On the previous sketch,

permanent magnets create the excitation field without iron. Rare-earth and ferrite magnets have a small permeability, like 1.05, which I neglect. They are equivalent to a current sheath I = e×H flowing at their rim. This lets arrange them as Helmholtz coils

https://en.wikipedia.org/wiki/Helmholtz_coil10mm thick Nd-Fe-B magnets with H=1MA/m are as strong as 10kA×turn, or 100A in 100 turns, wow. With D=60mm, the maximally flat induction distribution would result from 30mm spacing between the magnets if they were thin. This creates B=0.30T at the centre, dropping by 1% at +-5mm axial distance and a similar radial one. A tiny induction dip at the centre would slightly widen the good volume, but the maximally flat condition is algebraically simple:

dB/dx = d

^{2}B/dx

^{2} = d

^{3}B/dx

^{3}where the odd derivatives vanish by symmetry and the second by adjustment.

The magnets' thickness e spreads the equivalent current by the amount e. It's equivalent to a convolution of the current distribution by a square function of length e, and this square function is the integral of two opposed Dirac spaced by e. The convolution at the created induction incorporates some d

^{2}B/dx

^{2} from x around the centre. For reasonable e, this can be compensated by increasing a bit the spacing. I expect an algebraic solution by this Diracs convolution method, writing d

^{2}B/dx

^{2} with the degree change due to the integral, and solving for the maximally flat condition. I didn't check if this method is new, and don't plan to detail it. It applies to coils too.

Permanent magnets are strong but dangerous.

Electromagnets could replace them, with cores to achieve an interesting induction. The cores must allow easy flux variations. A uniform induction would result from FEM optimization and CNC machining of the poles. The measurement coils can surround the exciting coils; I'd keep them distinct to reduce noises.

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The

measurement coils too are arranged according to ol' Hermann. A sine current flowing in them would create a uniform induction in the measurement volume, hence induce a voltage in a small loop independent of the position there. Because the

*mutual* induction is the same from object A to B and B to A, the same sine current in the small loop would induce the same voltage in the measurement coils from all positions in the measurement volume. This way, all points of the sample within this volume contribute equally to the measurement signal.

To compute the signal induced by the sample, of which all points are equivalent, one can integrate over the measurement coils the vector potential A created by a small magnetic dipole, which has an algebraic expression. Instead, I compute here the mutual induction from the measurement coils in the sample. 1A×t in each R=50mm coil create 18µT so a 1cm

^{2}×1cm sample receives 1.8nWb and the mutual induction is 1.8nH, from the sample to the measurement coils too. If the 1cm

^{2}×1cm sample has a susceptibility χ=10

^{-5} in 0.30T, it's equivalent to a dipole of 24mA and 2.4µA×m

^{2} that induces 43pV×s per turn in the coil pair, for instance 14nV over 3ms transitions of the sample in and out the excitation field.

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If each coil has 2000 turns, the

signal is 86nV×s or 29µV over 3ms. D=0.29mm enameled copper make 340Ω resistance and about as much reactance at 40Hz. Integrated over 40Hz and with 2dB noise by the differential amplifier, the thermal background is 19nV, so the measure can be accurate even for smaller samples. Good amplifiers can cope with a smaller source resistance, that is, fewer turns.

The measurement coils (and their feed cables) demand shielding against slow electric fields, especially at 50Hz and 60Hz. A nonmagnetic metal sleeve can surround each coil if the torus it makes is not closed, but its ends can overlap if insulated. It's better symmetric starting from the feed cable and connected to the feeder's shield. The situation is much easier than for electrocardiograms.

100mA×2mm×0.5m at 1m distance and 50Hz induce 0.2µV in the coil pair, so a decently calm environment needs no magnetic shielding.

Averaging successive measures would further filter out interferences by the mains. The sample's cycle can be desynchronized from the mains or have a smart frequency ratio with it.

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An electric motor can rotate the sample if it's shielded by construction or is far enough. An adjustable counterweight is useful. Or let the sample oscillate as a pendulum.

The nacelle, or bottle for liquids and powders, should contribute little signal. Thin construction of polymer fibres is one logical choice. Its contribution can be measured separately and subtracted by computation.

The

parts holding the magnets and coils must be strong and also stiff. A 2000 turns coil moving by 1nm versus the magnet gets roughly 100nV×s, as big as the signal, so this shall not happen at the measurement timescale. Not very difficult, but needs computational attention. The nacelle's movements must be isolated from the magnets and coils.

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If a sample keeps a permanent magnetization, it should be measured without the excitation magnets.

Metals conduct too much, but most electrolytes fit.

Marc Schaefer, aka Enthalpy