Consider a toroidal electromagnet filled with a magnetic material of large permeability μ. The torus contains a small vacuum gap of length h. Over most of its length the torus has a circular cross section of radius R, but towards the gap the torus is tapered on both of its ends, i.e., its radius is decreased from R to r over a distance s towards the gap. The electromagnet has N windings through which a current of I is flowing.Explain why the magnetic flux across the cross section of the torus is conserved along the total length of the torus and within the gap.

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Claire Love · Accepted Answer

This follows from the magnetic boundary conditions. The component of the magnetic field normal to the surface (or interface) is conserved. Whereas the tangential component is not conserved. Due to the latter the magnetic flux across the cross section of the torus is conserved along the total length of the torus (decreasing cross-sections of the torus are accompanied with increasing tangential component). Due to the former condition the magnetic flux is conserved within the gap.

Antwan Perez · Accepted Answer

This follows from integrating the current conservation in a stokesian loop that traverses the interface of the highly permeable media and vacuum (or, for the case, air). Using the Stokes theorem you obtain as conclusion the continuity of the magnetic induction normal component and the discontinuity of its tangential component. Since the material is highly permeable, the tangential component of the field increases in area reduction sections.

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