# Does making a magnet move require more energy than a non-magnet?

I know that electric fields store energy, with their energy in an infinetesimal volume being proportional to $$E^2$$ at that point. I also know that a moving magnet creates an electric field (consequence of Faraday's law of induction).

Together to me this seems to imply that if I have two pieces of iron, identical except for the first one being magnetised and I apply a force on them to make them move and give them $$1\text{J}$$ of energy each, the second one will end up with a higher velocity because some of the energy given to the first one will go to setting up the electric field around it so it'll end up having less kinetic energy, hence I'd need to do more work upon the magnet to give it the same velocity compared to the non-magnet. This seems very weird and non-intuitive to me so I was wondering if my intuitions need updating or I'm making an error somewhere.

More generally, for non-relativistic electrons, the conjugate momentum, i.e. the bit that is conserved if space is translationally symmetric, is $$m\mathbf{v}-q\mathbf{A}/c$$ (libretext), where $$m$$ is mass, $$\mathbf{v}$$ is velocity, $$c$$ is speed of light, $$q$$ is charge and $$\mathbf{A}$$ is vector potential (of the electromagnetic field). So conservation of momentum for electron literally involves electromagnetic field. I am pretty sure similar expressions can be derived for particles with no electric charge but present magnetic dipole.