Ferromagnetic materials do not generally respond to magnetic fields through one reversible linear relation. Their current state depends partly on their magnetic history. This behavior is called hysteresis.

Hysteresis loop

If the magnetizing field is cycled, the magnetic flux density $B$ traces a loop rather than retracing the same path.

After the applied magnetizing field returns to zero, the material can retain a nonzero flux density. This is remanence or remanent magnetization.

A reverse field is required to bring the magnetic response back to zero. The required magnitude characterizes the coercivity.

Saturation

At sufficiently strong magnetizing fields, most available magnetic domains are aligned and further increases in the applied field produce relatively little additional magnetization. The material approaches magnetic saturation.

Energy loss

Cycling around the hysteresis loop dissipates energy as heat. The energy lost per cycle per unit volume is related to the area enclosed by the loop.

Soft and hard magnetic materials

Soft magnetic materials have narrow loops and low coercivity, making them useful for transformer and machine cores that reverse magnetization repeatedly. Hard magnetic materials have larger coercivity and retain magnetization, making them useful as permanent magnets.