Cooling magnets is important when those magnets are superconducting electromagnets. An electromagnet produces a magnetic field by passing electricity through coils of wire. If you want a powerful field you need a lot of electricity, and to do this efficiently you want the coils of wire to have low electrical resistance.

The magnets on the ITER use huge amounts of electricity, enough to just melt normal wires into a puddle. Instead they are made out of superconducting materials which have basically zero electric resistance. They need this in order to work, but the only materials we know of that can be superconducting are only that way when very cold. Keeping the magnets cold then is crucial to them working at all.

Those are not regular magnets, they're superconductive magnets. Superconductive materials need to be cold to stay superconductive, if they warm up they stop being so and this leads to electric resistance reappearing. If there is resistance in the wire you're losing a lot of electricity to heating the wire instead of producing magnetic field.

It uses electromagnets, that run insane ammount of electric current in order to generate required magnetic field. The only way to achieve it, that we currently know of, is by using superconductors - materials that allow electricity flow with very very little losses. (Electrical resistance of conductors increases with temperature) Superconductivity is achieved by cooling the material to as close to absolute zero as possible.

The electrical resistance of various metals varies with temperature. This is how electrical thermometers work and in fact how many heaters are self regulating. For copper the colder it is the less resistance there is. Until you get to a point at extremely cold temperatures where the resistance is zero. So you can send a current through the wire and there is nothing in the wire stopping it. It does still build up a magnetic field which stores the energy and can be used for various other things but you can maintain this magnetic field using very little power.

Of course this is not the entire truth, they do not make the magnets superconducting to save on the power bill. The problem is that any energy that is lost to electrical resistance gets turned into heat. And with the amount of current they need for the huge magnetic fields they need the amount of heat generated by any resistance is enough to melt the copper conductors. The Large Hadron Collider did this during an accident, called the quench incident. It was caused by a bad connector between the magnets which had a tiny bit of resistance. This caused the entire thing to explode in huge ball of green and blue sparks. Green was the vaporized copper and blue was the helium plasma. There are no footage of the fireball but there are images of the aftermath. So best keep those magnets cold and superconducting.

As the other answers say, it is because they are using superconductive magnets.

When a superconductive material is cooled below its critical temperature (different for each substance and we'll ignore magnetic field limits) ALL electrical resistance is removed.

Above this critical temperature the material may have a low, but finite resistance. Given the amount of current required to create the high magnetic fields even a very small resistance will result the release of a considerable amount of heat energy--very likely resulting in significant damage.

MRI machines make use of superconductive magnets to generate huge magnetic fields as well. If even a small part of the superconducting magnet is heated above the critical temperature a magnetic quench can occur. The high current will greatly heat this section leading to larger areas being exposed to higher than critical temperature with them going non-superconducting, and so on. This is very bad. The machines are built with safe guards to shut down the machine as quickly as possible, but there'll be some very loud noises and potential damage.

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