Magnetised Iron

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The piece of iron becomes magnetic itself because its inner structure changes. The model of the elementary magnets describes this concept: Usually, elementary magnets in iron face all directions; close to a magnet, however, they align themselves after the magnet. After being rubbed with a magnet multiple times, the elementary magnets stay in this position for long, so that the piece of iron keeps the magnetic effects without having a magnet closeby.
As you can see in the animation, the piece of iron itself forms a north and a south pole by aligning its elementary magnets.