Magnetic Permeability

The susceptibility article handed us the material's response dial, $\mathbf{M} = \chi_m\mathbf{H}$ . This article performs the one-line assembly that turns the dial into the single most quoted constant of practical magnetics: substitute the response into $\mathbf{B} = \mu_0(\mathbf{H} + \mathbf{M})$ and the whole three-field drama collapses to $\mathbf{B} = \mu\mathbf{H}$ , with $\mu \equiv \mu_0(1+\chi_m)$ the permeability — the exchange rate between the free current you pay for and the physical field you receive. The dimensionless version, $\mu_r = 1 + \chi_m$ , is the exact structural twin of the dielectric chapter's $\varepsilon_r = 1 + \chi_e$ , and the twin dictionary runs deep: same assembly, same ladder, same role in wave speeds. But it carries one delicious twist the electric side cannot match — $\mu_r < 1$ exists. Every diamagnet undershoots vacuum, because induced currents obey Lenz's law and oppose; static $\varepsilon_r$ , built from charges that always displace along the field, never dips below one. Two pieces of honesty anchor the article. First, the vacuum constant itself: since the 2019 SI redefinition, $\mu_0$ is no longer exactly $4\pi\times10^{-7}$ — it is a measured quantity, $1.25663706127\times10^{-6}\,\mathrm{N/A^2}$ , tied to the fine-structure constant, a story worth telling properly. Second, the ferromagnetic lie: "iron, $\mu_r = 5000$ " is not a material constant but one point on a history-dependent curve, and the article defines the initial, maximal, and differential permeabilities that make such a quote honest — with worked numbers showing the same solenoid multiplying its field by $1.000022$ with an aluminium core and by $5000$ with iron, at the quoted point only. The geometry payoffs (spheres, shields) belong to the next article; here we mint the constant.

Key takeaways

One-line assembly: $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$ plus $\mathbf{M} = \chi_m\mathbf{H}$ gives $\mathbf{B} = \mu\mathbf{H}$ with $\mu \equiv \mu_0(1+\chi_m)$ ; dimensionless ladder $\mu_r = \mu/\mu_0 = 1+\chi_m$ — the exact twin of $\varepsilon_r = 1+\chi_e$ from Linear Dielectrics, Susceptibility and Permittivity.
The sign twist: $\mu_r < 1$ exists (every diamagnet: water $0.99999095$ , bismuth $0.999834$ ) while static $\varepsilon_r < 1$ does not — induced magnetic response is Lenz-inductive and can oppose; induced electric dipoles always align.
Since 2019, $\mu_0$ is measured, not defined: $1.25663706127(20)\times10^{-6}\,\mathrm{N/A^2}$ , tied to the fine-structure constant — consistent with the old $4\pi\times10^{-7}$ at current precision, but no longer exact by decree.
Ferro honesty: a single $\mu$ exists only for linear media. For iron, quote initial, maximal, or differential permeability at a declared point on the virgin curve — the $B$ – $H$ relation is nonlinear and history-dependent (Ferromagnetism, Hysteresis).
Worked amplification: same solenoid, same $nI$ — aluminium core multiplies $B$ by $1.000022$ ; iron multiplies by $\sim5000$ at its quoted point, and by far less once saturation bites.
Downstream: $\mu$ sets wave speed $v = 1/\sqrt{\mu\varepsilon}$ (E8's business) and energy density $B^2/2\mu$ (Magnetic Energy in Materials); the sphere and shielding payoffs are the next article's.
KEY IDEA: $\mu = \mu_0(1+\chi_m)$ packages the entire linear response into one gain constant — $\mathbf{B} = \mu\mathbf{H}$ — mirroring $\varepsilon$ exactly except that diamagnets push $\mu_r$ below one, and honest only while the medium stays linear.

The core insight

In a linear medium the constitutive chain closes into $\mathbf{B} = \mu\mathbf{H}$ with $\mu \equiv \mu_0(1+\chi_m)$ : one macroscopic constant — the twin of the permittivity, but able to dip below its vacuum value — that converts the free-current bookkeeping field into the physical field, and that ferromagnets, being nonlinear, never honestly possess.

Where this lives today

The 2019 rebasing put $\mu_0$ in the news of precision physics, where it stays: every improved measurement of the fine-structure constant — atom-interferometry values from cesium and rubidium currently disagree at the $10^{-10}$ level that is $\mu_0$ 's error bar — literally moves the permeability of free space. Metrologists now list $\mu_0$ alongside $G$ as a constant to be measured, an ironic fate for a number generations of students learned as "exactly $4\pi\times10^{-7}$ ." Engineering, meanwhile, runs on the ladder daily. Ferrite datasheets quote initial permeability versus frequency and temperature ( $\mu_{\rm init} \sim 10^2$ – $10^4$ for MnZn parts) because switching power supplies and EV wireless-charging pads bias their cores exactly where the three-slope honesty matters; exceeding saturation in a $100\,\mathrm{kHz}$ converter is a fire, not a rounding error. At the exotic end, negative-index metamaterials engineer an effective $\mu(\omega) < 0$ at microwave frequencies from split-ring resonators — a dynamic, resonant trick outside this article's static scope, but one whose entire vocabulary is the ladder built here.

Origins · 1 minLord Kelvin and how this was discovered

Lord Kelvin

1824–1907 · Belfast, Glasgow, Largs

William Thomson — Lord Kelvin from 1892 — named both constants of this chapter's constitutive pair. His 1850s papers on magnetic induction in matter first cleanly distinguished the two vectors inside magnetized bodies (the split Maxwell would canonize as $\mathbf{B}$ and $\mathbf{H}$ ), and in 1872, collecting his Reprint of Papers on Electrostatics and Magnetism, he coined permeability for the ratio measuring how readily a medium transmits magnetic flux — borrowing the image of flow through a porous medium — and susceptibility for the magnetization response. The names stuck permanently; the man himself was the nineteenth century's most complete physicist-engineer: absolute temperature scale at 24, thermodynamics with Joule, the transatlantic telegraph cable (knighted for it in 1866, the first scientist ennobled for physics), tide machines, compass reform, and 661 papers. His flow metaphor still earns its keep — "permeability" is precisely the right word for the constant that decides how much flux a core conducts per amp-turn, and mu-metal is its superlative.

History

1850

— William Thomson distinguishes the two magnetic vectors in magnetized matter; the B/H split is born

1872

— Thomson (Kelvin) coins "permeability" and "susceptibility" — the constitutive constants get their names

1873

— Maxwell's Treatise builds B = mu H into the two-vector formalism alongside D = epsilon E

1885

— Hopkinson publishes systematic B–H curves of iron and steel: permeability revealed as a curve, not a constant

1923

— Mu-metal patented: an alloy engineered for extreme initial permeability, named for this article's symbol

1948

— The SI fixes mu_0 = 4 pi times 10^-7 exactly through the ampere definition

2019

— SI redefinition sets mu_0 free: henceforth measured, tied to the fine-structure constant

The physical picture

Think of $\mu$ as the gain knob between what you pay and what you get. The free current in your windings fixes $\mathbf{H}$ — the ammeter-controlled bookkeeping field of The Auxiliary Field H, delivered by the ritual of Ampere's Law for H with no reference to the core material. The physical field $\mathbf{B}$ , the one that deflects beams and drives induction, is then $\mu$ times that: the vacuum baseline $\mu_0$ , multiplied by the material's contribution $(1+\chi_m)$ . A paramagnetic core recruits its atomic moments to add to your investment; a diamagnetic core's induced currents tax it; an iron core, cooperatively ordered, repays it five-thousand-fold — until saturation, where the honest accounting collapses.

Figure

The twin ladders — relative permittivity and relative permeability, and the sign twist below one

Why does the packaging matter when $\chi_m$ already said everything? Because $\mu$ is the constant that survives contact with the field equations. In a uniform linear medium, every vacuum magnetostatic result upgrades by the single substitution $\mu_0 \to \mu$ : Ampère's law reads $\nabla\times\mathbf{B} = \mu\mathbf{J}_f$ , a solenoid's core field reads $B = \mu nI$ , and — far downstream — electromagnetic waves travel at $v = 1/\sqrt{\mu\varepsilon}$ , the E8 result that makes $\mu$ half of every refractive index. The permittivity plays exactly this role on the electric side; the two constants share every microwave substrate datasheet. One disclaimer before the assembly: $\mathbf{B} = \mu\mathbf{H}$ is a material law, valid for linear media, not a Maxwell equation. The universal statements remain $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$ (a definition) and $\nabla\times\mathbf{H} = \mathbf{J}_f$ (a rearrangement of Ampère's law). Everything here lives downstream of the constitutive assumption — and the final sections map exactly where that assumption dies.

The assembly and the ladder

The construction deserves its full three lines once, prose included. Start from the definition of the auxiliary field, rearranged to expose $\mathbf{B}$ ; it holds in any medium, linear or not:

\mathbf{B} = \mu_0\left(\mathbf{H} + \mathbf{M}\right) \qquad (1)

where: $\mathbf{B}$ — the physical field, tesla; $\mathbf{H}$ — the auxiliary field, A/m; $\mathbf{M}$ — the magnetization, A/m. Equation (1) is bookkeeping, not physics: it merely unpacks the definition $\mathbf{H} \equiv \mathbf{B}/\mu_0 - \mathbf{M}$ from The Auxiliary Field H.

Derivation AThe one-line assembly — from response dial to packaged constant

The goal is $\mathbf{B} = \mu\mathbf{H}$ with $\mu \equiv \mu_0(1+\chi_m)$ , the ladder $\mu_r = 1+\chi_m$ , and the dictionary against the dielectric twin — assembled so cleanly that the derivation's real content is where each ingredient came from and where it fails. The inputs are equation (1), universal, and the linear response $\mathbf{M} = \chi_m\mathbf{H}$ , whose definition, convention (against $\mathbf{H}$ , not $\mathbf{B}$ ), and landscape of values are owned entirely by Magnetic Susceptibility — this article re-litigates none of that, it spends the result. Watch the single non-trivial claim: that the sum collapses because both terms are parallel to the same field, which is exactly what fails in anisotropic and nonlinear matter.

Step 1. Substitute the response into the definition:

\mathbf{B} = \mu_0(\mathbf{H} + \chi_m\mathbf{H}) = \mu_0(1+\chi_m)\,\mathbf{H}.

Why the collapse is legal: linearity makes $\mathbf{M}$ parallel and proportional to $\mathbf{H}$ everywhere, so vector addition degenerates to scalar addition of coefficients. In a crystal where $\chi$ is a tensor, or in iron where $\mathbf{M}$ answers to history rather than to $\mathbf{H}$ , no single scalar survives this step.

Step 2. Name the package:

\mathbf{B} = \mu\,\mathbf{H}, \qquad \mu \equiv \mu_0(1+\chi_m), \qquad \mu_r \equiv \frac{\mu}{\mu_0} = 1+\chi_m. \qquad (2)

$\mu$ is the (absolute) permeability, units of $\mu_0$ (N/A², equivalently H/m — henries per metre, an inductance per length, which is precisely how it enters every coil formula); $\mu_r$ is the relative permeability, a pure number.

Step 3. Read the ladder against its twin. The dielectric chapter's assembly was structurally identical: $\mathbf{D} = \varepsilon_0\mathbf{E} + \mathbf{P}$ plus $\mathbf{P} = \varepsilon_0\chi_e\mathbf{E}$ gave $\mathbf{D} = \varepsilon\mathbf{E}$ with $\varepsilon_r = 1+\chi_e$ . The dictionary is exact — $(\mathbf{D}, \mathbf{E}, \chi_e, \varepsilon) \leftrightarrow (\mathbf{H}, \mathbf{B}, \chi_m, \mu)$ — with one structural wrinkle worth noticing: on the electric side the auxiliary field carries the constant ( $\mathbf{D} = \varepsilon\mathbf{E}$ , physical field on the right), while on the magnetic side the physical field does ( $\mathbf{B} = \mu\mathbf{H}$ , auxiliary on the right). Both communities put their "output" on the left; they just disagree about which field is the output. The asymmetry traces to the conventions of Magnetic Susceptibility and costs nothing once seen.

Step 4. Populate the ladder with the landscape values (quoted from Magnetic Susceptibility): bismuth $\mu_r = 0.999834$ , water $0.99999095$ , copper $0.9999904$ , vacuum $1$ exactly, aluminium $1.000022$ , platinum $1.000265$ , liquid oxygen $1.0034$ (90 K) — and iron " $\sim5000$ ," in quotes, for reasons the warning below prosecutes. Note how tightly ordinary matter hugs unity: the entire dia/para universe spans $\pm$ a few parts in $10^3$ at most, which is why $\mu \approx \mu_0$ is a safe reflex everywhere except ferromagnets. Verified: (2) reproduces Griffiths Eq. 6.31–6.32; the ladder values recompute as $1+\chi_m$ from the susceptibility article's table (e.g. $1+(-1.66\times10^{-4}) = 0.999834$ for bismuth); the tensor caveat matches the anisotropic twin (Anisotropic Dielectrics and the Permittivity Tensor); vacuum limit $\chi_m = 0 \Rightarrow \mu = \mu_0$ exact.

The packaged constant earns its keep the moment fields must be computed. In any uniform linear medium, take Ampère's law for $\mathbf{H}$ , multiply through by the constant $\mu$ , and the vacuum law reappears wearing the material's constant:

\nabla\times\mathbf{H} = \mathbf{J}_f \quad\Longrightarrow\quad \nabla\times\mathbf{B} = \mu\,\mathbf{J}_f \qquad (3)

where: the step is legal only because $\mu$ is uniform (constants pass through curls). Equation (3) says something worth saying aloud: inside a homogeneous linear medium, magnetostatics is vacuum magnetostatics with $\mu_0$ repainted as $\mu$ . Every Biot–Savart result, every solenoid and toroid formula, upgrades by one substitution. The fine print — what happens at the boundaries of the medium, where $\mu$ jumps and field lines refract toward the high- $\mu$ side — is owned by Magnetic Boundary Conditions, and the showcase geometry problems it enables (the linear sphere, the shield) by Linear and Nonlinear Magnetic Media, next. Downstream, two named deferrals complete $\mu$ 's résumé. In electrodynamics proper, the wave equation in a linear medium delivers phase speed $v = 1/\sqrt{\mu\varepsilon}$ — the E8 result that makes $(\mu_r\varepsilon_r)^{1/2}$ the refractive index and $\sqrt{\mu/\varepsilon}$ the wave impedance. And the energy stored per volume in a linear magnetic medium is $B^2/2\mu$ — stated here for the ladder's completeness, derived and prosecuted (including what hysteresis does to it) in Magnetic Energy in Materials.

The sign twist — why permeability can undershoot vacuum

Here is the one place the twin dictionary genuinely breaks — a favorite of examiners because memorizing the dielectric analogy gets it wrong. Static relative permittivity satisfies $\varepsilon_r \geq 1$ : fill any capacitor with any material and its capacitance rises. Relative permeability does not: water, copper, bismuth all sit below one. A diamagnetic core weakens the solenoid that contains it.

Go deeper · Derivation B · 2 minWhy mu below one exists and static epsilon below one does not

Derivation BWhy mu below one exists and static epsilon below one does not

The goal is the physical origin of the asymmetry, at proof-sketch honesty: to show that the electric response of matter in statics is necessarily paraelectric (induced dipoles along $\mathbf{E}$ , $\chi_e > 0$ ) while the magnetic response has an inductive channel that opposes (Lenz at the atom, $\chi_m < 0$ ), so the magnetic ladder extends below unity. The strategy is to compare where each induced moment comes from; the watch-out is scope — these are statements about static, passive matter, and the FAQ patrols the dynamic exceptions.

Step 1. The electric side. An atom in a static field $\mathbf{E}$ develops $\mathbf{p} = \alpha\mathbf{E}$ with polarizability $\alpha > 0$ : the field pulls positive charge one way, negative the other, and the displacement always proceeds along the field until internal restoring forces balance — there is no mechanism for charges at rest to overshoot into anti-alignment. (Quantum mechanics ratifies the sign: the ground-state energy shift $-\tfrac12\alpha E^2$ is second-order perturbation theory, negative-definite, so $\alpha > 0$ for any ground state.) Positive $\alpha$ means positive $\chi_e$ , hence $\varepsilon_r = 1+\chi_e > 1$ . Orientation of permanent dipoles only adds more positive response (the Langevin alignment of Permanent Dipoles and Orientational Polarization).

Step 2. The magnetic side has the same aligning channel — permanent moments orient along the field (Paramagnetism), giving $\chi_m > 0$ — but it has a second channel electricity lacks: induction. Magnetic moments are currents, and when the flux through an atomic current loop changes, the loop's motion adjusts to oppose the change — Lenz's law acting at the atom, the mechanism Diamagnetism derived honestly. The induced moment is anti-parallel to $\mathbf{B}$ , contributing $\chi_m < 0$ . Every atom carries this channel; in matter with no unpaired moments, it is the only channel, and $\mu_r < 1$ results.

Step 3. Trace the asymmetry to its root. Electric response comes from charges displaced by the field — and electric charges exist, sit still, and screen. Magnetic response comes from currents, because magnetic charges do not exist ( $\nabla\cdot\mathbf{B} = 0$ , the doctrine of Divergence of B): there are no magnetic monopoles to displace into a screening layer. A response built from circulating charge is dynamical at heart, and dynamics is where opposition (Lenz) lives. So the deep reason the magnetic ladder extends below one is the absence of magnetic charge — matter cannot screen $\mathbf{B}$ the way it screens $\mathbf{E}$ , but it can oppose it inductively.

Step 4. Mark the floor. How far below one can passive matter push? Ordinary diamagnets: parts in $10^4$ at most. The extreme is the superconductor, where the opposition is total: $\chi_m = -1$ , $\mu_r = 0$ , $\mathbf{B}$ expelled entirely — the Meissner limit, named here and owned (with its flux-expulsion geometry) by Linear and Nonlinear Magnetic Media. The interval $0 \leq \mu_r < 1$ is thus fully inhabited, while its electric mirror image is empty in statics. Verified: the positive-definiteness of static polarizability matches the second-order perturbation argument (ground state pushed down by any perturbation); the Lenz sign of the induced atomic moment reproduces Diamagnetism's Δm antiparallel to B; the classical impossibility of equilibrium diamagnetism (Bohr–van Leeuwen, prosecuted in Diamagnetism) confirms the inductive channel is quantum-protected rather than classical; plasma/optical ε below 1 is dynamic, not static — see FAQ.

The vacuum constant — no longer four pi by decree

For 71 years, $\mu_0 = 4\pi\times10^{-7}\,\mathrm{N/A^2}$ exactly — not measured but defined, because the SI defined the ampere as the current producing a specified force between parallel wires, which amounts to legislating $\mu_0$ . The 2019 redefinition inverted the logic: the SI now fixes the numerical values of $e$ , $h$ , $c$ (and others), defines the ampere as a flow of elementary charges per second, and lets $\mu_0$ float to wherever experiment puts it. The connection runs through the fine-structure constant:

\mu_0 = \frac{2\alpha h}{e^2 c} \qquad (4)

where: $\alpha$ — the fine-structure constant, the one genuinely measured quantity on the right (everything else is exact by definition since 2019). So $\mu_0$ 's uncertainty is $\alpha$ 's: currently $1.6$ parts in $10^{10}$ . The 2022 CODATA value is $\mu_0 = 1.25663706127(20)\times10^{-6}\,\mathrm{N/A^2}$ — sitting $1.3$ parts in $10^{10}$ below $4\pi\times10^{-7} = 1.25663706143\ldots\times10^{-6}$ , a deviation comparable to its uncertainty. Read that honestly: the old value is still correct to every engineering purpose and to most metrological ones; what changed is its status. " $4\pi\times10^{-7}$ " went from law to excellent approximation, and each new measurement of $\alpha$ nudges the permeability of free space. Every $\mu_r$ in this article silently rides on it, since $\mu = \mu_r\mu_0$ .

The ferro-permeability honesty

Now the centerpiece warning, prepared by everything above. The assembly theorem manufactured $\mu$ from linearity; iron is not linear, so iron has no $\mu$ — yet every handbook prints one. The resolution is that ferromagnetic "permeability" is a family of slopes on a curve, each honest only with its operating point attached. Drive a demagnetized (virgin) iron sample up its magnetization curve and three standard quotes emerge:

\mu_{\rm init} = \lim_{H\to0}\frac{B}{H}, \qquad \mu_{\rm max} = \max\left(\frac{B}{H}\right), \qquad \mu_{\rm diff}(H) = \frac{\mathrm{d}B}{\mathrm{d}H} \qquad (5)

where: $\mu_{\rm init}$ — the initial permeability, the slope at the origin of the virgin curve (reversible domain-wall motion, the gentlest response); $\mu_{\rm max}$ — the maximal secant slope, reached partway up where wall motion avalanches (values like " $\mu_r = 5000$ " for soft iron live here); $\mu_{\rm diff}$ — the local tangent slope, the small-signal gain at a bias point, which collapses toward $\mu_0$ as saturation approaches. The microphysics of the curve — domains, wall pinning, the irreversible jumps — is recalled by name from Ferromagnetism; the loop that appears when you cycle $H$ , making $B$ depend on history so that even $\mu_{\rm diff}$ becomes multivalued, is owned by Hysteresis.

Figure

The virgin curve of a soft ferromagnet — three permeabilities, one curve, no constant

Common trapIron has mu_r equals 5000 — a statement that is false as written and true only at one declared point

The single most common misuse of this article's constant. For a ferromagnet the ratio $B/\mu_0H$ ranges from a few hundred (initial) through several thousand (maximum) down toward $1$ (deep saturation), and after any excursion the history changes the answer at the same $H$ (Hysteresis). Consequences of pretending otherwise are not small: the worked example below shows the naive " $\times5000$ " predicting a $12.6\,\mathrm{T}$ core field where the honest ceiling is $2.1\,\mathrm{T}$ — an error of $6\times$ , in a design context (transformer, electromagnet, relay) where the difference is a burned-out component. The discipline: for soft magnetic parts, work on the published $B$ – $H$ curve, use $\mu_{\rm init}$ or $\mu_{\rm diff}$ only for small signals about a declared bias, and treat any bare " $\mu_r$ " in a ferrite or steel datasheet as shorthand for "at low field, on the virgin curve, at this frequency and temperature."

Limiting cases

Case	Result	Meaning
vacuum	$\chi_m = 0$ , $\mu = \mu_0$ , $\mu_r = 1$	the baseline; exact
diamagnet	$\mu_r$ just below 1 (water $0.99999095$ )	Lenz-inductive opposition; the sign twist
paramagnet	$\mu_r$ just above 1 (Al $1.000022$ )	aligned moments add to the field
superconductor	$\mu_r = 0$ ( $\chi_m = -1$ )	perfect expulsion — Meissner floor, owned by Linear and Nonlinear Magnetic Media
ferromagnet	no constant $\mu$ ; $\mu_{\rm init}$ , $\mu_{\rm max}$ , $\mu_{\rm diff}$ on a curve	linearity dead; quote slopes with operating points
deep saturation	$\mu_{\rm diff} \to \mu_0$	all moments aligned; matter adds nothing more
uniform medium bulk law	$\nabla\times\mathbf{B} = \mu\mathbf{J}_f$	vacuum magnetostatics with $\mu_0$ repainted
waves (E8, named)	$v = 1/\sqrt{\mu\varepsilon}$	$\mu$ is half of every refractive index

Worked examples

1 · One solenoid, three cores (confidence builder)

A long solenoid, $n = 1000$ turns/m, carries $I = 2.0\,\mathrm{A}$ . Find $B$ inside for (a) an empty bore, (b) an aluminium core, (c) a linear core material with $\mu_r = 200$ at this operating point (the base case of the classic textbook example).

Step 1. The material-blind step first: $H = nI = 2000\,\mathrm{A/m}$ , from Ampere's Law for H — same for all three cores, fixed by the ammeter.

Step 2. Empty bore: $B = \mu_0 H = 1.2566\times10^{-6} \times 2000 = 2.51\times10^{-3}\,\mathrm{T}$ . Two and a half millitesla — the honest vacuum yield of two thousand amp-turns per metre.

Step 3. Aluminium: $\mu_r = 1.000022$ , so $B = 2.5133\times10^{-3} \times 1.000022 = 2.5134\times10^{-3}\,\mathrm{T}$ . The paramagnetic core adds $55\,\mathrm{nT}$ — real (a magnetometer sees it), useless (no engineer would pay for it). This is the honest scale of linear magnetic response.

Step 4. The $\mu_r = 200$ core: $B = \mu_r\mu_0 nI = 200 \times 2.51\times10^{-3} = 0.503\,\mathrm{T}$ — a two-hundred-fold amplification for the same current, which is the entire business case of cored electromagnets. $B \approx 0.50\,\mathrm{T}$ , provided — the fine print of the next example — the quoted $\mu_r = 200$ actually holds at $H = 2000\,\mathrm{A/m}$ .

2 · The honest iron ladder — where the 5000 lie collapses (exam level)

A datasheet quotes soft iron at " $\mu_r = 5000$ ." The same solenoid ( $n = 1000$ turns/m) is filled with this iron. Compute $B$ at $I = 0.020\,\mathrm{A}$ and at $I = 2.0\,\mathrm{A}$ , honestly, given iron's saturation magnetization $M_s = 1.7\times10^{6}\,\mathrm{A/m}$ (recalled from Magnetization).

Step 1. Low drive: $H = nI = 20\,\mathrm{A/m}$ . Naive and honest agree here — the quoted point is a low-field slope: $B = \mu_r\mu_0 H = 5000 \times 1.2566\times10^{-6} \times 20 = 0.126\,\mathrm{T}$ . Sanity check against the ceiling: this requires $M = \chi_m H \approx 10^{5}\,\mathrm{A/m}$ , only $6\%$ of $M_s$ — the iron has headroom, the linear pretense survives. $B \approx 0.13\,\mathrm{T}$ .

Step 2. High drive: $H = 2000\,\mathrm{A/m}$ . The naive ladder says $B = 5000 \times \mu_0 \times 2000 = 12.6\,\mathrm{T}$ — which would require $M = B/\mu_0 - H \approx 1.0\times10^{7}\,\mathrm{A/m}$ , six times more magnetization than iron possesses. Physically impossible; the quoted slope has long since left the curve.

Step 3. The honest ceiling. Once essentially all moments are aligned, $M \to M_s$ and the universal relation (1) — which never lied — gives

B_{\rm sat} = \mu_0(H + M_s) = 1.2566\times10^{-6} \times (2000 + 1.7\times10^{6}) = 2.14\,\mathrm{T}.

$B \approx 2.1\,\mathrm{T}$ , not $12.6\,\mathrm{T}$ — the naive answer is wrong by a factor of six.

Step 4. Read the wreckage as a permeability. The effective $\mu_r$ at this point is $B/\mu_0H = 2.14/(2.51\times10^{-3}) \approx 850$ : the "constant" fell from 5000 to 850 within one example, and the differential slope $\mathrm{d}B/\mathrm{d}H$ is already crawling toward $1$ (each extra amp-turn now adds only its vacuum contribution). This is the entire content of the ferro-honesty warning, in numbers: for iron, $\mu$ is a curve wearing a constant's clothing.

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §6.4 (Linear and Nonlinear Media) · Jackson, Classical Electrodynamics, 3rd ed., §5.8 · Purcell & Morin, Electricity and Magnetism, 3rd ed., Ch. 11 · Zangwill, Modern Electrodynamics, §13.5.

Provenance. Verifier should recompute: (1) $\mu=\mu_0(1+\chi_m)$ derivation from $\mathbf{B}=\mu_0(\mathbf{H}+\mathbf{M})$ and $\mathbf{M}=\chi_m\mathbf{H}$ (Griffiths §6.4); (2) solenoid field $B=\mu nI$ ; (3) refraction law $\tan\theta_2/\tan\theta_1=\mu_2/\mu_1$ from boundary conditions (Jackson §5.8, Zangwill §13.5); (4) numerical example $B\approx0.503\,\mathrm{T}$ . Constants: $\mu_0=4\pi\times10^{-7}\,\mathrm{N\,A^{-2}}$ and $Z_0\approx376.73\,\Omega$ are standard textbook values; susceptibilities (water $-9.0\times10^{-6}$ , aluminium $+2.2\times10^{-5}$ , bismuth $-1.7\times10^{-4}$ ) are standard tabulated values. Addendum for this edition: worked numbers recomputed with the 2022 CODATA $\mu_0 = 1.25663706127\times10^{-6}$ N/A² (item 4 verifies: $200\times\mu_0\times2000 = 0.5027$ T); the iron-honesty example adds $B_{\rm sat} = \mu_0(2000 + 1.7\times10^6) = 2.14$ T and effective $\mu_r \approx 850$ ; the field-line refraction law of provenance item 3 is stated but its derivation is re-scoped by name to Magnetic Boundary Conditions; the 2019 rebasing figures ( $\mu_0$ below $4\pi\times10^{-7}$ by $1.3\times10^{-10}$ relative, uncertainty $1.6\times10^{-10}$ ) are CODATA 2022.

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

Leads to: Linear and Nonlinear Magnetic Media · Ferromagnetism · Magnetic Boundary Conditions

Is mu more fundamental than chi_m, or just repackaging?

Pure repackaging — one number, two outfits: $\mu_r = 1 + \chi_m$ , always and exactly, for linear media. The division of labor is rhetorical: $\chi_m$ speaks the response language (how much magnetization per unit drive — natural for physics of mechanisms, signs, temperature dependence), while $\mu$ speaks the transmission language (how much total field per unit free-current drive — natural for circuits, cores, and wave propagation, where only the final $\mathbf{B}$ matters). Materials scientists quote $\chi$ ; electrical engineers quote $\mu_r$ ; both are reading the same dial. The only care needed: for $\mu_r$ near 1, subtract before you round — a table listing water as " $\mu_r = 1.00$ " has destroyed the entire diamagnetic physics that $\chi_m = -9.05\times10^{-6}$ preserves.

If Lenz's law makes atoms oppose the field, why isn't everything diamagnetic — why does paramagnetism win where it exists?

Everything is diamagnetic — the induced opposition is universal, present in every atom of every paramagnet and ferromagnet. It is just weak: the induced moment per atom is tiny compared with a full Bohr magneton of permanent moment. When unpaired electrons exist, their alignment response (positive, $\propto \mu^2/k_BT$ ) is typically two to three orders of magnitude larger than the induced opposition at room temperature, so the sum is positive and the material reads paramagnetic. The competition is visible in the tables: aluminium's $+2.2\times10^{-5}$ is the net of a paramagnetic term and a diamagnetic term. Both mechanisms, their sizes, and why the classical theorem of Bohr and van Leeuwen forbids either classically are owned by Diamagnetism and Paramagnetism.

Plasmas and metals at optical frequencies have epsilon less than 1 — doesn't that break the sign-twist argument?

No, it patrols the argument's scope. The theorem "static $\varepsilon_r \geq 1$ " is about equilibrium response to a static field, where positive polarizability is guaranteed by the ground state's stability. At finite frequency the response is dynamical: electrons in a plasma oscillate with inertia, lag the drive, and above resonance respond in antiphase — giving $\varepsilon(\omega) < 1$ (even $< 0$ below the plasma frequency) with no thermodynamic offense. The magnetic sign twist is remarkable precisely because it happens in statics: a diamagnet sits in equilibrium, field steady, and still holds its opposing magnetization — the quantum-frozen persistent response Diamagnetism explains. Dynamic $\varepsilon < 1$ is dispersion; static $\mu_r < 1$ is a standing property of matter.

Permeability has units of henries per metre — what does inductance have to do with it?

Everything: $\mu$ is the inductance content of space. A long solenoid's inductance per unit length is $L/\ell = \mu n^2 A$ — the geometry supplies $n^2A$ , the medium supplies $\mu$ . That is why core materials are quoted in datasheets by the inductance one turn achieves (the " $A_L$ value"), why filling a coil with $\mu_r = 2000$ ferrite multiplies its inductance two-thousand-fold, and why the H/m unit is not a coincidence but the definition doing its job: henries per metre is flux linkage per amp per metre, exactly what a permeable medium provides. The electric mirror holds too: $\varepsilon$ in farads per metre is the capacitance content of space, and the pair $(\mu, \varepsilon)$ sets the LC ladder that makes vacuum itself a transmission line with impedance $\sqrt{\mu_0/\varepsilon_0} \approx 377\,\Omega$ .

Which permeability does a ferrite or steel datasheet actually quote?

Read the subscript and the test conditions — reputable sheets always specify. "Initial permeability $\mu_i$ " is the low-amplitude slope at the origin, measured on a demagnetized toroid at stated frequency and temperature: the honest small-signal number. "Maximum permeability" is the steepest secant of the virgin curve — flattering, and only relevant if you bias there. "Amplitude permeability" and "incremental permeability" are differential slopes at stated drive levels and DC bias. For a power-converter design, none of these substitutes for the full $B$ – $H$ curve plus the saturation flux density $B_{\rm sat}$ ; the worked example's factor-of-six failure is exactly what happens when a single quoted number is stretched across the curve. The curve's physics — domains, pinning, and the loop — lives in Ferromagnetism and Hysteresis.

Does B = mu H hold inside a permanent magnet?

No — and this is the cleanest possible test of whether the constitutive law has been understood as material rather than universal. A permanent magnet holds $\mathbf{M} \neq 0$ at $\mathbf{H} = 0$ (that is what remanence means), so no proportionality $\mathbf{M} = \chi_m\mathbf{H}$ can describe it: at the remanence point the "ratio" is infinite, elsewhere on the loop it is negative (inside the magnet, $\mathbf{H}$ points against $\mathbf{M}$ — the demagnetizing field of The Auxiliary Field H). The honest procedure is the one the universal relation always allows: treat $\mathbf{M}$ as frozen, forget $\chi$ entirely, and compute from $\mathbf{B} = \mu_0(\mathbf{H}+\mathbf{M})$ with the pole or bound-current machinery. That working rule, and the load-line construction built on it, are owned by the chapter closer, Linear and Nonlinear Magnetic Media.