As far as I know, it\'s possible to create a radially polarised ring magnet, whe

Question

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.

Radially polarised ring magnet

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

Radially polarised ring magnets torus

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

Impossible torus magnet

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?

Explanation / Answer

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).

enter image description here

It's obvious that the toroid is symmetrical under rotation along ?^ direction. Thus we can't have magnetic field along ?^. 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:

B? is symmetrical under reflection over x^ axis and z^ axis.

?

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