Editing Halbach array (section)

===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>