Magnetization, H-field, and magnetic materials: diamagnets, paramagnets, ferromagnets

Intuition Beginner

When you bring a magnet near a piece of iron, the iron becomes magnetised — it develops its own magnetic field. The iron is not creating new magnetic charges. Instead, the atomic currents inside the iron (electrons orbiting nuclei, and the intrinsic spin of electrons) are being aligned by the external field.

The magnetisation $\mathbf{M}$ of a material measures how much magnetic dipole moment per unit volume the material contains. Think of each atom as a tiny current loop producing a small magnetic field. In an unmagnetised material, these loops point in random directions and cancel out. An applied field lines them up, and their individual fields add together to produce a net field.

The picture parallels the polarisation story from electrostatics 10.01.04. In dielectrics, the electric field aligns electric dipoles. In magnetic materials, the magnetic field aligns magnetic dipoles. The difference is that electric dipoles come from charge separation, while magnetic dipoles come from current loops.

Three classes of magnetic material behave differently. Diamagnets (copper, water, bismuth) develop a magnetisation that opposes the applied field — they are weakly repelled by a magnet. Paramagnets (aluminium, platinum, oxygen gas) develop a magnetisation that aligns with the applied field — they are weakly attracted. Ferromagnets (iron, nickel, cobalt) develop enormous magnetisation that can persist even after the external field is removed — permanent magnets are ferromagnetic.

Visual Beginner

Material type	Response to applied field	Typical susceptibility	Example
Diamagnet	Opposes (weak)	${\chi }_m\sim -10^{-5}$	Copper, water
Paramagnet	Aligns (weak)	${\chi }_m\sim +10^{-3}$	Aluminium, oxygen
Ferromagnet	Aligns (strong)	${\chi }_m\sim 10^3$	Iron, nickel

Worked example Beginner

A long solenoid with $n=1000$ turns per metre carries current $I=1$ A. The interior field is $B_0={\mu }_{0}nI=4\pi \times 10^{-7}\times 1000\times 1=1.26\times 10^{-3}$ T.

Now insert an iron core with relative permeability ${\mu }_r=5000$. The field inside becomes $B={\mu }_r{B}_0=5000\times 1.26\times 10^{-3}=6.3$ T.

The iron amplifies the field by a factor of 5000. This is why electromagnets use iron cores: a modest coil current produces an enormous magnetic field. The iron atoms align their magnetic moments with the applied field, and each atom contributes its own small field, all adding up.

Check your understanding Beginner

Formal definition Intermediate+

Magnetisation. The magnetisation $\mathbf{M}(\mathbf{r})$ is the magnetic dipole moment per unit volume:

$$\mathbf{M}(\mathbf{r}):=\frac{d\mathbf{m}}{d\tau }$$

where $d\mathbf{m}$ is the total magnetic dipole moment in the volume element $d\tau$ at position $\mathbf{r}$. For $N$ atoms per unit volume, each with magnetic moment ${\mathbf{m}}_i$: $\mathbf{M}=N⟨{\mathbf{m}}_i⟩$ where the average is over a volume large compared to the atomic spacing but small compared to macroscopic variations.

Bound currents. A magnetised material produces the same magnetic field as a distribution of bound currents:

$${\mathbf{J}}_b=\nabla \times \mathbf{M}\text{(volume bound current density)}$$ $${\mathbf{K}}_b=\mathbf{M}\times \hat{\mathbf{n}}\text{(surface bound current density)}$$

where $\hat{\mathbf{n}}$ is the outward unit normal to the surface of the material. The total vector potential is:

$$\mathbf{A}(\mathbf{r})=\frac{{\mu }_0}{4\pi }{\int }_V\frac{{\mathbf{J}}_b({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d{\tau }^{\prime }+\frac{{\mu }_0}{4\pi }{\oint }_S\frac{{\mathbf{K}}_b({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}d{a}^{\prime }$$

The auxiliary H-field. The bound current adds to the free current in Ampere's law. Define the auxiliary field (or H-field):

$$\boxed{\mathbf{H}:=\frac{1}{{\mu }_0}\mathbf{B}-\mathbf{M}}$$

Then Ampere's law in matter becomes:

$$\nabla \times \mathbf{H}={\mathbf{J}}_f$$

where ${\mathbf{J}}_f$ is the free current density (the current you actually control, as opposed to the bound currents in the material). In integral form:

$$\oint \mathbf{H}\cdot d\mathbf{l}=I_{f,\text{enc}}$$

This is the magnetostatic analogue of the electric displacement $\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}$.

Linear materials. For linear, isotropic magnetic materials, the magnetisation is proportional to $\mathbf{H}$:

$$\mathbf{M}={\chi }_m\mathbf{H}$$

where ${\chi }_m$ is the magnetic susceptibility. Then:

$$\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M})={\mu }_0(1+{\chi }_m)\mathbf{H}=\mu \mathbf{H}$$

where $\mu ={\mu }_0(1+{\chi }_m)={\mu }_0{\mu }_r$ is the permeability and ${\mu }_r=1+{\chi }_m$ is the relative permeability.

Diamagnets have ${\chi }_m<0$ (${\mu }_r<1$), **paramagnets** have ${\chi }_m>0$ (${\mu }_r>1$), and ferromagnets have ${\chi }_m\gg 1$ (and $\mathbf{M}$ is not proportional to $\mathbf{H}$; the linear relation breaks down).

Counterexamples and caveats

• B and H are different fields with different units. The relation $\mathbf{B}=\mu \mathbf{H}$ holds only in linear isotropic materials. Even when it holds, $\mathbf{B}$ is the physical field (the one that determines the Lorentz force), while $\mathbf{H}$ is an auxiliary quantity useful in problems with free currents and specified boundary conditions.
• Ferromagnets are not linear. In a ferromagnet, $\mathbf{M}$ depends on the history of the sample (hysteresis), not just the current value of $\mathbf{H}$. There is no single-valued function $\mathbf{M}(\mathbf{H})$.
• The bound currents are real. The volume current ${\mathbf{J}}_b=\nabla \times \mathbf{M}$ and surface current ${\mathbf{K}}_b=\mathbf{M}\times \hat{\mathbf{n}}$ produce the same magnetic field as the actual magnetisation. They are not fictions: they represent the macroscopic effect of the aligned atomic current loops.

Key derivation Intermediate+

Derivation (Bound currents from magnetisation).

Theorem. A magnetised body with magnetisation $\mathbf{M}(\mathbf{r})$ produces the same vector potential as the bound current distribution ${\mathbf{J}}_b=\nabla \times \mathbf{M}$ (volume) and ${\mathbf{K}}_b=\mathbf{M}\times \hat{\mathbf{n}}$ (surface).

Proof. The vector potential of a single magnetic dipole $\mathbf{m}$ is ${\mathbf{A}}_{\text{dip}}=({\mu }_0/4\pi )\mathbf{m}\times \hat{\mathbf{R}}/R^2$ where $\mathbf{R}=\mathbf{r}-{\mathbf{r}}^{\prime }$. For a distribution of dipoles with density $\mathbf{M}$:

$$\mathbf{A}(\mathbf{r})=\frac{{\mu }_0}{4\pi }{\int }_V\frac{\mathbf{M}({\mathbf{r}}^{\prime })\times \hat{\mathbf{R}}}{R^2}d{\tau }^{\prime }$$

Using the identity $\hat{\mathbf{R}}/R^2=-{\nabla }^{\prime }(1/R)$ and the vector calculus identity ${\nabla }^{\prime }\times (f\mathbf{M})=f({\nabla }^{\prime }\times \mathbf{M})-\mathbf{M}\times {\nabla }^{\prime }f$ with $f=1/R$:

$$\mathbf{A}=\frac{{\mu }_0}{4\pi }{\int }_V\frac{{\nabla }^{\prime }\times \mathbf{M}}{R}d{\tau }^{\prime }-\frac{{\mu }_0}{4\pi }{\int }_V{\nabla }^{\prime }\times \left(\frac{\mathbf{M}}{R}\right)d{\tau }^{\prime }$$

The first integral is the contribution from ${\mathbf{J}}_b=\nabla \times \mathbf{M}$. For the second integral, use the curl theorem: ${\int }_V\nabla \times \mathbf{F}d\tau =-{\oint }_S\mathbf{F}\times d\mathbf{a}$. This gives:

$$-\frac{{\mu }_0}{4\pi }{\int }_V{\nabla }^{\prime }\times \left(\frac{\mathbf{M}}{R}\right)d{\tau }^{\prime }=\frac{{\mu }_0}{4\pi }{\oint }_S\frac{\mathbf{M}\times {\hat{\mathbf{n}}}^{\prime }}{R}d{a}^{\prime }$$

This is the contribution from ${\mathbf{K}}_b=\mathbf{M}\times \hat{\mathbf{n}}$. Combining:

$$\mathbf{A}(\mathbf{r})=\frac{{\mu }_0}{4\pi }{\int }_V\frac{{\mathbf{J}}_b({\mathbf{r}}^{\prime })}{R}d{\tau }^{\prime }+\frac{{\mu }_0}{4\pi }{\oint }_S\frac{{\mathbf{K}}_b({\mathbf{r}}^{\prime })}{R}d{a}^{\prime }\square$$

Bridge. The bound current decomposition builds toward a complete description of magnetic fields in matter by separating the contribution of the material (through ${\mathbf{J}}_b$ and ${\mathbf{K}}_b$) from the free currents. The foundational insight is that the magnetised body is equivalent to a distribution of currents, which appears again in the boundary conditions 10.02.04 where the surface bound current produces the discontinuity in the tangential component of $\mathbf{M}$. The central message is that the H-field formulation eliminates the bound currents from Ampere's law, generalises the electrostatic $\mathbf{D}$-field to magnetostatics, and is dual to the electric displacement $\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}$ 10.01.04. Putting these together, the H-field provides the practical framework for solving magnetostatic problems in the presence of magnetic materials, while the bound currents give the physical picture of how atomic dipoles produce the macroscopic field.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib has bilinear forms and basic integration theory but does not contain the magnetisation field, the bound current decomposition, the H-field, the magnetic susceptibility tensor, or the theory of ferromagnetic hysteresis. Formalising the bound currents would require the curl theorem on bounded domains and the identification of surface terms, which is not yet available. lean_status: none.

Advanced results Master

Ferromagnetism and hysteresis. Ferromagnetic materials exhibit a nonlinear, history-dependent relationship between $\mathbf{M}$ and $\mathbf{H}$. Starting from a demagnetised state and increasing $\mathbf{H}$, the magnetisation follows an initial magnetisation curve until it reaches the saturation magnetisation ${\mathbf{M}}_s$ (all atomic moments aligned). Reducing $\mathbf{H}$ to zero leaves a remanent magnetisation ${\mathbf{M}}_r\ne 0$ — this is a permanent magnet. Reversing $\mathbf{H}$ reduces $\mathbf{M}$ to zero at the coercive field $-{\mathbf{H}}_c$. The closed loop is the hysteresis curve.

The energy dissipated per cycle is the area enclosed by the hysteresis loop: $W=\oint \mathbf{H}\cdot d\mathbf{B}$ per unit volume. This energy loss heats transformer cores and limits the efficiency of electromagnetic devices.

Domain structure. Ferromagnetic materials are divided into Weiss domains — regions where all atomic moments are aligned. Adjacent domains are separated by Bloch walls (transition regions where the moments rotate between domain orientations). An unmagnetised ferromagnet has randomly oriented domains whose net magnetisation cancels. Applying a field causes domain walls to move (favourable domains grow at the expense of unfavourable ones) and, at higher fields, domain rotation. This explains why ferromagnets can be magnetised below the Curie temperature even in weak fields.

Quantum mechanical origin. The magnetic moments in ferromagnets arise from electron spin (not orbital motion). The exchange interaction (a quantum mechanical effect with no classical analogue) aligns neighbouring spins, producing the enormous susceptibilities of ferromagnets. Above the Curie temperature $T_C$, thermal energy overcomes the exchange interaction and the material becomes paramagnetic, following the Curie-Weiss law ${\chi }_m=C/(T-T_C)$.

Anisotropic and nonlinear media. In a general anisotropic material, the permeability is a tensor: $B_i={\mu }_{ij}{H}_j$. In nonlinear materials, $\mu$ depends on $\mathbf{H}$. The most general constitutive relation is $\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M}(\mathbf{H}))$ where $\mathbf{M}$ is a nonlinear function encoding the hysteresis.

Synthesis. The theory of magnetisation provides the material-side description of how matter responds to and modifies magnetic fields. The foundational insight is that the atomic current loops in matter produce bound currents that are captured by the magnetisation $\mathbf{M}$ and decomposed into volume and surface contributions. The central message is that the H-field formulation eliminates bound currents from Ampere's law, providing a practical tool for engineering calculations. Putting these together, the magnetisation framework is dual to the polarisation framework of electrostatics 10.01.04, the boundary conditions follow from the bound currents 10.02.04, and the demagnetisation factor determines the internal field of finite samples 10.02.05. The full macroscopic Maxwell equations in matter 10.03.04 combine both electric and magnetic material responses.

Full proof set Master

Proposition (Bound surface current of a uniformly magnetised body). For a body $V$ with uniform magnetisation ${\mathbf{M}}_0$ bounded by a smooth surface $S$ with outward normal $\hat{\mathbf{n}}$, the bound volume current vanishes (${\mathbf{J}}_b=0$) and the bound surface current is ${\mathbf{K}}_b={\mathbf{M}}_0\times \hat{\mathbf{n}}$.

Proof. Since $\mathbf{M}={\mathbf{M}}_0$ is constant inside $V$: ${\mathbf{J}}_b=\nabla \times {\mathbf{M}}_0=0$ (the curl of a constant vector field vanishes). On the surface, ${\mathbf{K}}_b=\mathbf{M}\times \hat{\mathbf{n}}={\mathbf{M}}_0\times \hat{\mathbf{n}}$. This is the only source of magnetic field from a uniformly magnetised body.

The vector potential is then $\mathbf{A}(\mathbf{r})=({\mu }_0/4\pi ){\oint }_S({\mathbf{M}}_0\times {\hat{\mathbf{n}}}^{\prime })/|\mathbf{r}-{\mathbf{r}}^{\prime }|d{a}^{\prime }$. For the specific case of a sphere of radius $R$: the surface current ${\mathbf{K}}_b=M_0\sin \theta \hat{\mathbf{\varphi }}$ produces a uniform field ${\mathbf{B}}_{\text{in}}=(2/3){\mu }_0{\mathbf{M}}_0$ inside and a perfect dipole field ${\mathbf{B}}_{\text{out}}=({\mu }_0/4\pi )(2\mathbf{m}\cdot \hat{\mathbf{r}}\hat{\mathbf{r}}-\mathbf{m})/r^3$ outside, with $\mathbf{m}=(4/3)\pi R^3{\mathbf{M}}_0$ being the total dipole moment. $\square$

Connections Master

• Dielectrics and polarisation 10.01.04 is the electrostatic analogue: $\mathbf{P}$ plays the role of $\mathbf{M}$, $\mathbf{D}$ plays the role of $\mathbf{H}$, and bound charges replace bound currents.
• Boundary conditions 10.02.04 follow from the bound currents: the surface bound current ${\mathbf{K}}_b$ produces the tangential B discontinuity.
• Magnetic scalar potential 10.02.05 is useful in regions with no free current, where $\nabla \times \mathbf{H}=0$ allows writing $\mathbf{H}=-\nabla {\Phi }_m$.
• Maxwell equations in matter 10.03.04 combine $\mathbf{D}$ and $\mathbf{H}$ into the complete set of equations with material response.
• Biot-Savart and Ampere 10.02.01 provide the fundamental laws from which the bound current formalism is derived.

Historical & philosophical context Master

The distinction between diamagnetic and paramagnetic response was established by Faraday in 1845, who discovered that bismuth is repelled by a magnet (diamagnetism) while other substances are attracted. The term "diamagnetic" was coined by Faraday himself.

Weiss (1907) introduced the molecular field hypothesis to explain ferromagnetism, postulating an internal field proportional to the magnetisation. The quantum mechanical basis was established by Heisenberg (1928), who showed that the exchange interaction between electron spins produces the enormous alignment energy characteristic of ferromagnets.

The hysteresis curve was first systematically measured by Warburg (1881) and Ewing (1882), who also introduced the term "hysteresis" (from the Greek "to lag behind"). The connection between hysteresis and energy dissipation was established by Steinmetz (1892), who derived the empirical law $W\propto B_{\max }^{1.6}$ for energy loss per cycle in transformer cores.

The Langevin theory of paramagnetism (1905) was one of the first applications of statistical mechanics to magnetic systems, predating quantum mechanics. The classical treatment fails at low temperatures, where the quantum Brillouin function replaces the Langevin function — a result of the quantisation of angular momentum.

Bibliography Master

• Faraday, M., "Experimental Researches in Electricity," Phil. Trans. Roy. Soc. 136, 21-40 (1846).
• Weiss, P., "L'hypothese du champ moleculaire et la propriete ferromagnetique," J. Phys. Theor. Appl. 6, 661-690 (1907).
• Heisenberg, W., "Zur Theorie des Ferromagnetismus," Z. Phys. 49, 619-636 (1928).
• Langevin, P., "Magnetisme et theorie des electrons," Ann. Chim. Phys. 5, 70-127 (1905).
• Griffiths, D. J., Introduction to Electrodynamics, 4th ed. (Cambridge, 2017).
• Jackson, J. D., Classical Electrodynamics, 3rd ed. (Wiley, 1999).
• Landau, L. D. and Lifshitz, E. M., Electrodynamics of Continuous Media, 2nd ed. (Pergamon, 1984).