# 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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