Ubiquity of chaotic magnetic-field lines generated by three-dimensionally crossed wires in modern electric circuits

We investigate simple three-dimensionally crossed wires carrying electric currents which generate chaotic magnetic-field lines (CMFLs). As such wire systems, cross-ring and perturbed parallel-ring wires are studied, since topologically equivalent configurations to these systems can often be found in contemporary electric and integrated circuits. For realistic fundamental wire configurations, the conditions for wire dimensions (size) and current values to generate CMFLs are numerically explored under the presence of the weak but inevitable geomagnetic field. As a result, it is concluded that CMFLs can exist everywhere; i.e., they are ubiquitous in the modern technological world.