Vokoun, Beleggia, Heller and Šittner (2009): Calculating and Measuring the Pull Between Cylindrical Permanent Magnets

"Magnetostatic interactions and forces between cylindrical permanent magnets" is a six-page paper by David Vokoun, Luděk Heller and Petr Šittner (Institute of Physics ASCR, Prague) and Marco Beleggia (Institute for Materials Research, University of Leeds). It appeared in the Journal of Magnetism and Magnetic Materials, volume 321 (2009), pages 3758–3763. The paper gives formulas for the attraction between small cylindrical magnets and checks them against forces measured as the magnets were pulled apart. It never mentions Giovanni Battista Ferlini. It belongs in this dossier because it shows, in measured numbers, how ordinary magnetostatics expects the pull between two attracting magnets to change with distance. Any claimed "barrier" or sudden jump at a critical distance (Magnetic Barrier and Critical Distance) would have to stand out against that expected behaviour.

What problem the paper addresses

The authors note that permanent magnets are widely used in actuators, sensors and releasable fasteners, and that such devices can only be controlled reliably if the magnetic force is known. A common shortcut treats each magnet as a point dipole. According to the paper, this works when the magnets are far apart but fails close to contact, where the shape of the magnets matters. The paper's aim is a formula that keeps the full cylindrical shape and works at any separation.

How the calculation works

The force comes from how the stored magnetostatic energy changes as the magnets move: the paper differentiates that energy with respect to position. What limits the result is a short list of idealisations, which the authors spell out. Each cylinder is treated as magnetised evenly along its axis, and that magnetisation is held fixed even when another magnet is nearby (the paper compares this to a strong axial anisotropy built in during manufacture). The cylinders must also share one radius and keep their axes parallel. The authors say that, apart from these idealisations, nothing is approximated, so the shape of the magnets is fully accounted for. For two magnets on a common axis, the force reduces to an integral that the authors express exactly in terms of complete elliptic integrals. A more general version covers magnets whose axes are offset sideways, and sums of such terms handle groups of several magnets.

From this, the paper derives a closed formula for the contact force, the pull when there is no gap at all. It compares this with an earlier contact-force formula by Agashe and Arnold (2008), after correcting what the authors identify as a typographical error in that paper.

How the force falls with distance

When the magnets are several radii apart, the paper's expressions simplify. For two identical magnets on a shared axis, the force falls off as 1/x⁴, where x is the gap; the authors say this matches the expected dipole behaviour and serves as a check on the full formula. They also warn that this far-field approximation is badly wrong at small gaps, where it predicts forces growing very large.

The paper also analyses a set of four touching magnets whose neighbours point in opposite directions, facing a matching set. When the two sets are far apart, the force falls much faster, as 1/x⁸. Like the 1/x⁴ law for a single pair, this is a far-field approximation that the authors derive only for large gaps; near contact, the full expressions must be used. The authors note that the attraction between two such sets can be more than four times the pull between a single pair, because the alternating arrangement is energetically much more stable than one with all magnetizations aligned.

What was measured

The quantity recorded was the pulling force between two sets of magnets as a function of how far apart they were, starting from contact and moving outward. Four arrangements were tested: stacks of two, four and six magnets on a common axis (equivalent to heights of 3, 6 and 9 mm), and the alternating four-magnet set described above.

The building blocks were small NdFeB cylinders, 3 mm across and 1.5 mm tall, magnetised along their axis by the supplier, Elidis sro. Several features make the numbers checkable. The magnetisation that feeds the formulas was not assumed but measured, using a Hall-probe gaussmeter: the reading of (1.079 ± 0.008) T corresponds, according to the paper, to a saturation magnetisation of (0.859 ± 0.006) MA/m. Forces came from a 100 N load cell on a tensile testing machine, at room temperature, and the magnets sat in non-magnetic holders so the fixtures would not add their own pull. After the reading was zeroed with the sets well apart, zero displacement was fixed as the point on approach where the net force reading returned to zero; from there the sets were drawn apart slowly, with force and stroke logged throughout.

The authors also flag what could spoil the comparison. Clamping the holders so the axes stay parallel and the faces meet squarely takes care, and they note that even slight misalignment can open a gap between theory and measurement.

What the measurements showed

For the stacked cylinders, the authors describe "very good agreement" between the measured and calculated force-versus-distance curves, without putting a percentage on it. The clearest numbers for the stacks are the contact forces in the table below, where the paper's formula misses the measurement by roughly 5–7%: 2.76 N calculated against 2.63 N measured for the shortest stack (about 5% too high), and 3.20 N against 3.40 N for the tallest (about 6% too low).

For the alternating four-magnet set, the paper is more specific. It reports agreement below 1% across the separations tested, except when the sets were at or very near contact, where deviations were closer to 5%. Even that largest deviation, at contact, is described as barely visible in the plotted curves.

The contact forces were reported as follows. The paper labels each row by aspect ratio, the stack height as a multiple of the 3 mm diameter, so rows 1, 2 and 3 correspond to the two-, four- and six-magnet stacks (3, 6 and 9 mm tall):

The 1.4% uncertainty on the calculated values comes from the 0.7% error in magnetization, doubled because force depends on its square. The authors could not tell from these data which contact formula was better. Both overestimated the first two cases and underestimated the third. Extending the comparison to flatter magnets, they found that the Agashe–Arnold formula peaks (0.776, in units of the magnetostatic energy constant over radius, at aspect ratio 0.225) and then falls to 0.750, while their own rises steadily to about 0.883. Because there is no physical reason for that peak, they argue their formula is closer to reality.

They attribute the remaining contact discrepancies to imperfect alignment, the use of stacked magnets instead of single long ones, variation in magnetization between individual magnets, and the assumption of uniform magnetization.

What this means for the Ferlini account

The paper sets out the ordinary baseline; it does not test the Ferlini narrative:

• A steep rise near contact. The paper's calculated curves show the pull rising rapidly as the gap closes, and it reports that the measured curves agreed with them. On this wiki's reading, which the paper does not itself make, a sense that attraction "suddenly" strengthens when poles are inched together may be no more than this ordinary steep rise. The paper's measurements, however, run only from contact outward; it neither looks for nor rules out a critical-distance effect, and it does not show that a distinct barrier exists.
• The hardware differs. The paper studies millimetre-scale NdFeB cylinders with parallel axes. The account in Magnetic Barrier and Critical Distance describes mild-steel U-shaped magnets weighing several quintals. The paper does not address magnets of that kind, and its formulas do not apply directly to that geometry.
• Alternating poles: only a loose parallel. In the paper's four-magnet set, neighbouring magnets within one set point in opposite directions, and this changes how the force varies with distance and how strong it is, using nothing beyond standard magnetostatics. On this wiki's reading, that is at most a loose analogy to the Ferlini account's alternating poles at four cardinal points; the arrangements differ, and the paper shows no direct equivalence.
• Method as contrast. The paper reports calibrated magnetization, a load cell, continuous force-distance records and stated error budgets. The Ferlini material reviewed in Ferlini Portal Claims vs. Evidence: The Book, the Forum Retelling and the Video Disclaimer offers nothing comparable.
• Out of scope. The paper deals only with forces. It says nothing about the vibration, mist or ozone the account attributes to the critical region.

For the companion constraint on what static magnetic fields can do, see Berry and Geim's 1997 'Of Flying Frogs and Levitrons': Earnshaw's Theorem and the Exceptions That Allow Stable Magnetic Levitation.

Reference

Vokoun D, Beleggia M, Heller L and Šittner P 2009 "Magnetostatic interactions and forces between cylindrical permanent magnets", J. Magn. Magn. Mater. 321, 3758–3763 (received 19 February 2009; available online 17 July 2009). doi:10.1016/j.jmmm.2009.07.030. Copy: https://jontallen.ece.illinois.edu/Public/Allen/Noori-pdf/VokounBeleggiaHellerSittner.09.pdf