Magnetic Flux - what is an "equivalent plane"?

[741]

I'm reading about the Biot-Savart law WRT magnetic field and flux calculations. In some treatments, there is a concept called an "equivalent plane".
I don't really understand the diagrams for this: what is an "equivalent plane" in this context and why/how is this idea useful?

One example is here "Figure 6 shows the equivalent plane diagram of the coil and the track"
https://www.researchgate.net/publication/321688638_Combination_of_Compensations_and_Multi-Parameter_Coil_for_Efficiency_Optimization_of_Inductive_Power_Transfer_System#pf6

[mag_therm]

I think it is as follows:
In real coil packs, the conductors have a finite thickness (down into the plane of Fig 5, and horizontally, in Fig 6)

In Biot and Ampere's laws, wire can be  assumed to be
infinitely thin so that the B at a distant point from the contributions of wire lengths dl along the wire length can be solved analytically as a simple first approximation.
 
So Fig 6 is being solved analytically for (Bx,y) as if the coil packs are infinitely thin, lying in the vertical plane shown in Fig 6 ( along the blue vertical axis) Fig 6 has rotated Fig 5 by 90 degrees.
They call the vertical blue line "equivalent plane" of the coil packs.
Then they get (9) and (10) contributions  analytically using  8

In a 2D or more accurate 3D FEM, the real thickness of the packs would be drawn, and the numerical solvers would consider the J in elements inside the copper conductors across the thickness.

Back in early 20th C they were solving jobs like Fig 5 and including the finite thickness, analytically with slide rules, log tables and adders.
Eg Dwight H B "Coils and Conductors" 1945

[MarkT]

I'm reading about the Biot-Savart law WRT magnetic field and flux calculations. In some treatments, there is a concept called an "equivalent plane".
I don't really understand the diagrams for this: what is an "equivalent plane" in this context and why/how is this idea useful?

I think its just the words "equivalent" and "plane diagram" used in conjuction, not a notion of "equivalent plane"