Editing Halbach array (section)

==Linear arrays==
{{Multi image
| image1            = Linear Halbach Array (Strong side up).png
| image2            = Linear Halbach Array (Weak side up).png
| direction         = vertical
| total_width       = 220
| footer            = Orientation of strong and weak sides in a linear Halbach array
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===Magnetization===
[[File:HalbachArray1.png|thumb|Cancellation of magnetic components resulting in a one-sided flux|220x220px]]

The magnetic flux distribution of a linear Halbach array may seem somewhat counter-intuitive to those familiar with simple magnets or [[solenoids]]. The reason for this flux distribution can be visualised using Mallinson's original diagram (note that it uses the negative ''y'' component, unlike the diagram in Mallinson's article).<ref name=":0" /> The diagram shows the field from a strip of [[ferromagnetic material]] with alternating magnetization in the ''y'' direction (top left) and in the ''x'' direction (top right). Note that the field above the plane is in the ''same'' direction for both structures, but the field below the plane is in ''opposite'' directions. The effect of superimposing both of these structures is shown in the figure.

The crucial point is that the flux will ''cancel below the plane and reinforce itself above the plane''. In fact, any magnetization pattern where the components of magnetization are <math>\pi/2</math> out of phase with each other will result in a one-sided flux. The mathematical transform that shifts the phase of all components of some function by <math>\pi/2</math> is called a [[Hilbert transform]]; the components of the magnetization vector can therefore be any Hilbert-transform pair (the simplest of which is simply <math>\sin(x) \cos(y)</math>, as shown in the diagram above).

[[File:InfiniteHalbachArray.JPG|thumb|The magnetic field around an infinite Halbach array of cube magnets. The field does not cancel perfectly due to the discrete magnets used.|220x220px]]
The field on the non-cancelling side of the ideal, continuously varying, infinite array is of the form<ref>{{cite web |url=http://www.uta.edu/physics/main/resources/ug_seminars/papers/HalbachArrays.doc |title=Concerning the Physics of Halbach Arrays |first=James R. |last=Creel |date=2006 |access-date=31 August 2008  |archive-url=https://web.archive.org/web/20110604160523/http://www.uta.edu/physics/main/resources/ug_seminars/papers/HalbachArrays.doc |archive-date=4 June 2011  }}</ref>

: <math>F(x, y) = F_0 e^{ikx} e^{-ky},</math>
where
: <math>F(x, y)</math> is the field in the form <math>F_x + i F_y</math>,
: <math>F_0</math> is the magnitude of the field at the surface of the array,
: <math>k</math> is the [[wavenumber]] (i.e., the spatial frequency) <math>2\pi / \lambda.</math>

===Applications===
The advantages of one-sided flux distributions are twofold:
* The field is twice as large on the side on which the flux is confined (in the idealized case).
* There is no [[stray field]] produced (in the ideal case) on the opposite side. This helps with field confinement, usually a problem in the design of magnetic structures.

Thus they have a number of applications, ranging from flat [[refrigerator magnet|refrigerator magnets]] through industrial applications such as the brushless [[DC motor]], [[voice coil]]s,<ref>{{Cite web|url=https://patents.google.com/patent/US7368838B2/en|title=High efficiency voice coil motor}}</ref> magnetic drug targeting<ref name="Sarwar2012"/> to high-tech applications such as [[Wiggler (synchrotron)|wiggler]] magnets used in [[particle accelerator]]s and [[free-electron laser]]s.

The [[Inductrack]] [[maglev train]]<ref>{{cite web |author1=Richard F. Post |date=10 October 2005 |title=Toward More Efficient Transport: The Inductrack Maglev System |url=https://gcep.stanford.edu/pdfs/ChEHeXOTnf3dHH5qjYRXMA/09_Post_10_11_trans.pdf |access-date=1 December 2017 |publisher=Lawrence Livermore National Laboratory |archive-date=4 April 2023 |archive-url=https://web.archive.org/web/20230404162449/https://gcep.stanford.edu/pdfs/ChEHeXOTnf3dHH5qjYRXMA/09_Post_10_11_trans.pdf |url-status=dead }}</ref> and Inductrack rocket-launch system<ref name="Tung2001"/> utilize the Halbach array to lift the train by repelling loops of wire in the track.
[[File:Magnetic_viewing_film.jpg|thumb|[[Magnetic viewing film]] showing a flat refrigerator magnet's magnetization|220x220px]]
Flat flexible (not [[Ferrite (magnet)#Hard ferrites|hard ceramic ferrite]]) refrigerator magnets are created with a Halbach magnetization pattern for a stronger holding force when attached to a flat [[ferromagnetic]] surface (e.g. a fridge door) than the holding force from a uniform magnetization. They're made from powdered ferrite mixed in a flexible binder (e.g. plastic or rubber) that is exposed to a Halbach magnetization field pattern as it is [[Extrusion|extruded]], permanently giving the ferrite particles in the magnetic compound this one-sided flux distribution (which can be viewed with [[magnetic viewing film]]).

[[File:Fridge Magnet Halbach.svg|thumb|Flux distribution for a flat refrigerator magnet|220x220px]][[File:HalbachArrayFEL2.png|thumb|Schematic diagram of a free-electron laser|220x220px]]
Scaling up this design and adding a top sheet gives a [[wiggler magnet]], used in [[synchrotron]]s and [[free-electron laser]]s. Wiggler magnets wiggle, or oscillate, an electron beam perpendicular to the magnetic field. As the electrons are undergoing acceleration, they radiate electromagnetic energy in their flight direction, and as they interact with the light already emitted, photons along its line are emitted in phase, resulting in a "laser-like" monochromatic and coherent beam.

The design shown above is usually known as a Halbach wiggler. The magnetization vectors in the magnetized sheets rotate in the opposite senses to each other; above, the top sheet's magnetization vector rotates clockwise, and the bottom sheet's magnetization vector rotates counter-clockwise. This design is chosen so that the ''x'' components of the magnetic fields from the sheets cancel, and the ''y'' components reinforce, so that the field is given by

: <math>H_y \approx \cos(kx),</math>

where ''k'' is the [[wavenumber]] of the magnetic sheet given by the spacing between magnetic blocks with the same magnetization vector.

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===Variable linear arrays===

[[File:HalbachArraySchematic1.png|thumb|Schematic of a Halbach array consisting of a series magnetized rods|220x220px]]
[[File:HalbachArraySchematic2.png|thumb|Equal-gearing arrangement for a variable Halbach array|220x220px]]
A series of magnetic rods, magnetized perpendicular to their axes, can be arranged into a Halbach array. If each rod is then rotated alternately through 90°, the resultant field moves from one side of plane of the rods to the other, as shown schematically in the figure.

This arrangement allows the field to effectively be turned on and off above or below the plane of the rods, depending on the rotation of the rods. Such a device makes an efficient mechanical magnetic latch requiring no power. A detailed study of this arrangement has shown that each rod becomes a subject to a strong torque from its neighboring rods when rotated.<ref name="Hilton2012"/> However, a simple and efficient solution, providing both stabilization and the ability to rotate each rod alternately, is simply to provide an equal-gearing arrangement on each rod, as shown in the figure.
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