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We know that we can artificially manipulate magnetic flux in space combining different discrete magnetic sources to create complex magnetostatic fields on air like chiral magnets or even helical magnetic fields.

Assuming two discrete magnetic helical field sources of same strength B (i.e. magnetic flux density per unit area), brought close together in space, would the $attraction force$ $F$ between their unlike N-S poles:

$$ \mathbf{F}=\boldsymbol{\nabla}(\mathbf{m} \cdot \mathbf{B}) $$

and also torque produced:

$$ \boldsymbol{\tau}=\mathbf{m} \times \mathbf{B} $$

be stronger than that of the unlike poles of two normal dipole bar magnets generating the same field strength B, or less or the same and why?

helix field vs axial field

Also, is it possible the case (a) illustrated in figure above, to actually emulate the elastic Yakawa potential $V_{(r)}$, used for the calculation of the strong force between subatomic particles:

$$ F=-V^{\prime}(r) $$ Where, $$ V(r)=-\frac{4}{3} \frac{\alpha_{s}(r) \hbar c}{r}+k r $$

Where $α_{s}$ the coupling between the particles that varies with forced separation distance $r$ from their initial instantaneous rest positions where $α_{s}=1$ (i.e. fine structure constant rest value for strong force) and $k$ the string tension constant.

Meaning, is it possible that if we stretch out the configuration of case (a) shown above to a larger separation, the force $F$ would actually increase similar to the strong force characteristic between two bounded quarks?

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  • $\begingroup$ Can you specify better your "helical field source"? On your picture there's a single induction line. If I consider some other $B$-field lines inside the cone, where do they go to when they mustn't cross with the one visualized? $\endgroup$
    – Hoody
    Jul 14, 2022 at 18:24
  • $\begingroup$ Can you specify better what you mean by the two sources generating the same-strength $\vec B$? The magnetic field around a finite magnet is always non-homogeneous, so at which point in space do you want to compare them? @Markoul11 $\endgroup$
    – Hoody
    Jul 14, 2022 at 18:27
  • $\begingroup$ @Hoody "The magnetic field around a finite magnet is always non-homogeneous, so at which point in space do you want to compare them? " At the center point of the air gap between the separated N-S poles. In that area the field is approximate homogeneous. "Can you specify better what you mean by the two sources generating the same-strength B ? " Each magnet alone without interacting with another magnet, at distance d/2, half the separation gap shown above, generates at that center point the same B value. $\endgroup$
    – Markoul11
    Jul 15, 2022 at 8:35
  • $\begingroup$ @Hoody "Can you specify better your "helical field source"? On your picture there's a single induction line. If I consider some other B-field lines inside the cone, where do they go to when they mustn't cross with the one visualized?" Well, the illustration of the helical field is not the best. Better imagine it like the spirals of an electrical solenoid. The solenoidal flux lines are not criss-crossing of course. $\endgroup$
    – Markoul11
    Jul 15, 2022 at 8:41
  • $\begingroup$ @Hoody Helical magnetic fields polarization is possible: mdpi.com/2075-4434/7/1/5/htm Apart of synthetically generating a helical flux magnetic field there are also physical magnets that generate this kind of field like conical magnets en.wikipedia.org/wiki/Helimagnetism $\endgroup$
    – Markoul11
    Jul 15, 2022 at 8:50

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I don't know how you plan to produce such a field, but it wouldn't be with simple north and south poles sitting some distance apart like you have pictured.

One way you might do it is to take a bunch of short permanent magnets and connect them end to end, making a long flexible permanent magnet. Bend that into the shape you have pictured.

Another way: A long straight current creates a cylindrical field. A Magnet with wide flat poles creates a uniform field between them. Combine the two by putting a wire from one pole to the other. This gives a helical field. You probably can change the shape of the current to make the field some sort of helical shape. But you have drawn the field like the 2-D surface of a cone. You need to think about the field in 3-D.

As for $F = \nabla (m \cdot B)$ and $\tau = m \times B$, those apply to a magnetic dipole $m$ in a non-uniform field $B$. They are not the force/torque of one pole on the other.

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