Experiment: What's the shape of a magnetic field?

The electric charge passing through the wire seemed to create a magnetic field!
This was a discovery made by Hans Christian Ørsted in 1820.

When this was first discovered (1819) it was assumed that it might be traveling along the wire:

Or maybe it is radiating outwards?

Can you disprove these initial assumptions using a wire, a compass and a battery?

Here is an actual page from Ørsted's laboratory notebook contains sketches of the experiments he performed in July 1820.

Here he clearly shows that he repeatedly carried out the experiment with the conducting wire to the West and to the East of the magnetic needle (upper drawings), before putting the wire exactly over the magnetic needle.