Cube Of Magnetically Coupled Coils

Figure 1 shows a cubic frame with side length \(s\). The frame is made of a material with magnetic permeability \(u\). Each arm of the frame has cross sectional area \(A\). Also, \(s^2 >> A\), and \(u >> u_\circ \), where \(u_\circ\) is the magnetic permeability of free space.

An \(N\) turn coil (\(N >> 1\)) is wound around each arm of the frame. Figure 2 shows, for all arms of the frame, how a coil is wound around an arm. Also in Figure 2, pay particular attention to the direction of the magnetic field produced by the current in the coil.

Figure 3 shows how the coils are electrically connected. The shaded dots indicate the terminals of the system.

The inductance of the system as seen across its terminals is \[ \dfrac XY u N^2 A/s , \]

where \(X\) and \(Y\) are coprime positive integers. Determine \(X+Y\).

Details and Assumptions:

  • In each arm, the magnetic field is uniform and also parallel to the arm.

  • The magnetic field does not leak to the outside of the material of the cubic frame.


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