Magnetic Hysteresis
When a ferromagnet is magnetized and the applied field is reversed, the magnetization does not retrace its original path — it lags behind, forming the classic hysteresis loop. Drag the applied field H left and right to watch the B–H curve emerge in real time. Below the curve, magnetic domains grow and shrink as domain walls sweep through the material. Adjust material properties to see how soft iron differs from hard steel.
B = μ0(H + M) · M = Ms tanh[(H ± Hc) / a]
The hysteresis loop
The word hysteresis comes from the Greek “to lag behind.” When you magnetize a ferromagnet by increasing the applied field H, then reverse the field, the magnetization M does not return along the same path. Instead it traces a loop — the hysteresis loop — whose area represents energy dissipated as heat in each magnetization cycle. This energy loss is why transformer cores use soft magnetic materials with narrow loops, while permanent magnets use hard materials with wide loops that resist demagnetization.
Domains and domain walls
A ferromagnet is divided into magnetic domains — regions where atomic magnetic moments are aligned in a common direction. Between domains lie domain walls, thin transition regions where the magnetization rotates from one orientation to another. When an external field is applied, favorably oriented domains grow at the expense of others as domain walls sweep through the material. In soft materials, walls move easily; in hard materials, defects and grain boundaries pin the walls, requiring stronger fields to move them. This pinning is what gives hard magnets their large coercivity.
Coercivity and remanence
Two key parameters define a hysteresis loop. The coercivity Hc is the reverse field needed to reduce the magnetization to zero — it measures how “hard” the magnet is. The remanence Br is the flux density remaining when the applied field returns to zero — it measures the strength of the residual magnetism. A permanent magnet wants both large Hc (hard to demagnetize) and large Br (strong field). A transformer core wants small Hc (low loss) but can tolerate moderate Br.
The Jiles–Atherton model
This simulation uses a simplified Jiles–Atherton-inspired approach where the anhysteretic magnetization follows a Langevin-like function, and irreversible domain wall motion introduces the lag that creates hysteresis. The shape parameter “squareness” controls how abruptly the magnetization switches — a small value gives a square loop (like a recording medium), while a large value gives a gradual, sloped loop (like a soft ferrite). Real materials exhibit far more complex behavior including minor loops, accommodation, and aftereffect, but this model captures the essential physics beautifully.