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Setup: we have a large number of thin magnets shaped such that we can place them side by side and eventually form a hollow ball. The ball we construct will have the north poles of all of the magnets pointing toward the center of the ball, and the south poles pointing away from the center. The magnets in this case are physically formed such that in this hollow ball arrangement they are space filling and there are no gaps between them.

Is such as construction possible? If so, what is the magnetic field (B-field) inside and outside the ball?

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Possible duplicate: physics.stackexchange.com/q/15655/2451 –  Qmechanic Jun 7 at 5:00
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@Qmechanic, close but not exactly the same question. The setup is the same but I was specifically asking about the B-field inside the sphere. –  ThomasMcLeod Jun 7 at 12:57

4 Answers 4

up vote 12 down vote accepted

This is interesting. You would definitely have to 'nail down' the magnets to the sphere, because it will be an unstable configuration. Also in the real world, edge-effects will destroy any chance of perfect radial field lines, so let's assume we're in an ideal scenario.

Outside the sphere, the magnetic field would be that of a source monopole placed at the sphere's center. But we need $\nabla\cdot B=0$, so as a result there is no B-field on the outside.

Inside the sphere, there is nowhere the magnetic field lines can end, especially when they are all pointed towards the center... In fact, such a magnetic field would have divergence less than zero (the center of the sphere being a 'sink'), and this is a property that magnetic fields cannot have (since $\nabla\cdot B=0$). As a result, my answer is that there is no $B$-field on the inside either.

The real reason the B-field must have zero divergence: If there are no physical source monopoles in the vicinity, then any configuration is made of dipoles, and there is no way mathematically (I think) for a collection of dipoles to produce a monopole.

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My intuition was also that there is no B-field on the inside. But then what happens to the b-field lines that come in from the outside? –  ThomasMcLeod Dec 11 '11 at 1:59
    
Are you saying you can create something that looks just a monopole using bar magnets? I think I have to vote you down. Sorry. –  Peter Shor Dec 11 '11 at 5:23
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Yea you're perfectly right, my statement was incomplete, I should have demonstrated the contradiction that we need $\triangledown\cdot B=0$ and as a result there is NO magnetic field outside. –  Chris Gerig Dec 11 '11 at 7:19
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@PeterShor Fixed, so renege the downvote! :) –  Chris Gerig Dec 11 '11 at 10:55
    
Surely there would be a breakdown of the magnetism somewhere on the sphere resulting in the magnetic flux lines re-entering? –  JoeHobbit Oct 24 '12 at 2:27

I think this kind of set up is similar to Halbach Cylinders and Halbach Spheres.

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I just wanted to point out that the linked page shows that for a cylinder with $k=1$ (all magnets pointing inwards or outwards) the field inside the cylinder is zero. –  Arnoques Dec 11 '11 at 19:16

Theoretically, if everything was perfectly in balance, you'd have no magnetic field either inside or outside. The magnets would all cancel one another out.

In practice I imagine you'd wind up with a significant magnetic field, because some of the magnets would be weaker than the others. The north and south poles would be arranged randomly, although I suspect you'd have more north poles than south poles (with the latter being correspondingly stronger).

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What do you mean more north poles than south poles? How can than be? –  ThomasMcLeod Dec 12 '11 at 1:34
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"More" is imprecise in this context, I guess. I meant that, if you divided the area of the sphere up into north-pole areas and south-pole areas, the latter would be smaller (but correspondingly stronger). –  Harry Johnston Dec 12 '11 at 23:40

In some ways is this correct but take this on a much larger scale and it's incorrect. Yes the epicenter would be zero but now the magnetic field would still produce a field. It is cancelled out but you are making it sound as if there were no forces acting anymore, that it would simply become a sphere of polarized metal. Take the large magnetic structured Earth. Yes it's not completely hollow but because of the inner fields the magnetic pressure on the center would still be relevant. This is a magnetic field that pushes the core particles together. Therefore saying that the magnetic field is dissipated is false. The magnetic field is still there, we still can find magnetic north correct?

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If you assume ideal magnets perfectly fitted together, the magnetic field would be restricted to the inside of the magnets themselves. –  Harry Johnston Jul 22 '13 at 20:21

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