A current-carrying wire won’t exert a net force on itself.
Work Step by Step
A current-carrying wire can’t exert a net force on itself, because of Newton’s third law. If one part of the object pushed on another part of the same object, this internal force would be canceled by an oppositely-directed reaction force. When considering the force on one of the wires, we only needed to consider the force on it due to the magnetic field of the other conductor.