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Since the electric and the magnetic fields are interrelated, can a material have high permittivity and low permeability or the other way around? Would it be possible to have a material that would be highly magnetized upon an applied magnetic field but have low levels of electric polarizability, or the inverse?

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It all depends from the temperature of the material and of course from the existence of unpaired electrons in the outer shell of the atoms.

For permanent magnets below the Curie temperature a stable magnetic field do to the alignment of the magnetic dipoles of the involved subatomic particles exist and on higher temperatures this alignment gets destroyed by the kinetic energy (vibrations) of the subatomic particles. Such magnets could be an isolator or a good electric conductor.

For other materials immobile electrons in the outer shell are in interrelation with the subatomic particles of the atom or molecule and this determines the alignment of their magnetic dipoles. For some materials the interrelation can be changed by an external magnetic field easily and for others not. Independent from this the mobility of outer electrons can be good or bed or be zero.

In short, search for insulators with electron configurations which magnetic dipoles are easy to influence and for conductors with high interacting dipole moments which are hard to change.

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