If them magnetic field is creating electric field and they combine to
form EM waves, why does a compass show a magnetic field around the
To produce EM waves, we do need the current to be time varying such that the magnetic field is time varying which induces a time varying electric field etc.
However, it isn't that case that the entire magnetic field produced by the time varying current is associated with EM radiation.
In the reactive near field close to the wire, there are time varying electric and magnetic fields that are not associated with EM waves (which transport energy away) but are, rather, associated with energy storage. From the linked article:
For example, current flowing in the antenna creates a purely magnetic
component in the near-field, which then collapses as the antenna
current begins to reverse, causing transfer of the field's magnetic
energy back to electrons in the antenna as the changing magnetic field
causes a self-inductive effect on the antenna that generated it. This
returns energy to the antenna in a regenerative way, so that it is not
This is all quite complicated in general but, for low frequency, e.g., 60Hz AC current along a wire in a circuit, the sinusoidally time varying magnetic field dominates.
Energy is alternately stored in this field (as the current magnitude increases) and returned (as the current magnitude decreases).