There is a model of magnetic field taught in New South Wales schools that predicts how two magnets or a magnet and wire will behave:
Step 1: For two devices draw independent field.
Step 2: Draw the resultant fields.
Step 3: The resultant diagram has area of high and low field density, the field lines are given the properties of gas molecules that exert some sort of packing pressure against items in the diagram.
So in the above diagram there are
(A) 6 lines outside 4 lines inside ....outside wins the wires attract.
(B) 6 lines outside 12 inside ..... to crowded inside repel.
Here is for a motor from
Electrical Science Fundamentals Handbook
This is a neat visualization guide, if the lines where springs they would push the conductor.
Comment 1: The maths F = BilSin(theta) considers the field only at the conductor.
Comment 2: The fields do not interact, but sum.
Comment 3: Faraday was completely non mathematical, and offered "lines of force" as a non scientific guide. Maxwell tried to develop this into "tubes of force" to attempt a mathematical explanation. This was abandoned some 150 years ago.
Comment 4: The discovery of the electron and special relativity means magnetic fields from currents are understood as an extension of electrostatics. Magnetic fields from permanent magnets are explained by Quantum Mechanics and special relativity. This presents a problem to a high school teacher as this is outside the breadth of school physics.
Comment 5: This path back to electrostatics still does not clear away for deep understanding. Force at a distance can be managed by the construct of fields, which is a mathematical device that overlays an answer field. The exchange of virtual particles radiating over distance is used by Quantum Field Theory and Feymann
diagrams. The curvature of space, created by mass or charge, then telling the other particle how to move has been developed by Einstein's General Relativity. The most up to date theory is 'bane' theory, string theory or M theory.
This a to look at
