — where exactly does the gathering-versus-screening twist sneak in?

In the last multiplication. Both matchings give the same interior auxiliary answer: $E_{\rm in} = 3E_0/(\varepsilon_r+2)$ and $H_{\rm in} = 3H_0/(\mu_r+2)$ — both die as the material constant grows. The fork is which field is physical. Electrically, $\mathbf{E}$ itself is physical, so screening ends the story. Magnetically, the physical field is $\mathbf{B} = \mu_r\mu_0\mathbf{H}$ , and the $\mu_r$ upstairs overwhelms the $(\mu_r+2)$ downstairs: $B_{\rm in} \to 3B_0$ . Microscopically the twist is the chapter's sign story: polarized matter builds surface charges whose field opposes $\mathbf{E}_0$ inside; magnetized matter builds surface currents whose field aids $\mathbf{B}_0$ inside — the electret's $-\mathbf{P}/3\varepsilon_0$ versus the magnet sphere's $+2\mu_0\mathbf{M}/3$ , made constitutive.

Why does a solid high-mu sphere concentrate flux but a hollow shell shield it — aren't they the same trick?

Same trick, opposite beneficiary. The permeable material itself always gathers flux — that is what $\mu_r \gg 1$ means. In a solid sphere, the point of interest sits inside the material, so it enjoys the concentration ( $3B_0$ ). In a shell, the point of interest is the cavity: the wall gathers the flux and conducts it around the hollow, as a copper wire carries current around, rather than through, its insulation. The cavity is starved precisely because the wall feasts — the worked example's wall field ( $6.3\,\mathrm{mT}$ ) is over four orders above the cavity's ( $0.23\,\mathrm{\mu T}$ ). That is also why shields die at saturation: a wall that can no longer swallow more flux stops starving the cavity.

Is a superconductor really a linear medium, or is chi = -1 a cartoon?

For boundary-value purposes below its critical field, it is honestly linear — the most linear material there is: $\mathbf{B} = 0$ in the bulk enforced exactly, independent of field strength, i.e. $\chi_m = -1$ with no higher-order corrections. The cartoon lies elsewhere: the expulsion is carried by real surface currents in a thin skin (the London penetration depth, tens of nanometres), so " $B = 0$ " is a bulk statement; the state dies above a critical field; and type-II superconductors admit quantized flux vortices, where the response is neither $-1$ nor linear. The mechanism — and why it is thermodynamically distinct from perfect conductivity (the warning above) — belongs to superconductivity; this chapter uses only the $\mu_r = 0$ endpoint it certifies.

How do I compute anything for a permanent magnet if chi is meaningless there?

Use the rule the nonlinear half issues: freeze $\mathbf{M}$ , forget $\chi$ . A hard magnet's magnetization barely responds below its coercivity — that is what "hard" means — so take $\mathbf{M}$ as a given constant vector and compute with the universal kit: bound currents $\mathbf{J}_b = \nabla\times\mathbf{M}$ , $\mathbf{K}_b = \mathbf{M}\times\hat{\mathbf{n}}$ (Bound Currents), or poles $\sigma_m = \mathbf{M}\cdot\hat{\mathbf{n}}$ with the scalar potential (The Auxiliary Field H), then $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$ . The magnetized sphere of Field of a Magnetized Object is the worked prototype. The refinement — the magnet's own demagnetizing field $-N\mathbf{M}$ pushing its operating point down the demagnetization curve, the load-line construction — lives in Hysteresis and Magnetic Energy in Materials.

What are ferrofluids, constitutively — linear or nonlinear?

Both, on different axes. Each suspended magnetite nanograin is a single ferromagnetic domain — internally saturated, maximally nonlinear. But the fluid's response comes from grains orienting against Brownian disorder, mathematically identical to Paramagnetism's alignment problem with a giant moment ( $\sim10^4$ Bohr magnetons per grain): superparamagnetism. At modest fields the fluid is an honest linear medium with $\chi \sim 1$ – $5$ — five orders above aluminium — and it saturates at accessible fields ( $\sim50\,\mathrm{mT}$ -scale), putting the tanh ceiling within desktop reach. Engineering exploits the linear regime (seals, dampers) and the approach to saturation (the spike instability of the demonstration sculptures) — one material straddling the article's dividing line.

Does the J_b = chi_m J_f theorem mean a wire inside iron carries extra current I can measure?

The bound current is real — the chapter's standing doctrine — but audit what "extra" means. Wrap a wire carrying $I_f$ in a homogeneous linear sleeve: the theorem puts volume bound current $\chi_m I_f$ alongside the wire inside the sleeve, so a loop in the sleeve encloses $\mu_r I_f$ total — exactly why $B$ there is $\mu_r$ times the vacuum value. But the sleeve's outer surface carries $\mathbf{K}_b$ totalling $-\chi_m I_f$ (bound current never transports net charge; the closed-ledger theorem of Bound Currents), so any loop drawn outside encloses just $I_f$ again: the external field is unchanged, and a clamp ammeter reads only the free current. The amplification is an inside story — measurable there (a Hall probe in a gap sees $\mu_r B_{\rm vac}$ ), invisible outside.