The well known Helmholtz coil can create a relatively uniform magnetic field near the center of the two coils. The field uniformity is improved upon by the Maxwell coil, consisting of three coils, with a larger diameter central coil (see https://en.wikipedia.org/wiki/Maxwell_coil).
A drawback to the Maxwell coil is that there is no direct radial access to the center, blocked by the central coil, which would be useful in testing and calibrating H-field sensing coils.
The first part of my question is: what geometry of four coils and relative currents would create an even more uniform internal field, and allow both radial and axial access to the field? I notice that the Maxwell coils are on a spherical surface, so it seems that this relationship would still hold for a quad coil arrangement.
Furthermore, can the concept be extended to any number of coils, ultimately to a current carrying spherical surface?
It's not 100% clear which exactly kind of homogeneity you need: +-1%? In how large region? Which direction is more important in terms of homogeneity? Would it be DC field and what is its magnitude (bare coils are usually limited by 100s of Oe)? If you would imagine solenoid as the source of the field ( https://en.wikipedia.org/wiki/Solenoid ) it would give you perfectly constant (theoretically, for infinite solenoid) field distribution in the axial direction. But it's radial distribution expected to be slightly worse than for Helmholtz pair.
In theory, both Helmholtz and Maxwell coils supposed to have a circular cross-section with diameter d many times smaller, than the diameter of coil ring. However, technically it's usually not 100% true: cross-section is rather square and its linear sizes (of the cross-section) not very much smaller than coil diameter. All this would lead to some disturbances of the field distribution in comparison with theoretical (usually not very significant). The idea to make 2 central coils with some split between it, instead of making 1 central coil in case of Maxwell design could work, but i expect that it would lead to significant field rearrangement in a case of the wide split. Of course, the number of loops (or current) should be reduced twice in each of pair of central coils in comparison to Maxwell coil. I would say that you should make some finite element simulation, for example in COMSOL. Only in this case you would know exact field distribution.
After all, why do you use coils for sensing the field? The signal, that they sense is somehow integrated over volume, isn't it? And they are bulky. Honeywell produces magnetoresistive sensors, that are way more compact and with field resolution of 10-100 uOe, take a look: https://aerocontent.honeywell.com/aero/common/documents/myaerospacecatalog-documents/Missiles-Munitions/Sensors_Product__Catalog.pdf.
Thanks for your reply.
A Helmholtz coil works well for my application of 'calibrating' (more of a functional test) small H-field probes used for locating sources of EMI. However, I noticed in a Wikipedia article that the Maxwell coil "...removes variations in magnetic field, up to its 6th-order derivative with respect to position, near the centre of the virtual sphere."
I was largely curious how much this could be further improved with 4 coils, giving radial as well as axial access to the field (and what the geometry and currents would be), and whether it would be theoretically possible to create an absolutely uniform field inside a sphere with a suitable current distribution on the surface and, if so, what that current distribution would be.
I am reasonably sure there are several published algorithms for solenoid design, including split coil designs. Fine tuning for " extreme" homogeneity in a given volume has to be optimised numerically . Do note that helmholtz etc ignore any pitch to the winding .. that gives extra small azimuthal fields .
Best places to look in the literature are
1 MRI magnet design ( some machines have/had split coils to allow radial access to humans ( surgeons!)
and
2. Accelerator bending magnets etc ( Cern / Fermilab etc )
For sensing, using coils ,don't forget the reciprocity theorem .it makes calculations so much easier
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