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 \(\mu\). Each arm of the frame has cross sectional area \(A\). In each arm, the magnetic field is uniform and also parallel to the arm. Also, \(s >> \sqrt{A}\), and \(\mu >> \mu_0 \), where \(\mu_0 \) 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 \[ \frac{X}{Y} \frac{\mu N^2 A}{s} \]

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


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