There is the boundry condition that E -> 0 faaar frpm all chhrges.  We also add the divergence and the curl of B which are part of magnetostatcis.

Electric field lines always orginate on postive charges and terminate on negative charges, while magnetic field lines dont have a begin and an end.They typically form closed loops. It is also believed thus that magentic monopoles do not exist.
Electric forces are enormously larger than magnetic ones. This has to do with the size of the fundamenta constanst.

The currnet matters. So if we have a current with a smallish velocity by pouring huge amounts of charge down the wire. This charge would simulateounsly have a bigggg electric force as tp swamp the magentic one. But if our wire is neutral and by embedding an equal amount of oppposite charges than the electric field will cnacel out. THis leaves the magnetic field to stand alone.