So electromagnetism has two components, basically – one that is dependent on how many electrons are where (E, electric) and one on how fast the electrons are moving (B, magnetic). The two operate at right angles to each other – if you make a coil of wire with electrons going around it due to the E field down the wire, the B field is at right angles to this. The laws you are looking for are as other respondents have noted Maxwell’s equations.
* So why do magnetic fields pick up ferrous metals? Because if moving electrons make the field then the field makes electrons move.
* So why don’t all materials get picked up by magnets? Because some electrons basically ‘aren’t free enough’ to experience this force – it’s like the rest of the atom shields them.
* So what about permanent magnets? (At ELI5 level? Hoo boy) Electrons are always spinning and this gives them their own magnetic field. Left to their own devices they’ll pair up to make this cancel out to zero, but in ferrous materials they don’t get the chance, leading to materials that can be ‘trained’ to have a standing magnetic field by putting them in a magnetic field for a while.
I realise that a lot of these answers say ‘it just does’ to your questions. That’s honestly the best we’ve got — welcome to physics. Maxwell and those who went before him made a bunch of observations he was able to unify into a theory that explains so much that we can treat it as true – a physical law.
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