Can magnetic fields have any geometry we want?

[physea]

Is it possible to have a magnetic field of a any geometry we want, or there are only few types of geometries that can be achieved with permanent magnets and electromagnets? If the former, how do we produce a magnetic field of specific geometry? For example, can the magnetic field be cylindrical or can it be swirl? If the latter, can I see what are the available geometries of all the possible magnetic fields?

 

[Orodruin]

No. By Maxwell's equations the magnetic field must be divergence free.

 

[rumborak]

That said, I would think that in a limited space you should be able to approximate any field distribution arbitrarily closely, with enough surrounding magnets.

 

[Orodruin]

That said, I would think that in a limited space you should be able to approximate any field distribution arbitrarily closely, with enough surrounding magnets.

No. This fails due to your "arbitrarily closely". This cannot be true based on post #2. You cannot come arbitrarily close to a field with non-zero divergence using just divergence free fields.

 

[rumborak]

That depends on the definition of "close". True, of course you will never have a non-zero divergence, but if the incoming field lines are bundled very close together and then fan out radially, you are very close to emulating non-zero divergence.

 

[Orodruin]

That depends on the definition of "close". True, of course you will never have a non-zero divergence, but if the incoming field lines are bundled very close together and then fan out radially, you are very close to emulating non-zero divergence.

No you are not. The field will still have exactly zero divergence - just like the electric field of a point charge away from the point charge. As long as your definition of "close" is reasonable, you will always be able to find a field with non-zero divergence that is closer to the target field (with non-zero divergence) than any divergence free field and so you cannot get arbitrarily close.

If you are thinking of emulating a monopole field in some finite region that excludes the pole - that field is divergence free in that region.

 

[rumborak]

I am suggesting an "engineering approximation" here, where you try to confine the errors to an arbitrarily small region of space.

It's the same as with the Gibbs phenomenon when approximating a signal with sine waves. You will never get rid of the overshoot because it is a mathematical consequence, but if your definition of "close" is "minimize the amount of space with errors", you can get arbitrarily close.

 

[Orodruin]

I am suggesting an "engineering approximation" here, where you try to confine the errors to an arbitrarily small region of space.

It's the same as with the Gibbs phenomenon when approximating a signal with sine waves. You will never get rid of the overshoot because it is a mathematical consequence, but if your definition of "close" is "minimize the amount of space with errors", you can get arbitrarily close.

In the case of Gibbs' phenomenon, you will not be able to find a function that better approximates the target function than all superpositions of sine waves because the sine waves form a dense basis and closeness here has a definite meaning in terms of the $L^2$ norm. This stands in stark contrast to the case of a divergence free field approximating a field with non-zero divergence.