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?
Answer
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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