Magnetic field in the gap of a tapered torus

[Alettix]

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!

 

[Charles Link]

The magnetic material with the magnetic fields does not behave quite like the electric field of a conductor, and there is no requirement that the $H$ field at the surface of the material needs to be normal to the surface. [Editing: There may however be a tendency for the $H$ field to be somewhat normal to the surface in the region of the gap]. $\\$ [Editing: What I originally posted may not be correct, so I'm revising my response]. $\\$ The magnetic circuit equations have a provision in them for when the cross sectional area changes. A refined solution would perhaps make use of this. See https://en.wikipedia.org/wiki/Magnetic_reluctance $\\$ If the total flux $\Phi$ was first computed, a more refined calculation might then determine how the $\Phi$ gets distributed across the gap. One number that seems to be somewhat subjective here is the number to use in the cross-sectional area formula for the reluctance in crossing the air gap=do you use the smallest tapered area there, or do use a somewhat larger area, such as the average between the area before the taper and the area at the end of the taper?...$\\$ .............................. $\\$ Additional editing: I think you could compute the flux $\Phi$ from the reluctance equations, and then assume this flux crosses the air gap and is uniform inside the radius $r$ and is zero outside of that radius. That way, the cross-sectional area in the air gap would be $A=\pi r^2$. The total reluctance for the circuit is computed as $R=\oint \frac{1}{\mu_o \mu_r A} \, dl$, and flux $\Phi=\frac{MMF}{R}$. (With a little algebra, the cross-sectional area can be computed in the region of the taper as a function of position). I think you may even find, (part c), that the taper helps funnel the flux $\Phi$, and the magnetic field in the gap is enhanced by the taper. $\\$ ................................ $\\$ You could certainly write the equation that just considers the path of a straight line through the air, and determine $H$ as a function of position. [Editing: Scratch the previous sentence=the above solution (in the "Additional editing" section) of computing the reluctance is far better than this latter calculation]. These equations I believe are not exact solutions, but rather are simply reasonably good approximate solutions, so there may be no perfect answer... [ Editing: I believe the "Additional editing" in the previous paragraph may be a good solution ]. $\\$ See also Feynman's lecture notes, (he uses different units from SI, but it should still be easy enough to follow ), for a treatment of the problem=equation (36.26) http://www.feynmanlectures.caltech.edu/II_36.html $\\$ @jim hardy What might you recommend for this tapered air gap?

 

[Alettix]

. $\\$ Additional editing: I think you could compute the flux $\Phi$ from the reluctance equations, and then assume this flux crosses the air gap and is uniform inside the radius $r$ and is zero outside of that radius. That way, the cross-sectional area in the air gap would be $A=\pi r^2$. The total reluctance for the circuit is computed as $R=\oint \frac{1}{\mu_o \mu_r A} \, dl$, and flux $\Phi=\frac{MMF}{R}$. (With a little algebra, the cross-sectional area can be computed in the region of the taper as a function of position). I think you may even find, (part c), that the taper helps funnel the flux $\Phi$, and the magnetic field in the gap is enhanced by the taper. $\\$ ................................ $\\$

Thank you for your answer Charles Link, I am sorry to say it leaves me rather confused. I am doing a basic EM course at University and we have not learned about magnetic reluctance. This being a past exam question, it must be solvable by other means. Do you have any idea how this could be done without overcomplicating?

I agree with you that $\vec{B} = \mu \mu_0 \vec{H}$ is not valid in strongly magnetic medium but we must use $\vec{B} = \mu_0 (\vec{H} + \vec{M})$ instead. However, provided that there is no free current flowing at the surface, I cannot see why the standard derivation of $\vec{H}_{parallel} =$ constant doesn't work. And this predicts the fieldlines to turn away from the gap, not to concentrate in it.

And now I have realized I am confused about something more, is the magnetic flux density or the magnetic flux conserved along the torus? I thought they meant the B field but now I am realising it might be Φ... although I cannot see why that holds in the gap!

 

[Charles Link]

Here is a simpler problem that appeared about a year ago on Physics Forums: https://www.physicsforums.com/threads/absolute-value-of-magnetization.915111/ The one you are trying to work is somewhat more advanced.

 

[Alettix]

Here is a simpler problem that appeared about a year ago on Physics Forums: https://www.physicsforums.com/threads/absolute-value-of-magnetization.915111/ The one you are trying to work is somewhat more advanced.

Thank you! I think I am fine with the straight edge gap, what I need help with understanding is what happens in the case with the tapered ends.

 

[Charles Link]

Thank you! I think I am fine with the straight edge gap, what I need help with understanding is what happens in the case with the tapered ends.

The tapered case is new to me as well. The way I described the solution in the Additional editing" portion should work. See also the "link" to what I think is the Wikipedia write-up on Reluctance concepts. https://en.wikipedia.org/wiki/Magnetic_reluctance They give a formula for the total reluctance for a section for the case of uniform cross-sectional area $A$, but the integral formula I gave (in the "Additional editing" portion) should work for this tapered case. (In that integral formula $A=A(l)$).

 

[Alettix]

The tapered case is new to me as well. The way I described the solution in the Additional editing" portion should work. See also the "link" to what I think is the Wikipedia write-up on Reluctance concepts. They give a formula for the total reluctance for a section for the case of uniform cross-sectional area $A$, but the integral formula I gave (in the "Additional editing" portion) should work for this tapered case.

Yes I have read your previous post, but as expressed in my reply (post #3) I am not convinced by all of it, or rather have further questions. As mentioned in the same post, I have not studied magnetic reluctance and this being a past exam question must be solvable without that concept. Of course, I could immense myself in learning it, but as exams are quickly approaching I would rather like to focus on learning the things I am expected to know and be able to apply that effectively.

So as asked in #3, do you have any idea how this could be solved without magnetic reluctance?

Thank you this far!

 

[Charles Link]

So as asked in #3, do you have any idea how this could be solved without magnetic reluctance?

That very topic just came up the other day on a thread that began a couple of months ago, and the latest inputs discuss the Biot-Savart/magnetic surface current approach approach in some detail, and how difficult it would be to do the computation in this manner: https://www.physicsforums.com/threads/magnetic-flux-is-the-same-if-we-apply-biot-savart.927681/ See posts 10, 11, and 12 in particular.$\\$ If such an approach were taken, as described in post 11 of the "link", a "first guess" for $B$ could be the solution for $B$ that is found by using reluctance techniques.

 

[Alettix]

With the approximation that field lines concentrate and that we have a linear medium the question was solved with ampere's only.

Thanks for the help!

 

[Charles Link]

@Alettix You may also find posts 28, 29, and 30 of this thread of interest: https://www.physicsforums.com/threa...he-same-if-we-apply-biot-savart.927681/page-2 Also, see in particular the [Edit] comments of post 29. This problem can indeed be solved by Biot-Savart techniques.