Hysteresis

Intuition

Apply a slowly varying field $H(t)$ to a ferromagnet and watch its magnetization. Unlike a paramagnet, the response is history-dependent: $M$ at a given $H$ depends on whether you arrived from $+H_{\max }$ or from $-H_{\max }$. Plot $M$ versus $H$ over one full cycle and you get a closed loop — the hysteresis curve.

That single picture is the identity card of a ferromagnetic material. Its shape decides whether the material is good for a transformer core, a permanent magnet, a read head, or a memory cell. Three numbers on the loop carry almost all the engineering information, and the area of the loop is literally the energy lost per cycle.

Formal definition

Take an initially demagnetized sample, sweep $H$ from $0$ to $+H_{\max }$, back to $-H_{\max }$, and back to $+H_{\max }$. The resulting trajectory $M(H)$ has:

• a virgin curve that leaves the origin on first magnetization,
• two saturation plateaux at $\pm M_s$ for $|H|\gtrsim H_K$,
• a descending branch from $+M_s$ at $+H_{\max }$ to $-M_s$ at $-H_{\max }$,
• a symmetric ascending branch on the way back.

The three quantities that label the loop are:

Symbol	Name	Definition
$M_s$	Saturation magnetization	$M$ at $H\to \infty$ — an intrinsic property of the material
$M_r$	Remanence	$M$ at $H=0$ on the descending branch — what the magnet “remembers”
$H_c$	Coercivity	$H$ at which $M=0$ on the descending branch — the field needed to erase the magnet

The squareness $M_r/M_s$ tells how well the magnet retains its remanence; the area enclosed by the loop is the energy dissipated per unit volume per cycle.

Key results

1. Loop area = energy loss per cycle

For one closed loop, the work done by the source per unit volume is

$$W={\mu }_0\oint HdM={\mu }_0(\text{area of the loop}).$$

This energy is irreversibly converted into heat — through domain-wall friction, eddy currents, and microscopic Barkhausen jumps. In a transformer running at 50 Hz, that loss happens 50 times per second and is the dominant source of core heating.

2. The soft–hard dichotomy

Two extreme designs dominate applications:

	Soft magnets	Hard magnets
Coercivity $H_c$	$<1$ kA/m	$>10$ kA/m, up to $10^6$
Loop shape	thin, low-area	wide, high-area
Remanence	low to moderate	high
Anisotropy $K_u$	small	large
Typical materials	soft Fe, permalloy (Ni${}_{80}$Fe${}_{20}$), Mn–Zn ferrite	NdFeB, SmCo${}_5$, AlNiCo, Sr/Ba ferrite
Used for	transformers, read heads, sensors	permanent magnets, motors, loudspeakers, MRAM bits

The compromise is fundamental: a soft magnet minimizes $H_c$ and loop area (low loss, high permeability); a hard magnet maximizes $M_r$ and $H_c$ (strong, stable remanence). No single material wins both.

3. What sets the coercivity

In an ideal Stoner–Wohlfarth single-domain particle,

$$H_c=\frac{2K_u}{{\mu }_0{M}_s}=H_K$$

is the anisotropy field — purely intrinsic. In real materials, domain-wall nucleation and pinning determine $H_c$, and measured values are typically 10–100× smaller than $H_K$. This gap, Brown’s paradox, makes coercivity an extrinsic property sensitive to microstructure, defects, grain size and surface treatment. The same alloy can have $H_c$ varying by a factor of 100 depending on processing.

4. Minor loops and the virgin curve

A loop swept between $\pm H_{\max }<\pm H_K$ is a minor loop, with smaller area, smaller remanence, and a smaller effective coercivity. The virgin curve (first magnetization from a demagnetized state) is inside the major loop because the as-cooled domain structure has $M=0$ on average.

These distinctions matter in practice: a permanent magnet’s “remanence” is whatever it was last saturated to, which need not be $M_s$ if the magnet was never fully saturated during manufacturing.

Limits

• Loops shown here assume quasi-static sweeping ($\omega \to 0$). At finite frequency, eddy currents and viscous wall motion broaden the loop further. The full frequency dependence is captured by the LLG equation coupled to Maxwell’s equations.
• Real samples have demagnetizing fields that shear the loop along the $H$ axis. The “intrinsic” loop $M$ vs the internal field $H_{\text{int}}=H_{\text{ext}}-NM$ is what we plot here; the experimental loop $M$ vs $H_{\text{ext}}$ is always sheared.

Summary

Hysteresis is a ferromagnet’s identity card. Three numbers carry the engineering content:

• $M_s$ — how much magnetization is available,
• $M_r$ — how much survives at zero field,
• $H_c$ — how big a reverse field is needed to wipe it out.

Their product, together with the loop area, dictates whether the material belongs in a transformer core (small $H_c$, small area) or in a permanent magnet (large $M_r$, large $H_c$). Everything else in this wiki — domains, anisotropy, dynamics — eventually feeds back into the shape of this curve.

Connections

• stoner-wohlfarth — the minimal model of a hysteretic single-domain particle
• magnetocrystalline-anisotropy — sets the maximum possible $H_c$
• magnetic-domains — domain walls and pinning are what makes real $H_c$ smaller than $H_K$
• magnetic-materials — where ferromagnets sit among the five families
• llg-equation — what happens to $\vec{m}(\vec{r},t)$ during a single irreversible jump

References

• B. D. Cullity & C. D. Graham, Introduction to Magnetic Materials (Wiley, 2009), Chs. 11–13.
• A. Hubert & R. Schäfer, Magnetic Domains (Springer, 1998).
• J. M. D. Coey, Magnetism and Magnetic Materials (Cambridge, 2010), Ch. 11.