a hard magnetic material

A hard magnetic material is one that, once magnetized, stubbornly stays magnetized, this is what we simply call a permanent magnet. As with soft magnets, hard does not mean mechanically hard; it means magnetically stubborn, resistant to being demagnetized by stray fields, heat, or rough handling. The horseshoe magnet, the motor magnet, the little disc holding a note to the fridge: all are hard magnets doing the one job a soft magnet cannot, holding a field on their own.

On the hysteresis loop a hard magnet is fat and wide: high remanence so it keeps a strong field after the magnetizer is switched off, and above all high coercivity, meaning you must apply a large reverse field to demagnetize it. A single number that captures how much useful magnetism you can pack in is the maximum energy product, written BHmax; the bigger it is, the smaller and lighter the magnet for a given pull. You make a magnet hard by pinning domain walls hard, using fine single-domain particles and strong crystal anisotropy that resists reversal. Modern rare-earth magnets, neodymium-iron-boron and samarium-cobalt, have enormous energy products; older alnico and cheap hexagonal ferrites are weaker but each has its niche.

Hard magnets do work without being plugged in, which is why they hide inside almost every electric motor, generator, loudspeaker, headphone, hard-disk actuator, and MRI machine. The tradeoffs are practical and honest: the strongest neodymium magnets are expensive, mechanically brittle, corrode without a coating, and lose strength as they warm toward their Curie temperature, so a design that runs hot may prefer the weaker but more heat-stable samarium-cobalt. And because rare-earth supply is concentrated and costly, the humble ferrite magnet, weaker but cheap and corrosion-proof, still wins wherever raw strength is not the priority.