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When we place an iron near to a magnet, the induced pole nearest to the magnet would be the opposite, therefore there is an attraction. When we take it away, iron loses all its induced magnetism. What if it's a unmagnetized steel this time? My approach would be the same thing will happen, except that it will be permanently magnetized even though it is removed from the magnet. If we place the North Pole of the steel near the magnet, it would then repel. So is the polarity of a permanent magnet even possible to change after it becomes permanently magnetized? If yes, with what methods? Solanoid maybe?

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"Permanent" magnets aren't really permanent. Ferromagnetic materials have a property called Remanence which is a measure of how much field strength remains in the material after an external field has been removed. Materials that have high remanence make good "permanent" magnets: They retain a strong magnetic field after being exposed to intense external field in the factory where they were made.

You can always demagnetize and/or re-magnetize any ferromagnetic object.

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