How can a magnetic field be produced without using a magnet?
A magnetic field is a field which is present around magnet and has magnetic properties. So, to produce magnetic field a magnet is surely needed. But we have learnt from the Oersted's experiment that whenever a current is passed through a conductor, a magnetic field is produced around it.
The current carrying conductor behaves like a magnet. And a magnetic field is produced around it. The magnetic field produced depends on the magnitude of the current and the length or number of loop of current. The direction of the produced magnetic field from a current carrying conductor is given by Right-Hand Thumb Rule.