electromagnetism - Current, Current density

edit: Hi I'm trying to find the magnetic field generated by a time dependent oscillating current in the quasistatic case ($|z|,r <

The current is flowing through a long, thing wire that is laying on the z-axis. If $z=0$, we can write the current as

$$I(t)=I_0\sin \omega t$$ and now I am trying to find the magnetic field $B(r,t)$ at $z=0$ with this current. How can we calculate $\vec B$?

Possibly, can we calculate $\vec J$ to calculate $\vec A$ to calculate the magnetic field?

THanks a lot for your help on the last problem too.