Question

how can it be shown that an electric current in wire produces a magnetic field around it?

Answer 1

Since compasses work by pointing along magnetic field lines, this means that there must be a magnetic field near the wire through which the current is flowing. The magnetic field produced by an electric current is always oriented perpendicular to the direction of the current flow.

Answer 2

Place several magnetic compass needles in a circle , around a conducting wire which can carry current . Turn the current on . If all the compasses needle will align in a direction which is tangential to the circle at respective point of location then the wire carries current .

But if all the needles point in earth's north pole direction then no magnetic field is produced .