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If I approched a magnet to a coil does it moves it by the effect of induced electric current so it repells away from the magnet or it just induces the current in the coil without the coil making a flux that makes a repulsion force that can make the coil move away?

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you can think of the repulsion due to a coil and a magnet in two ways

way 1: a coil and a magnet both have a magnetic field.A magnetic pressure force between them causes a force of repulaion/attraction

or 2, the lorentz force on each of the e.g moving charges in the coil q( vxb) cause it to repell

You are over complicating it, yes technically there is an induced electric field if i were to MOVE a magnet near a coil of current. however there is still a repulsion force in a magnostatic case

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