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Homework Statement
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.
a)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.
b)Determine the magnetic field strength inside the gap.
c)Calculate the ratio of the magnetic field strength inside the gap of an electromagnet with tapered ends to that of an untapered, but otherwise identical electromagnet. Explain the benefit of the tapered ends in the limit in which s L << r R , and what might limit it in practise.
Homework Equations
Ampere's law: ##\nabla x \vec{H} = \vec{J} + d\vec{D}/dt ##
## \vec{B} = \mu \mu_0 \vec{H} ##
The Attempt at a Solution
a) We may argue that the circular symmetry of the torus is conserved even with the gap. This implies that the field must be purely tangential and is hence perpendicular to all internal surfaces (and also the gap except for the tapered parts). Because we have the boundary condition that the perpendicular component of the B field is continuous, it must be the same in all of these regions.
b) We are looking for the H field in the gap (I am not sure if this includes the tapered parts too or not), so I would like to employ Ampere's law in integral form and integrate around an Amperian loop in the torus.
However, when looking at the gap in more detail, I believe the field lines behave in a more complicated way. Because of the condition that the parallel component of ##\vec{H}## ought to be conserved across the boundary (assuming no free current), the field lines will refract at the tilted boundary. This appears to break the symmetry of the problem (altought the lines must form closed circles, so I am not sure what really happens).
Could anybody help me get on the right track?
Many thanks in advance!