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I wish to light a LED by moving a permanent magnet relative to an inductor consisting of a coil of copper wire. Can I increase efficiency (current produced for a given magnet motion) by using an iron or ferrite core in my magnetic coil?

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Of course, that is what is regularly done in all common AC generators - the coils are wound on special steel poles with high permeability. Well placed ferromagnetic core can increase the induced emf in wires by three and even more orders of magnitude.

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