---
title: Magnetic Materials
module: Magnetic Sources
moduleNumber: 7
lessonNumber: 6
order: 706
summary: >
  Put matter in a magnetic field and its atoms respond, each acting as a tiny current
  loop; the aligned moments per unit volume are the magnetization $\vec M$, whose bound
  currents add to the field. Separating what we control (the free current) from what the
  material supplies leads to $\vec H$ and the relation $\vec B=\mu_0(\vec H+\vec M)$. We
  sort materials into diamagnets, paramagnets, and ferromagnets by how $\vec M$ answers,
  follow a ferromagnet around its hysteresis loop, and see why the loop's area is the
  energy dissipated per cycle and why a sample's shape changes the field it actually
  feels.
topics: [Magnetic Sources]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 27 — Sources of the Magnetic Field; §27-5 Magnetism in Matter"
---

## Magnetic moments and bound currents

Atoms and molecules can carry magnetic dipole moments through electronic orbital
motion and intrinsic electron spin. A macroscopic specimen contains an enormous
number of such moments. Their vector sum, rather than the magnitude of one
atomic moment, determines the magnetic response observed outside the material.

A small permanent moment $\vec m$ in an applied magnetic field
$\vec B_{\mathrm{app}}$ has interaction energy

$$
U=-\vec m\mathbin{\cdot}\vec B_{\mathrm{app}}.
$$

Parallel alignment lowers this energy; antiparallel alignment raises it. Thermal
motion competes with the alignment tendency. In an unmagnetized paramagnetic
sample, moment directions are broadly distributed and the vector average is
close to zero. A field produces a slight excess of moments pointing along the
applied direction. The individual moments remain large on the atomic scale even
when their macroscopic average is small.

$$
% caption: Atomic moments in two equal volumes. With no applied field the directions
% are scattered and their vector sum is near zero; an applied field biases the
% population toward its own direction, leaving a net magnetization.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,fill=acc!8] (-3.35,0.30) rectangle (-0.55,2.70);
\draw[black,fill=acc!8] (0.55,0.30) rectangle (3.35,2.70);
% random moments (left)
\draw[acc,thick,->] (-2.95,1.90)--(-2.55,2.20);
\draw[acc,thick,->] (-2.05,2.00)--(-2.35,1.65);
\draw[acc,thick,->] (-1.30,1.75)--(-0.95,1.95);
\draw[acc,thick,->] (-2.90,0.85)--(-2.55,0.70);
\draw[acc,thick,->] (-1.95,0.75)--(-2.25,1.05);
\draw[acc,thick,->] (-1.15,0.95)--(-1.45,0.70);
\draw[acc,thick,->] (-1.70,1.45)--(-1.35,1.30);
% aligned fraction (right)
\draw[acc,thick,->] (0.85,2.05)--(1.45,2.05);
\draw[acc,thick,->] (1.75,2.10)--(2.35,2.10);
\draw[acc,thick,->] (0.90,1.45)--(1.50,1.45);
\draw[acc,thick,->] (1.80,1.45)--(2.40,1.45);
\draw[acc,thick,->] (0.85,0.80)--(1.45,0.80);
\draw[acc,thick,->] (1.80,0.80)--(2.40,0.80);
\draw[black,->,thick] (0.75,3.05)--(3.15,3.05);
\node[above] at (1.95,3.05) {applied B};
\node[below] at (-1.95,0.24) {random moments};
\node[below] at (1.95,0.24) {aligned fraction};
\end{tikzpicture}
$$

Magnetization is the magnetic dipole moment per unit volume. A small volume
$\Delta V$ containing moments $\vec m_i$ has

$$
\vec M=\lim_{\Delta V\rightarrow0}
\frac{\sum_i\vec m_i}{\Delta V}.
$$

Its SI unit is ampere per metre. A moment has units $\mathrm{A\,m^2}$, so
division by volume leaves $\mathrm{A\,m^{-1}}$. Magnetization is a vector
field: both magnitude and direction can vary from point to point. A material
with spatially varying moment alignment has spatially varying $\vec M$ even
when its chemical composition is uniform.

The term saturation magnetization refers to the largest magnetization available
when the relevant microscopic moments are nearly fully aligned. If a number
density $n$ of atoms each contributes a moment of magnitude $m$, the limiting
scale is

$$
M_{\mathrm s}=nm.
$$

The expression is an upper scale, not a general room-temperature result.
Thermal agitation, competing atomic configurations, and interactions among
neighbouring moments commonly keep the measured magnetization far below
saturation.

### Bound-current interpretation

Magnetization can be represented macroscopically by effective currents. The
representation does not require a literal tiny wire loop at every atom. It
captures the magnetic field of the summed microscopic orbital and spin
moments. The associated volume and surface current densities are

$$
\vec J_{\mathrm b}=\nabla\mathbin{\times}\vec M,
\qquad
\vec K_{\mathrm b}=\vec M\mathbin{\times}\hat n,
$$

where $\hat n$ is the outward normal of a material surface.
$\vec J_{\mathrm b}$ has units of $\mathrm{A\,m^{-2}}$ and
$\vec K_{\mathrm b}$ has units of $\mathrm{A\,m^{-1}}$.

Uniform magnetization has $\nabla\mathbin{\times}\vec M=0$ in
the interior. Adjacent microscopic loops cancel there: a segment of current on
one loop is opposed by a neighboring segment. At the exterior surface, there
is no neighboring loop beyond the boundary to supply the cancellation. The
remaining effective surface current accounts for the material’s external
magnetic field in the bound-current description.

A long cylinder with uniform $\vec M=M\hat z$ has a cylindrical side
with
$\hat n=\hat r$ and therefore

$$
\vec K_{\mathrm b}
=M\hat z\mathbin{\times}\hat r
=M\hat\phi.
$$

The effective current circulates around the cylinder. Its magnitude per unit
axial length is $M$. End faces carry no surface bound current in this ideal
case because their normal is parallel or antiparallel to $\vec M$. A finite
cylinder still has end-field structure; the simple long-cylinder picture
describes the central region.

$$
% caption: Uniform axial magnetization as an effective surface current. Adjacent
% atomic loops cancel in the interior; only the outermost contribution survives, so a
% sheet of current circulates around the side, its current per unit length equal to M.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[black,fill=acc!8] (0,0) ellipse (1.40 and 0.45);
\draw[black] (-1.40,0)--(-1.40,2.40);
\draw[black] (1.40,0)--(1.40,2.40);
\draw[black,fill=acc!8] (0,2.40) ellipse (1.40 and 0.45);
\draw[->,acc,very thick] (0,0.60)--(0,1.95);
\node[right] at (0.10,1.75) {M};
\draw[acc,thick,->] (1.40,0.85) arc (0:-180:1.40 and 0.45);
\draw[acc,thick,->] (1.40,1.75) arc (0:-180:1.40 and 0.45);
\node[right] at (1.50,1.30) {surface current};
\end{tikzpicture}
$$

The bound-current picture separates internal magnetization from free current
carried through an externally connected circuit. A copper wire feeding a coil
has a transport current set by the circuit. A uniformly magnetized insulating
sample can have a nonzero $\vec K_{\mathrm b}$ without any charge crossing
its exterior surface from a power supply. Both forms of current contribute to
magnetic fields, but their physical origins and measurement controls differ.

## Susceptibility and field variables

In a weak, reversible response regime, magnetization is often proportional to
the applied field. Tipler and Mosca express that relation as

$$
\vec M=\chi_m\frac{\vec B_{\mathrm{app}}}{\mu_0}.
$$

The dimensionless susceptibility $\chi_m$ gives the sign and approximate size
of the response. In the ideal central region of a long sample placed in a long
solenoid,

$$
\vec B=\vec B_{\mathrm{app}}+\mu_0\vec M
=(1+\chi_m)\vec B_{\mathrm{app}}.
$$

The factor $\kappa_m=1+\chi_m$ is the relative permeability in this linear
model. Geometry, field strength, temperature, and magnetic history determine
whether a constant susceptibility is an adequate approximation.

Diamagnetic matter has a small negative susceptibility. Its response is
induced by the applied field and opposes that field. Atoms with no permanent
net magnetic moment can still develop a diamagnetic response because the
applied field changes electronic orbital motion. Typical ordinary diamagnetic
susceptibilities have magnitudes near $10^{-5}$, so the effect is usually weak.

Paramagnetic matter has a small positive susceptibility. Permanent atomic or
molecular moments exist, but thermal motion randomizes their directions when
the applied field is absent. An applied field biases the orientation
distribution. The alignment remains partial when the dipole-field energy is
small compared with thermal energy. In the weak-field limit, Curie’s law gives

$$
\frac{M}{M_{\mathrm s}}
\approx\frac{mB_{\mathrm{app}}}{3kT}.
$$

The response increases with applied field and decreases with absolute
temperature. Paramagnetism usually leaves no remanent magnetization after the
applied field is removed.

Ferromagnetic matter has a strong positive response arising from cooperative
interactions among neighboring moments. Microscopic regions called domains can
have nearly aligned moments even when the macroscopic specimen has little net
magnetization, because different domains point in different directions. A
modest applied field can move domain boundaries or rotate domain alignment,
producing a much larger net response than ordinary paramagnetism.

Ferromagnetic response is nonlinear and history dependent. Increasing and
then decreasing the applied field traces a hysteresis loop. The loop can retain
a nonzero magnetization or magnetic field at zero applied level, and a reversed
applied field is required to bring the macroscopic response through zero.
Thermal agitation above the Curie temperature disrupts the cooperative
ordering, leaving a paramagnetic response.

$$
% caption: The three magnetic classes by how magnetization answers an applied field.
% A diamagnet has a small negative slope, a paramagnet a small positive reversible
% slope, and a ferromagnet traces a nonlinear loop that keeps a response at zero field.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% panel 1: diamagnetic
\draw[->,black] (0.10,1.10)--(1.90,1.10);
\draw[->,black] (1.00,0.25)--(1.00,2.10) node[above] {M};
\draw[acc,thick] (0.35,1.45)--(1.65,0.75);
\node[above] at (1.00,2.30) {dia};
% panel 2: paramagnetic
\draw[->,black] (3.30,1.10)--(5.10,1.10);
\draw[->,black] (4.20,0.25)--(4.20,2.10) node[above] {M};
\draw[acc,thick] (3.55,0.75)--(4.85,1.45);
\node[above] at (4.20,2.30) {para};
% panel 3: ferromagnetic loop
\draw[->,black] (6.50,1.10)--(8.30,1.10);
\draw[->,black] (7.40,0.25)--(7.40,2.10) node[above] {M};
\draw[acc,thick] plot[smooth,tension=0.7] coordinates {(6.60,0.45)(6.83,1.31)(7.28,1.67)(7.94,1.75)(8.20,1.43)(7.96,0.90)(7.43,0.53)(6.83,0.45)(6.62,0.77)(6.92,1.40)(7.45,1.72)};
\node[above] at (7.40,2.30) {ferro};
\node[below] at (4.20,0.20) {applied H};
\end{tikzpicture}
$$

The three classes describe response trends rather than a complete catalogue of
magnetic behavior. Measured susceptibility should always be paired with field
strength, temperature, orientation, and magnetic history. Those conditions
determine whether a simple linear relation describes the sample or whether domain
processes and hysteresis dominate the measurement.

### Measuring a magnetic response

A response measurement separates the source field from the sample’s added
magnetization. The sample magnetic moment can be found from a calibrated torque,
force-gradient, or flux measurement, then divided by sample volume to obtain
$M$. The applied source setting and the sample temperature must be recorded at
the same time. A value of magnetization without a field scale cannot determine
a susceptibility.

In a reversible weak-response regime,

$$
\chi_m=\frac{\mu_0M}{B_{\mathrm{app}}}.
$$

Repeating the measurement at positive and negative source settings checks the
sign. Diamagnetic response reverses with the source field and opposes it.
Paramagnetic response reverses with the source field and follows it. A
ferromagnetic sample can retain a response when the source setting returns to
zero, so measurements must state the prior sweep direction and maximum applied
level.

Finite sample shape changes the relation between an external source field and
the internal field experienced by the material. The simple
$\vec B=\vec B_{\mathrm{app}}+\mu_0\vec M$ expression used above
describes the central region of the long-cylinder idealization in the reference
text. Ends, corners, gaps, and irregular sample shapes introduce spatial
variation. A susceptibility reported from a finite specimen therefore includes
the stated geometry and measurement location unless an appropriate shape
correction has been made.

Temperature scans provide a second classification check. A weak paramagnetic
response commonly decreases as temperature rises because random thermal motion
weakens moment alignment. Ordinary diamagnetic susceptibility changes much less
strongly with temperature. Ferromagnetic response can change abruptly near its
Curie temperature, where domain-scale cooperative ordering is lost. A single
room-temperature measurement cannot establish a complete magnetic class when
the material composition or thermal history is unknown.

The response also depends on the timescale of the source sweep. Slow
measurements that trace a ferromagnetic hysteresis loop give different values
on increasing and decreasing branches. Reproducible reports identify the
initial state, sweep range, sweep direction, and whether the specimen was
demagnetized before the run. Those conditions turn susceptibility and
magnetization from isolated numbers into reproducible material-response data.

### B, H, and M

The magnetic flux density $\vec B$ is the field that appears in the magnetic
force and flux laws. Its SI unit is the tesla. Magnetization $\vec M$ is the
material dipole moment per unit volume and has units of amperes per metre. The
magnetic field strength $\vec H$ separates the source-controlled part of a
magnetostatic problem from the material response:

$$
\vec H=\frac{\vec B}{\mu_0}-\vec M,
\qquad
\vec B=\mu_0(\vec H+\vec M).
$$

Both $\vec H$ and $\vec M$ have units of amperes per metre. Their sum
has the same unit; multiplication by $\mu_0$ converts it to tesla. The
definition separates a circuit's free current from the sample's bound-current
response. In steady magnetostatics, the
circulation of $\vec H$ counts free current:

$$
\oint_C\vec H\mathbin{\cdot}\d\vec\ell
=I_{\mathrm{free,enc}}.
$$

The microscopic material currents are incorporated through $\vec M$ rather
than appearing separately on the right side. This division makes coil currents,
magnetization, and total magnetic flux distinguishable in one calculation.

A linear, isotropic material in a reversible range has

$$
\vec M=\chi_m\vec H,
\qquad
\vec B=\mu_0(1+\chi_m)\vec H
=\mu\vec H,
\qquad
\mu=\mu_0(1+\chi_m).
$$

The relation uses a scalar $\chi_m$ only when the response has no preferred
direction and remains proportional to $\vec H$. Crystals can be anisotropic,
and ferromagnetic specimens can be nonlinear and history dependent. In those
cases, a single scalar permeability cannot represent every direction and every
point on a hysteresis loop. The slope of a local response curve may still
characterize a stated operating point.

The applied field in a long empty solenoid is often written
$\vec H_{\mathrm a}=nI\,\hat z$. The associated empty-space flux
density is $\mu_0\vec H_{\mathrm a}$. Once a sample is inserted, the
material magnetization adds to the total $\vec B$. Referring only to
“the magnetic field” obscures whether a numerical value denotes the source
setting, the material moment density, or the total flux density.

### Linear-media interfaces

Boundary conditions provide sharp tests of a linear-media model. Let
$\hat n$ point from medium 1 to medium 2. Magnetic flux has no
isolated sources, so the normal component of $\vec B$ is continuous:

$$
\hat n\mathbin{\cdot}
(\vec B_2-\vec B_1)=0.
$$

The tangential jump in $\vec H$ is set by free surface current density
$\vec K_{\mathrm{free}}$:

$$
\hat n\mathbin{\times}
(\vec H_2-\vec H_1)=\vec K_{\mathrm{free}}.
$$

At an interface carrying no free sheet current, tangential $\vec H$ is
continuous. If both media are linear and isotropic, the consequence is

$$
\frac{B_{1,\mathrm t}}{\mu_1}
=\frac{B_{2,\mathrm t}}{\mu_2},
\qquad
B_{1,\mathrm n}=B_{2,\mathrm n}.
$$

The tangential component of $\vec B$ can therefore change at a permeability
boundary even when no free surface current is present. The normal component
does not jump. A map that crosses the boundary at different angles separates
these two conditions and exposes a sensor alignment error.

$$
% caption: A plane interface between two linear media. The normal component of B
% crosses continuously; with no free surface current the tangential component of H is
% continuous too, so a change in permeability bends the field as it crosses.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\fill[black] (-3,-1.35) rectangle (0,1.35);
\fill[acc!10] (0,-1.35) rectangle (3,1.35);
\draw[black,thick] (0,-1.35)--(0,1.35);
\draw[acc,->,very thick] (-1.60,-0.20)--(1.60,-0.20);
\draw[acc,->,thick] (-1.70,-0.95)--(-1.70,0.85);
\draw[acc,->,thick] (1.70,-0.95)--(1.70,0.85);
\node[above] at (-1.50,1.36) {medium one};
\node[above] at (1.50,1.36) {medium two};
\node[below] at (0,-1.40) {interface};
\node[above] at (0.05,-0.20) {normal B};
\node[left] at (-1.92,-0.05) {tangent H};
\node[right] at (1.92,-0.05) {tangent H};
\end{tikzpicture}
$$

The interface equations concern local components. They do not guarantee that
the field is uniform within either medium. A finite specimen can have edge
fringing and spatially varying magnetization even when its bulk susceptibility
is well characterized. Boundary data are interpreted alongside the sample
shape, probe position, and source geometry.

### Demagnetization geometry and measurement protocol

Magnetization produces an internal field that often opposes the applied
magnetizing direction in a finite sample. This demagnetizing field is a shape
effect. An ellipsoid magnetized along one principal axis has it represented by

$$
\vec H_{\mathrm d}=-N\vec M,
\qquad
\vec H_{\mathrm{int}}=\vec H_{\mathrm a}-N\vec M,
$$

where $N$ is the dimensionless demagnetizing factor for that axis. The three
principal factors of an ellipsoid sum to one in SI conventions. A long rod
magnetized along its length has a small $N$. A thin disk magnetized
perpendicular to its face has a factor approaching one. The same material can
therefore show very different apparent response when cut into different
shapes.

A linear sample has

$$
\vec M=\chi_m\vec H_{\mathrm{int}},
\qquad
\vec M=
\frac{\chi_m}{1+N\chi_m}\vec H_{\mathrm a}.
$$

Ignoring $N$ overestimates magnetization whenever $\chi_m$ is positive. The
correction is modest for weak paramagnets and can dominate the apparent
response of a high-susceptibility specimen. A long-rod geometry is widely used
when a central region close to the applied field is required.

$$
% caption: Shape sets the internal magnetizing field. A long rod along the applied
% direction has a small demagnetizing factor, so the short blue opposing arrow barely
% cuts the field; a thin disk across the field has a large factor and a long opposing
% arrow, leaving much less H inside for the same source.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% long rod
\draw[black,fill=acc!8] (-3.15,-0.45) rectangle (-0.55,0.45);
\draw[acc,->,very thick] (-3.65,0)--(-0.35,0);
\draw[black,->,thick] (-1.05,0.24)--(-1.75,0.24);
\node[above] at (-1.85,0.75) {long rod};
\node[below] at (-1.85,-0.75) {small N};
% thin disk (flat across the field)
\draw[black,fill=acc!8] (1.75,-0.95) ellipse (0.32 and 0.95);
\draw[acc,->,very thick] (0.55,0)--(2.95,0);
\draw[black,->,thick] (2.60,0.30)--(1.15,0.30);
\node[above] at (1.75,1.20) {thin disk};
\node[below] at (1.75,-1.25) {large N};
\end{tikzpicture}
$$

A careful measurement protocol begins with an empty-coil calibration of
$H_{\mathrm a}$ at the intended current. Sample dimensions are measured before
mounting, including the axis selected for magnetization. The moment measurement
is converted to $M=m_{\mathrm{sample}}/V$. The demagnetizing factor is chosen
for the measured shape and orientation, after which
$H_{\mathrm{int}}=H_{\mathrm a}-NM$ is evaluated point by point.

Plotting $M$ against $H_{\mathrm{int}}$, rather than raw coil current, gives the
material response in the linear model. A straight low-field branch gives
$\chi_m$ from its slope. Reversing the source current removes a stationary
background signal. Ferromagnetic specimens require a controlled initial state,
full record of prior extrema, and separate increasing and decreasing sweeps.
The report should include temperature, sample dimensions, coil calibration,
probe location, field orientation, and the adopted demagnetizing factor.

> **Worked example (Demagnetizing correction for sample shape).** A reversible
> specimen has $\chi_m=0.200$ in an applied field $H_{\mathrm a}=500\ \mathrm{A/m}$.
> The internal field is reduced by the demagnetizing term $-N M$, and with
> $M=\chi_m H_{\rm int}$ this closes to $M=\chi_m H_{\mathrm a}/(1+N\chi_m)$. For a
> disk with demagnetizing factor $N=0.800$,
>
> $$
> M=\frac{(0.200)(500)}{1+(0.800)(0.200)}=86.2\ \mathrm{A/m},
> \qquad
> H_{\rm int}=H_{\mathrm a}-N M=431\ \mathrm{A/m},
> $$
>
> which check out as $M=\chi_m H_{\rm int}$. Multiplying the applied field by $\chi_m$
> directly gives $100\ \mathrm{A/m}$, a $16\%$ overestimate. The flux density at the
> centre is
>
> $$
> B=\mu_0(H_{\rm int}+M)=6.50\times10^{-4}\ \mathrm T,
> $$
>
> versus the empty-coil $\mu_0 H_{\mathrm a}=6.28\times10^{-4}\ \mathrm T$. The same
> material as a long rod along the field $(N=0.050)$ gives $M=99.0\ \mathrm{A/m}$: the
> intrinsic susceptibility is unchanged, only the shape moved the internal field. So
> comparing two samples demands matched shape, orientation, and temperature — or an
> explicit $N$ correction for each.

### Boundary measurement checks

A boundary experiment uses two probe orientations. A probe normal to a smooth
interface tests continuity of the normal B component. A probe tangent to the
same interface tests the permeability-dependent change in tangential B. The
probe centre must be farther from the boundary than the active sensor radius;
otherwise the reading averages material on both sides and cannot be compared
with either limiting value.

Free surface current and bound surface current have different roles in the
analysis. The tangential H jump condition uses free current supplied through a
circuit. Magnetization contributes through M and can change B even when no
free sheet current is present at the interface. Treating every observed B
change as a free-current signal produces an incorrect boundary inference.

Record a reference scan with the specimen removed, then repeat the same scan
with the specimen inserted and the same coil current. Current reversal removes
stationary environmental offsets from both scans. Comparing the reversal
differences isolates the material contribution while retaining a traceable
source calibration. The resulting data can test the linear-media equations
only over the stated field range and sample orientation.

## Domains and hysteresis

Ferromagnetism arises from cooperative interactions among neighboring magnetic
moments. The exchange interaction favors parallel alignment over microscopic
regions. A large specimen usually divides into magnetic domains rather than
maintaining one uniform direction everywhere. Within an individual domain the
moments are strongly aligned; neighboring domains can point in different
directions, leaving a small net moment for the whole specimen before an
external field is applied.

Domain formation balances several energy contributions. Exchange energy favors
alignment within a domain. Domain walls cost energy because the moment
direction changes across a finite region. Magnetostatic energy favors patterns
that reduce stray magnetic flux outside the specimen. The observed domain
structure minimizes the combined energy subject to sample shape, defects,
temperature, and prior magnetic history.

An applied magnetizing field changes that balance. Domains already oriented
near the applied direction can grow by domain-wall motion. Moments within a
domain can also rotate toward the applied direction. Both mechanisms increase
the macroscopic magnetization. Pinning by defects and grain boundaries makes
the wall motion partly irreversible, which produces hysteresis in a bulk
measurement.

$$
% caption: Domains respond to an applied field. A demagnetized specimen holds domains
% pointing in different directions with little net moment; an applied field grows and
% rotates the favorably aligned domains, so the net magnetization climbs.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% left: mixed domains
\draw[black,fill=acc!8] (-3.40,-1.15) rectangle (-0.60,1.15);
\draw[black] (-2.00,-1.15)--(-2.00,1.15);
\draw[black] (-3.40,0)--(-0.60,0);
\draw[acc,thick,->] (-3.00,0.55)--(-2.40,0.55);
\draw[acc,thick,->] (-1.00,0.55)--(-1.60,0.55);
\draw[acc,thick,->] (-2.70,-0.85)--(-2.70,-0.25);
\draw[acc,thick,->] (-1.30,-0.25)--(-1.30,-0.85);
% right: aligned domains
\draw[black,fill=acc!8] (0.60,-1.15) rectangle (3.40,1.15);
\draw[black] (2.00,-1.15)--(2.00,1.15);
\draw[acc,thick,->] (0.95,0.55)--(1.65,0.55);
\draw[acc,thick,->] (0.95,0)--(1.65,0);
\draw[acc,thick,->] (0.95,-0.55)--(1.65,-0.55);
\draw[acc,thick,->] (2.35,0.55)--(3.05,0.55);
\draw[acc,thick,->] (2.35,0)--(3.05,0);
\draw[acc,thick,->] (2.35,-0.55)--(3.05,-0.55);
\draw[black,->,thick] (0.85,1.55)--(3.15,1.55);
\node[above] at (2.00,1.55) {applied H};
\node[below] at (-2.00,-1.36) {mixed domains};
\node[below] at (2.00,-1.36) {aligned domains};
\end{tikzpicture}
$$

Thermal agitation weakens the domain-scale order. Above the Curie temperature,
the cooperative ferromagnetic ordering is lost and the material responds
paramagnetically. The transition temperature and domain behavior depend on
composition and microstructure. A magnetic classification therefore applies to
a stated temperature range, processing history, and field scale.

### Hysteresis, remanence, and coercivity

A hysteresis loop records the response during a complete forward-and-reverse
magnetizing cycle. Start from a demagnetized state and increase $H$ in the
positive direction. Magnetization rises rapidly while favorable domains grow,
then approaches saturation magnetization $M_{\mathrm s}$ as most moments align.
Further increase of the source mainly raises the flux density through the
source-field term once $M$ changes slowly.

Reducing $H$ from saturation follows a different branch. At $H=0$, a
ferromagnetic sample can retain a remanent flux density $B_{\mathrm r}$ or
remanent magnetization $M_{\mathrm r}$. A reversed field is required to reduce
the measured response to zero. Its magnitude is the coercive field
$H_{\mathrm c}$. Continuing the reverse sweep reaches negative saturation;
returning to positive field closes the loop.

$$
% caption: A quasistatic hysteresis loop. At zero applied field the sample keeps a
% remanent B; a reverse field of magnitude Hc, the coercive field, is needed to bring
% B back to zero. Which branch the sample follows depends on its sweep history.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (-3.00,0)--(3.00,0) node[right] {applied H};
\draw[->,black] (0,-2.25)--(0,2.55) node[above] {B};
\draw[acc,thick,->] plot[smooth] coordinates {(2.40,1.75)(1.00,1.55)(0,1.35)(-1.30,0.95)(-1.90,0)(-2.40,-1.75)};
\draw[acc,thick,->] plot[smooth] coordinates {(-2.40,-1.75)(-1.00,-1.55)(0,-1.35)(1.30,-0.95)(1.90,0)(2.40,1.75)};
\filldraw[draw=acc,fill=acc!12] (0,1.35) circle (0.06);
\filldraw[draw=acc,fill=acc!12] (-1.90,0) circle (0.06);
\node[right] at (0.15,1.90) {remanence};
\node[below] at (-2.15,-0.20) {coercivity};
\end{tikzpicture}
$$

The energy dissipated per unit volume in a quasistatic cycle is proportional to
the enclosed loop area. With $B$ plotted against $H$, the loss density is

$$
w_{\mathrm{loss}}=\left|\oint H\,\d B\right|.
$$

The expression has units $\mathrm{J\,m^{-3}}$. It measures irreversible domain
processes during the cycle. A narrow loop has lower hysteresis loss than a
wide loop at comparable operating amplitude.

### Soft and hard material tradeoffs

Magnetically soft materials have low coercivity and a narrow hysteresis loop.
Their magnetization changes readily when the source field changes, and their
cycle loss is relatively small. Such behavior suits transformer and inductor
cores, where the intended operating state repeatedly reverses. High
permeability can be valuable in a restricted low-field range, yet saturation
and frequency-dependent losses still set an operating limit.

Magnetically hard materials have larger coercivity and substantial remanence.
They retain a magnetized state after the source field is removed, making them
appropriate for permanent magnets and recorded magnetic states. Their larger
loop area represents a larger energy cost for repeated reversal. A material
chosen to preserve a stored magnetic state is therefore poorly suited to a
low-loss alternating cycle.

$$
% caption: Soft against hard magnetic material. The narrow soft loop has low
% coercivity and a small enclosed area, so little energy is lost per reversal; the
% wide hard loop keeps a large response at zero field and resists being switched.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
% soft (narrow loop), centred at x = -1.9
\draw[->,black] (-3.30,0)--(-0.50,0);
\draw[->,black] (-1.90,-1.45)--(-1.90,1.45);
\draw[acc,thick] plot[smooth cycle] coordinates {(-1.61,1.05)(-1.78,0.93)(-1.90,0.81)(-2.06,0.57)(-2.13,0)(-2.19,-1.05)(-2.02,-0.93)(-1.90,-0.81)(-1.74,-0.57)(-1.67,0)};
\node[above] at (-1.90,1.45) {soft};
% hard (wide loop), centred at x = 1.9
\draw[->,black] (0.50,0)--(3.30,0);
\draw[->,black] (1.90,-1.45)--(1.90,1.45);
\draw[acc,thick] plot[smooth cycle] coordinates {(2.98,1.05)(2.35,0.93)(1.90,0.81)(1.32,0.57)(1.05,0)(0.82,-1.05)(1.45,-0.93)(1.90,-0.81)(2.48,-0.57)(2.76,0)};
\node[above] at (1.90,1.45) {hard};
\node[below] at (-1.90,-1.55) {small loss area};
\node[below] at (1.90,-1.55) {large remanence};
\end{tikzpicture}
$$

Permeability and susceptibility require an operating-point label in a
ferromagnet. A secant estimate, such as $B/H$, depends on the initial state and
the endpoint of a sweep. A differential permeability,

$$
\mu_{\mathrm{diff}}=\frac{\d B}{\d H},
$$

describes a local slope on one specified branch. Near saturation,
$\d M/\d H$ becomes small and $\d B/\d H$ approaches $\mu_0$. A numerical
permeability quoted without field amplitude, branch direction, temperature,
and sample shape has limited predictive value.

A coercivity measurement uses a calibrated magnetizing coil and a signal
proportional to either $B$ or $M$. The sample is first driven to a stated
positive saturation condition. The current is then swept through zero at a
controlled rate while the response is sampled. Interpolation between the two
samples that bracket zero response gives $H_{\mathrm c}$. The intercept at
$H=0$ on the return branch gives remanence. Repeating the full loop after
reversing the initial saturation tests symmetry and can indicate drift in the
measurement chain.

### Saturation and data reduction

Saturation is a statement about magnetization, not a statement that flux
density becomes constant. As the available microscopic moments align,
$M$ approaches a limiting value $M_{\mathrm s}$. The total flux density still
contains the source contribution:

$$
B=\mu_0(H+M).
$$

Above magnetic saturation, an increase in $H$ produces little additional
magnetization, while $B$ continues to increase with a slope approaching
$\mu_0$. A graph of B alone can therefore appear to keep rising after the
material response has saturated. Separating M from B prevents the source-field
slope from being misidentified as additional moment alignment.

The initial magnetization curve is measured from a reproducibly demagnetized
state. It differs from a major hysteresis loop because domain populations and
wall positions begin in a different configuration. Minor loops trace only a
restricted range between previous extrema. They are relevant to low-amplitude
devices, but their slope and loss area cannot be substituted for a full
saturation-loop specification.

Data reduction begins with the coil calibration that converts current to
applied $H$. The sample geometry then sets the demagnetizing correction needed
for $H_{\mathrm{int}}$. A calibrated magnetic-flux or moment signal is
converted to B or M with the sensor scale and sample volume. Each point is
stored with the sweep direction and the preceding maximum field. A ferromagnet
has memory; two readings at the same applied current can represent different
states after different histories.

Noise treatment must preserve the loop shape. Averaging repeated points at a
fixed state reduces random sensor noise. Averaging values from opposite sweep
branches would erase the hysteresis that the experiment intends to measure.
Baseline offsets are measured with the sample absent or with a reversal
protocol appropriate to the sensor. Drift is checked by returning to a
previous reference current after a sweep.

The final report identifies whether a quoted permeability is an initial
slope, a differential slope at an operating point, a secant ratio over a stated
interval, or an effective value inferred from a particular geometry. These
quantities can differ by orders of magnitude in a ferromagnet. The field range,
temperature, frequency or sweep rate, sample shape, and magnetic prehistory
are part of the numerical value rather than optional experimental detail.

Traceable hysteresis data also retain the specimen dimensions and the selected
magnetization axis. Those records allow later comparison of coercivity,
remanence, and loss measurements made with different sample geometries.

## Magnetic energy and material selection

A reversible magnetization process stores magnetic energy in the coupled source
and material system. A quasistatic reversible branch has incremental energy density

$$
du=H\,\d B.
$$

Integrating from an unmagnetized reference state gives

$$
u(B)=\int_0^B H(B')\,\d B'.
$$

The integrand uses the actual constitutive branch. A linear medium with constant
permeability has

$$
B=\mu H,
\qquad
u=\frac{B^2}{2\mu}=\frac{\mu H^2}{2}.
$$

The units are joules per cubic metre. This result applies only while the
response is single-valued and reversible. A ferromagnetic path contains
history-dependent domain changes, so the energy supplied during an increase
and the energy returned during a decrease differ.

The B–H loop area gives this difference,

$$
w_{\mathrm{loss}}=\left|\oint H\,\d B\right|.
$$

The energy per unit volume becomes heat during one complete quasistatic
cycle. A small loop area reduces core heating in a repeated magnetic drive.
The loop area is independent of the graphical scale only when the axes are
calibrated in their physical units; an unscaled chart cannot determine energy
loss.

> **Worked example (Hysteresis heating of a transformer core).** A core of volume
> $V=1.50\times10^{-3}\ \mathrm{m^3}$ traces a $B$–$H$ loop enclosing
> $w_{\rm loss}=\left|\oint H\,\d B\right|=300\ \mathrm{J/m^3}$ per cycle, driven at
> $f=60.0\ \mathrm{Hz}$. Each cycle dissipates $w_{\rm loss}V$ as heat, so the
> hysteresis power is
>
> $$
> P=w_{\rm loss}\,V\,f=(300)(1.50\times10^{-3})(60.0)=27.0\ \mathrm W.
> $$
>
> Halving the loop area — a softer magnetic material — halves this loss. This is the
> quasistatic hysteresis loss only; at frequency, induced eddy currents add a separate
> dynamic loss that a slow-loop measurement does not capture.

The partition between stored and dissipated energy changes with the drive
amplitude. A minor loop far below saturation can have a much smaller loss
than a major loop driven close to saturation. Frequency also matters in a real
component because magnetic processes may lag the applied drive and circulating
currents can be induced within conductive material. A quasistatic loop provides
the baseline hysteresis loss; additional dynamic loss requires a measurement at
the intended waveform and frequency.

### Transformer-core choices

A transformer core guides magnetic flux through a closed low-reluctance path so
that the windings couple efficiently. The core material is usually magnetically
soft: low coercivity, low remanence, and a narrow loop reduce the energy needed
to reverse magnetization each cycle. A high low-field permeability reduces the
magnetizing current required for a specified operating flux density.

Core selection also requires a saturation margin. If the operating B approaches
the material’s saturation range, incremental permeability falls and the
magnetizing current rises sharply. The waveform can distort and losses increase.
Designs therefore specify a maximum B below saturation at the highest intended
temperature and the lowest expected supply frequency.

Conductive cores also support induced circulating currents. Laminating the core
into insulated thin sheets interrupts large current loops and reduces this
heating mechanism. Ferrite materials use high electrical resistivity for the
same reason at higher frequencies. These loss controls supplement, rather than
replace, the choice of a narrow hysteresis loop.

$$
% caption: A transformer core. Windings on the legs drive flux around a closed
% laminated soft-magnetic loop; the thin insulated plates block large circulating
% currents, and the working point is kept below saturation.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\fill[acc!8] (-2.30,-1.30) rectangle (2.30,1.30);
\fill[white] (-1.25,-0.55) rectangle (1.25,0.55);
\draw[black,thick] (-2.30,-1.30) rectangle (2.30,1.30);
\draw[black,thick] (-1.25,-0.55) rectangle (1.25,0.55);
\foreach \x in {-2.05,-1.75,-1.45,-1.15,-0.85,-0.55,-0.25,0.05,0.35,0.65,0.95,1.25,1.55,1.85,2.15} {
  \draw[black] (\x,0.55)--(\x,1.30);
  \draw[black] (\x,-1.30)--(\x,-0.55);
}
\draw[acc,very thick,->] (-0.70,0.925)--(0.70,0.925);
\draw[acc,very thick,->] (0.70,-0.925)--(-0.70,-0.925);
\foreach \y in {-0.75,-0.45,-0.15,0.15,0.45,0.75} {
  \draw[black,thick] (-1.78,\y) ellipse (0.60 and 0.12);
  \draw[black,thick] (1.78,\y) ellipse (0.60 and 0.12);
}
\node[above] at (0,1.36) {laminated core};
\node[left] at (-2.42,0) {coil};
\node[right] at (2.42,0) {coil};
\node[above] at (0,0.98) {B path};
\end{tikzpicture}
$$

Core loss is measured in watts per mass or volume at stated frequency, peak
flux density, waveform, and temperature. A quoted loss number without those
conditions cannot be transferred directly to another drive. Mechanical stress,
grain orientation, air gaps, and manufacturing heat treatment can alter both
permeability and loss. The finished component must be characterized in its
assembled geometry, especially when a deliberate air gap changes the magnetic
path.

### Temperature dependence and characterization errors

Weak paramagnets follow Curie-like behavior over an appropriate temperature and
field range:

$$
\chi_m=\frac{C}{T},
$$

where $C$ is a material-dependent Curie constant. Heating weakens the
orientation bias by increasing thermal agitation. Diamagnetic susceptibility
usually changes more weakly with temperature. Ferromagnetic order weakens
substantially as temperature rises toward the Curie temperature and the material
becomes paramagnetic above that transition.

$$
% caption: Ferromagnetic response against temperature. Spontaneous domain alignment
% holds up through the ordered region, falls steeply near the Curie temperature, and
% gives way above it to a weak paramagnetic tail.
\begin{tikzpicture}[>=stealth,font=\footnotesize]
\definecolor{acc}{HTML}{4A6FA5}
\draw[->,black] (0,0)--(5.40,0) node[right] {temperature};
\draw[->,black] (0,0)--(0,3.05) node[above] {response};
\draw[acc,thick] plot[smooth] coordinates {(0.25,2.65)(1.10,2.58)(2.05,2.35)(2.62,1.80)(2.92,0.70)(3.35,0.42)(4.20,0.28)(5.00,0.20)};
\draw[dashed,black] (2.92,0)--(2.92,2.20);
\node[below] at (2.92,0) {Curie point};
\node[above] at (1.20,2.72) {ordered};
\node[above] at (4.35,0.42) {para};
\end{tikzpicture}
$$

Temperature measurement requires a sensor thermally coupled to the specimen. A
sensor mounted on the coil former or chamber wall can miss the temperature gradient
between an internally heated core and the thermometer during alternating drive.
Stabilization time, drive amplitude, and temperature ramp direction belong in the
characterization record.

Several systematic errors recur in magnetic measurements. An incorrect sample
volume scales every reported magnetization. An uncertain demagnetizing factor
distorts the inferred internal H. A pickup coil with unknown effective area
miscalibrates B. Mechanical vibration, sensor offset, and trapped remanence
shift low-field readings. Electrical resistance in the drive winding changes
with temperature, so a nominal voltage source does not guarantee a constant
magnetizing current.

Characterization therefore uses a sequence of checks: empty-fixture baseline,
coil-current calibration, sample geometry measurement, controlled initial
state, temperature stabilization, and repeated forward/reverse sweeps. Results
are reported with the drive waveform, frequency, peak level, field direction,
sample orientation, and uncertainty method. This information separates a
material property measured under defined conditions from a response produced by
one particular fixture.

### Separating magnetic losses

Core loss is obtained from an electrical power balance with stated fixture terms. One
reduction is

$$
P_{\mathrm{core}}
=P_{\mathrm{in}}-I_{\mathrm{rms}}^2R_w-P_{\mathrm{empty}},
$$

where $R_w$ is the winding resistance at operating temperature and
$P_{\mathrm{empty}}$ is the no-sample or empty-fixture loss defined by the same
drive and measurement chain. The subtraction is valid only when those terms are
measured under the same frequency, waveform, and thermal conditions as the specimen.

- **Frequency series.** At fixed peak $B$, measure a sequence of frequencies
  rather than one power reading. Hysteresis loss per cycle follows loop area, so its
  power contribution rises approximately with cycle rate when loop shape is fixed.
  Induced-current loss rises more rapidly and depends on lamination thickness,
  resistivity, and magnetic-path geometry. The series identifies whether the core
  remains in its intended low-loss range.
- **Magnetic amplitude.** Determine peak $B$ from a calibrated sense winding or
  flux probe. Drive voltage alone is insufficient: it includes winding drops, and
  saturation changes the current-to-$B$ relation. Inspect the magnetic waveform
  before assigning a sinusoidal-loss specification to a distorted drive.
- **State and uncertainty.** Propagate core mass or volume, input power, winding
  resistance, temperature, frequency, and peak $B$ into the reported loss density.
  Repeat after a controlled demagnetization or saturation sequence as well as during
  repeated runs. Retracing the same minor loop can understate variation caused by
  magnetic history.
- **Matched specimens.** Compare samples with matched processing and geometry.
  Cutting, pressing, machining, and mounting stress alter domain-wall motion. An air
  gap changes the effective magnetic path and lowers apparent permeability even when
  bulk material is unchanged. Report intrinsic sample data separately from the
  finished magnetic-circuit response.

The record distinguishes a material property measured under defined conditions from
a loss value set by one fixture, waveform, and magnetic history.
## Measurement and model limits

A measured flux density becomes a material response only after the source field and
sample shape are accounted for.

> **Worked example (Recovering $M$ from a measured $B$).** An ellipsoid magnetized
> along a principal axis has demagnetizing factor $N=0.200$. An empty-coil calibration
> sets $H_{\mathrm a}=800\ \mathrm{A/m}$, and a pickup at the centre reads
> $B=1.60\ \mathrm{mT}$. Combining $B/\mu_0=H_{\rm int}+M$ with
> $H_{\rm int}=H_{\mathrm a}-NM$ eliminates $H_{\rm int}$:
>
> $$
> M=\frac{B/\mu_0-H_{\mathrm a}}{1-N}
> =\frac{1.27\times10^3-800}{0.800}=5.92\times10^2\ \mathrm{A/m},
> $$
>
> using $B/\mu_0=1.27\times10^3\ \mathrm{A/m}$. Then
> $H_{\rm int}=H_{\mathrm a}-NM=6.82\times10^2\ \mathrm{A/m}$, giving a secant response
>
> $$
> \chi_{\rm sec}=\frac{M}{H_{\rm int}}=0.868,\qquad
> \mu_{\rm sec}=\frac{B}{H_{\rm int}}=1.87\,\mu_0.
> $$
>
> "Secant" flags a value averaged over the measured interval — not a differential
> permeability, a saturation value, or a property of every direction. Substituting
> $H_{\mathrm a}$ for $H_{\rm int}$ here would misreport the material response by the
> full demagnetizing correction.

Uncertainty propagation follows the same sequence. The B calibration,
$H_{\mathrm a}$ calibration, and N estimate contribute separately to M. Near
$N=1$, the denominator $1-N$ amplifies every uncertainty, which makes a
thin-disk geometry a poor choice for extracting a high-susceptibility response.
A long axial specimen with small N reduces that amplification. Select and document
the specimen geometry during measurement design, then include its contribution in
the final uncertainty budget.

### Minor loops and magnetic history

The full saturation loop is one possible magnetic history. Components often
operate on minor loops after a limited change in drive rather than traversing
the major loop. Begin from a point on a major branch, reverse the applied field
before saturation, then return. The resulting minor loop encloses its own loss
area and has its own local slope. Its remanence and effective permeability
depend on the reversal point.

The same applied H can therefore give several possible B or M values. A
statement such as “the permeability at 200 amperes per metre” is incomplete
for a hysteretic material unless it includes the initial state, sweep direction,
previous extrema, and waveform amplitude. This history dependence is a
measurable property of domain-wall motion and pinning, not random instrument
scatter.

Minor-loop measurements use a fixed reset protocol. One sequence drives the
sample to a documented reference state, reaches the chosen upper drive level,
and cycles between the specified lower and upper values. The cycle is repeated
until consecutive loops agree within the measurement uncertainty. A different
reset or a larger previous excursion can change the loop even when the final
drive amplitude is identical.

The loss associated with a periodic minor loop is found from its own
$\left|\oint H\,\d B\right|$ area, multiplied by the operating frequency and
magnetic volume for a power estimate. Substituting the major-loop area
overstates loss for a small-signal component. Using a minor loop measured at
one temperature and frequency to predict another requires a separate
temperature and rate validation.

### Shape and calibration reconciliation

The B/H/M inversion can be checked by measuring the same material in two
shapes. A long rod and a thin disk have different raw slopes when M is plotted
against applied $H_{\mathrm a}$, because their demagnetizing factors differ.
After converting each point to

$$
H_{\mathrm{int}}=H_{\mathrm a}-NM,
$$

the low-field data should collapse onto one material-response curve within
uncertainty. Persistent separation after the correction indicates a problem
with the assumed N, sample volume, sensor calibration, anisotropy, or
history control.

Calibration reconciliation starts with an empty-fixture current sweep that
relates the actual coil current to applied H. A second reference checks the B
sensor or pickup coil against a known field or a traceable reference sample.
Sample dimensions are measured along the magnetization axis, and volume is
calculated from those dimensions rather than from a nominal stock size. The
same temperature sensor, drive waveform, and reset protocol are used for both
shapes.

Agreement is evaluated with residuals rather than visual overlap alone. At a
common corrected H, form

$$
z=\frac{M_{\mathrm{rod}}-M_{\mathrm{disk}}}
{\sqrt{u_{\mathrm{rod}}^2+u_{\mathrm{disk}}^2}}.
$$

Values of $|z|$ near unity support agreement within the stated uncertainty.
Large systematic residuals that grow with H often indicate nonlinear response
or a demagnetizing correction outside its valid range. Residuals that remain
constant can indicate a sensor zero offset or sample-volume bias.

### Model boundary statement

A concise material-data statement names the response branch, temperature,
frequency or sweep rate, field interval, sample geometry, magnetization axis,
demagnetizing correction, calibration references, and uncertainty convention.
It also says whether the reported quantity is B, H, M, secant permeability,
differential permeability, remanence, coercivity, or loop-loss density.

The linear B/H/M model applies only to the reversible interval used for the
inversion. Saturation, hysteresis, anisotropy, thermal gradients, large air
gaps, and dynamic loss require an expanded model or direct measurement. This
boundary statement makes the quoted data usable without claiming a wider range
than the experiment supports.

### Inversion uncertainty audit

The worked inversion contains three independent input classes: the B scale,
the applied-H scale, and the shape factor. Let their standard uncertainties be
$u_B$, $u_H$, and $u_N$. Differentiating

$$
M=\frac{B/\mu_0-H_{\mathrm a}}{1-N}
$$

gives the first-order uncertainty estimate

$$
u_M^2=
\left(\frac{u_B}{\mu_0(1-N)}\right)^2+
\left(\frac{u_H}{1-N}\right)^2+
\left(\frac{Mu_N}{1-N}\right)^2.
$$

For the numerical example, use
$u_B=0.020\ \mathrm{mT}$,
$u_H=8\ \mathrm{A\,m^{-1}}$, and
$u_N=0.010$. The three contributions to $u_M$ are approximately
$19.9\ \mathrm{A\,m^{-1}}$,
$10.0\ \mathrm{A\,m^{-1}}$, and
$7.4\ \mathrm{A\,m^{-1}}$. Their quadrature sum is

$$
u_M=23\ \mathrm{A\,m^{-1}}.
$$

The result is reported as
$M=(5.92\pm0.23)\times10^2\ \mathrm{A\,m^{-1}}$ for the stated
field, temperature, ellipsoidal shape, and reversible branch. The B
calibration dominates this particular uncertainty budget. Improving the
demagnetizing-factor estimate alone would not substantially improve the final
magnetization uncertainty.

The audit also catches inconsistent measurements. A B reading below the empty
coil value in a material expected to have positive low-field response may
indicate an incorrect sensor sign, a reversed coil lead, a different sample
orientation, or a diamagnetic contribution. A recovered M larger than an
independently measured saturation scale indicates that the linear central-field
model has been applied outside its domain or that volume and calibration data
need review.

Raw readings, calibration constants, geometry measurements, and all reduction
equations should remain with the published material curve. A future
measurement can then update one calibration factor or one shape estimate
without recreating the full experiment. This traceability supports later measurement
of the same material in another core, at another
temperature, or on a different hysteresis branch.

The uncertainty expression assumes independent input errors. A shared coil
calibration or common temperature drift introduces correlation between points
on a curve. Such effects are recorded as systematic terms rather than reduced
by averaging repeated readings. Separating random and systematic uncertainty
keeps an apparently precise fit from overstating the confidence of a
material-data inversion.

Independent replication with a second calibrated fixture gives the strongest
check on a reported response curve. Agreement across fixture geometries,
after the stated shape corrections, tests both the material model and the
calibration chain.