# What happens to the magnetic field in this case?

As far as I know, it's possible to create a radially polarised ring magnet, where one pole is on the inside, and the field lines cross the circumference at right angles.

So imagine if I made one which was shaped like a sector of a torus.

Then I forced a load of these magnets into a complete torus.

Clearly this magnet is impossible because there's no way for the field lines to get back into the middle. So what happens to the field in this case? Does it disappear completely? Do the magnets blow up?

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I think Emilio Pisanty's answer is good enough. But here is another longer, 'magnetic charge' approach. (

Let's specify the coordinates first (sorry I borrow your picture).

It's obvious that the toroid is symmetrical under rotation along $\hat{\phi}$ direction. Thus we can't have magnetic field along $\hat{\phi}$. Which means it is sufficient for us to find the magnetic field on the $xz$ plane, and we can generalize later by rotating this $xz$ plane.

We have some constrains to consider here due to the shape of torus:

• $\vec{B}$ is symmetrical under reflection over $\hat{x}$ axis and $\hat{z}$ axis.

• $\nabla\times B=0$.

• $\nabla.B=0$

The most general field lines in $xz$ plane that satisfy these conditions roughly looks like this.

Now we only have two possible directions, the one shown in the picture or the opposite of that(or zero everywhere). We can apply magnetic Gauss law here with the Gaussian surface marked with black dotted line(in 3D point of view this black dotted line is rotated along $\phi$ so that the product looks like a mountain).

The only part of the Gaussian surface which has magnetic flux through it is the top part. The remaining area is intentionally shaped to follow the field lines tangentially so that there's no flux through it. Now we only need to determine whether the magnetic charge inside the envelope is positive or negative(Positive charges are shown in blue, and negative charges in red). As we can see in the picture, the inner magnetic field which is represented by yellow lines may bend slightly from radial direction towards the direction which allows the envelope to catch more or less positive charge. So the total charge inside the envelope might be positive or negative. However if the yellow lines bend in a way like that, then we will meet a kink somewhere between the inner and outer field or we may also get a non-zero curl field and these are impossible. So we are only left with a radial inner field. And therefore the total charge inside the envelope is zero and there can't be any flux passing through it. So in conclusion the field outside must vanish everywhere.

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 This correctly determines that the only configuration in which a hollow torus can exist with N poles on its external surface and S poles on its internal surface is for there to be no field at all, since it the torus were to be closed, it would create a non-physical magnetic configuration. But that doesn't answer the question "what happens" when this configuration is attempted. It takes a lot of energy to assemble the $N-1$ pieces before the $N$th piece is added; what happens when we attempt to add the $N$th piece? The magnetic field doesn't just quietly go away... – KDN Feb 21 at 18:22 I don't see why it will create a non-physical magnetic configuration. Can you specify what rule might be violated ? I think it's okay for the magnetic field outside to disappear entirely, it doesn't mean that the interaction energy will also go away with it. Higher potential energy(depends on negative of the cross term of magnetic energy density) doesn't always mean higher total field energy, because only part of the field energy is responsible for interactions(force). The final configuration will have a very high potential energy and tends to blow up, but there's no singularity involved here. – Emitabsorb Feb 21 at 21:31 Also we don't need so much energy to insert the last piece of magnet, the field is already quite weak before the insertion. Can the downvoter please explain? – Emitabsorb Feb 21 at 21:50 It would be non-physical to have a ring of S poles inside with no path to connect to the N poles. By adding magnet segments to one another, the total magnetic flux passing from the inner tube of the torus (the S pole) to the outer surface (the N pole) is increasing, but the path through which it can do this remains the same size (the diameter of the inner tube). This means that the flux density at the exits to the partial torus increases as new segments are added. If the last segment could be added, it would cut off all path for the field lines. – KDN Feb 22 at 0:40 In the attempt to add the last segment, all of the magnetic flux from the S poles has to squeeze through the opening left, and as the last segment is inserted this hole is diminished. Eventually, the magnetic flux density will be so high that the system must break. – KDN Feb 22 at 0:42