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I understand that ferromagnets have domains that align parallel to the applied magnetic field, in which the spin of the unpaired electrons line up. But I am curious as to what happens to such magnets when placed under a electric field. Because, in cases of dielectrics, the electron polarization causes the cloud of electrons to move one way to counter the applied field. Would the same happen for permanent magnets? what would happen to its magnetic property?

Thanks

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It depends on the transport property of the ferromagnets. If the ferromagnet being considered is also a conductor, then the spin-up current will have a different resistivity than the spin-down current; this will result in a spin-polarized current in that ferromagnetic material. Details about this can be read in many spintronic articles.

If the ferromagnet is not a conductor, than the effects can range from nothing to very complicated. But the logic is the following the ferromagnetic phenomenon emerges from the interplay between (at least) two phenomena, namely unpaired electron spins and "relatively strong" Coulomb interactions (https://en.wikipedia.org/wiki/Exchange_interaction). That can happen because these two phenomena are embodied in one entity called electron. Having said that, you may realize that the part of magnetism that you could influence with your electric field is the charge of electrons, which is also the source of the coulomb interaction. If your electric field is strong enough that will "create" another coulomb interaction which interferes with the coulombs interactions that "create" the ferromagnetism in the considered material.

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