Question

how can you show that the magnetic field produced by a given electric current in the wire decrease as the distance from the increase

Answer 1

The strength of magnetic field is inversely proportional to the distance, which implies that the strength of the magnetic field reduces with distance from the wire.

This can be proved experimentally by taking a straight wire and inserting it through the center of a carboard piece. Sprinkle iron fillings around it and once you pass electricity through the wire, you will notice that the fillings arrange themselves as the field lines. So as the distance increases, the line go farther and farther apart.