Here is a question I encountered sometime ago.
For a current carrying wire of finite length the value of B is less than that of an infinitely long wire (u can look up online for formulae or their derivation) at a same distance from the wire. So consequently the circulation of B on the path shown falls short of the value u times i. The biot savart law suggests the equation written above. The puzzle is to find the mistake.
the cheesy way is absolutely correct because you would have to imagine charge source and sink at at the ends of the wire to maintain the current in wire.
so the changing charges cause electric field on a point on the path to change which would produce an additional magnetic field. Which could also be named as the magnetic field due to displacement current, due to changing charges.
in the inequality I could have written ic + id (as per biot savart law) instead of i but that would have been too many hints for the puzzle.
i had an idea to try to create symmetry in the asymmetrical case you were suggesting, to have the semi infinite wires in opposite directions but I’m wrong 99%