imported>Poocat at 14:24, 28 December 2025

2025-12-28T14:24:14Z

← Previous revision		Revision as of 14:24, 28 December 2025
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	[[File:Halbach cylinder2.png\|thumb\|right\|Cylinder magnetization]]		[[File:Halbach cylinder2.png\|thumb\|right\|Cylinder magnetization]]
	A '''Halbach cylinder''' is a magnetized cylinder composed of [[ferromagnetic]] material producing (in the idealized case) an intense magnetic field confined entirely within the cylinder, with zero field outside. The cylinders can also be magnetized such that the magnetic field is entirely outside the cylinder, with zero field inside. Several magnetization distributions are shown in the figures.		A '''Halbach cylinder''' is a magnetized cylinder composed of [[ferromagnetic]] material producing (in the idealized case) an intense magnetic field confined entirely within the cylinder, with zero field outside. The cylinders can also be magnetized such that the magnetic field is entirely outside the cylinder, with zero field inside. Several magnetization distributions are shown in the figures.


			[[File:K 1–4 cylindrical Halbach array.png\|thumb\|220x220px\|K 1–4 cylindrical Halbach array]]

	The direction of magnetization within the ferromagnetic material, in plane perpendicular to the axis of the cylinder, is given by		The direction of magnetization within the ferromagnetic material, in plane perpendicular to the axis of the cylinder, is given by
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	* Brushless motors or alternators typically use cylindrical designs in which all the flux is confined to the centre of the bore (such as ''k'' = 4 above, a 6-pole rotor) with the AC coils also contained within the bore. Such self-shielding motor or alternator designs are more efficient and produce higher torque or output than conventional motor or alternator designs.		* Brushless motors or alternators typically use cylindrical designs in which all the flux is confined to the centre of the bore (such as ''k'' = 4 above, a 6-pole rotor) with the AC coils also contained within the bore. Such self-shielding motor or alternator designs are more efficient and produce higher torque or output than conventional motor or alternator designs.
	* Magnetic-coupling devices transmit torque through magnetically transparent barriers (that is, the barrier is non-magnetic or is magnetic but not affected by an applied magnetic field), for instance, between sealed containers or pressurised vessels. The optimal torque couplings consists of a pair of coaxially nested cylinders with opposite +''k'' and −''k'' flux magnetization patterns, as this configuration is the only system for infinitely long cylinders that produces a torque.<ref name="Bjoerk2010"/> In the lowest-energy state, the outer flux of the inner cylinder exactly matches the internal flux of the outer cylinder. Rotating one cylinder relative to the other from this state results in a restoring torque.		* Magnetic-coupling devices transmit torque through magnetically transparent barriers (that is, the barrier is non-magnetic or is magnetic but not affected by an applied magnetic field), for instance, between sealed containers or pressurised vessels. The optimal torque couplings consists of a pair of coaxially nested cylinders with opposite +''k'' and −''k'' flux magnetization patterns, as this configuration is the only system for infinitely long cylinders that produces a torque.<ref name="Bjoerk2010"/> In the lowest-energy state, the outer flux of the inner cylinder exactly matches the internal flux of the outer cylinder. Rotating one cylinder relative to the other from this state results in a restoring torque.
	*Cylindrical Halbach Arrays are used in ~~Portable~~ [[MRI]] Scanners.<ref>{{cite web \|url=https://www.stanfordmagnets.com/halbach-magnets-history-types-and-uses.html \|title=Halbach Magnets: History, Types, and Uses \|last=Marchio \|first=Cathy \|date=May 23, 2024 \|website=Stanford Magnets \|access-date=July 31, 2024}}</ref> They offer the potential for a relatively lightweight, low to mid-field system with no [[cryogenics]], a small fringe field, and no electrical power requirements or heat dissipation needs.<ref>{{cite journal \|last1=Cooley \|first1=C. Z. \|last2=Haskell \|first2=M. W. \|year=2018 \|title=Design of Sparse Halbach Magnet Arrays for Portable MRI Using a Genetic Algorithm \|journal=IEEE Transactions on Magnetics \|volume=54 \|issue=1 \|pages=1-12 \|doi=10.1109/TMAG.2017.2751001\|pmc=5937527 }}</ref> The reduced stray fields also enhance safety and minimize interference with surrounding electronic devices.<ref>{{cite journal \|last1=Sarwar \|first1=A. \|last2=Nemirovski \|first2=A. \|year=2012 \|title=Optimal Halbach permanent magnet designs for maximally pulling and pushing nanoparticles \|journal=Journal of Magnetism and Magnetic Materials \|volume=234 \|issue=5 \|pages=742-754 \|doi=10.1016/j.jmmm.2011.09.008\|pmc=3547684 }}</ref>		*Cylindrical Halbach Arrays are used in [[MRI]] Scanners.<ref>{{cite web \|url=https://www.stanfordmagnets.com/halbach-magnets-history-types-and-uses.html \|title=Halbach Magnets: History, Types, and Uses \|last=Marchio \|first=Cathy \|date=May 23, 2024 \|website=Stanford Magnets \|access-date=July 31, 2024}}</ref> They offer the potential for a relatively lightweight, low to mid-field system with no [[cryogenics]], a small fringe field, and no electrical power requirements or heat dissipation needs.<ref>{{cite journal \|last1=Cooley \|first1=C. Z. \|last2=Haskell \|first2=M. W. \|year=2018 \|title=Design of Sparse Halbach Magnet Arrays for Portable MRI Using a Genetic Algorithm \|journal=IEEE Transactions on Magnetics \|volume=54 \|issue=1 \|pages=1-12 \|doi=10.1109/TMAG.2017.2751001\|pmc=5937527 }}</ref> The reduced stray fields also enhance safety and minimize interference with surrounding electronic devices.<ref>{{cite journal \|last1=Sarwar \|first1=A. \|last2=Nemirovski \|first2=A. \|year=2012 \|title=Optimal Halbach permanent magnet designs for maximally pulling and pushing nanoparticles \|journal=Journal of Magnetism and Magnetic Materials \|volume=234 \|issue=5 \|pages=742-754 \|doi=10.1016/j.jmmm.2011.09.008\|pmc=3547684 }}</ref>

	===Uniform fields===		===Uniform fields===
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	: <math>H = M_r \ln\left(\frac{R_\text{o}}{R_\text{i}}\right) \widehat{y},</math>		: <math>H = M_r \ln\left(\frac{R_\text{o}}{R_\text{i}}\right) \widehat{y},</math>

	where the inner and outer cylinder radii are ''R''<sub>i</sub> and ''R''<sub>o</sub> respectively. ''H'' is in the ''y'' direction. This is the simplest form of the Halbach cylinder, and it can be seen that if the ratio of outer to inner radii is greater than [[e (mathematical constant)\|''e'']], the flux inside the bore actually exceeds the [[remanence]] of the magnetic material used to create the cylinder. However, care has to be taken not to produce a field that exceeds the coercivity of the permanent magnets used, as this can result in demagnetization of the cylinder and the production of a much lower field than intended.<ref name="Bjoerk2015"/><ref name="Insinga2016"/>		where the inner and outer cylinder radii are ''R''<sub>i</sub> and ''R''<sub>o</sub> respectively. ''H'' is in the ''y'' direction. This is the simplest form of the Halbach cylinder, and it can be seen that if the ratio of outer to inner radii is greater than [[e (mathematical constant)\|''e'']], the flux inside the bore actually exceeds the [[remanence]] of the magnetic material used to create the cylinder. However, care has to be taken not to produce a field that exceeds the [[coercivity]] of the permanent magnets used, as this can result in demagnetization of the cylinder and the production of a much lower field than intended.<ref name="Bjoerk2015"/><ref name="Insinga2016"/>

	[[File:Halbach cylinder3.png\|thumb\|Three designs (A) (B) (C) producing uniform magnetic fields within their central air gap\|220x220px]]		[[File:Halbach cylinder3.png\|thumb\|Three designs (A) (B) (C) producing uniform magnetic fields within their central air gap\|220x220px]]
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	\| {{EquationRef\|8}}		\| {{EquationRef\|8}}
	}}		}}
	at the outer boundary. The potential gradient has non-vanishing radial component <math>\beta\cos\theta - M_0 \ln r \cos\theta - M_0 \cos\theta</math> in the cylinder walls and <math>\alpha\cos\theta</math> in the bore, and so the conditions on the potential derivative become		at the outer boundary. The [[potential gradient]] has non-vanishing radial component <math>\beta\cos\theta - M_0 \ln r \cos\theta - M_0 \cos\theta</math> in the cylinder walls and <math>\alpha\cos\theta</math> in the bore, and so the conditions on the potential derivative become
	:<math>(\beta\cos\theta - M_0 \ln r_\mathrm{i} \cos\theta - M_0\cos\theta) - \alpha \cos\theta = -M_0\cos\theta \implies \beta - M_0 \ln r_\mathrm{i} - \alpha = 0 \implies \alpha = \beta - M_0 \ln r_\mathrm{i}</math>		:<math>(\beta\cos\theta - M_0 \ln r_\mathrm{i} \cos\theta - M_0\cos\theta) - \alpha \cos\theta = -M_0\cos\theta \implies \beta - M_0 \ln r_\mathrm{i} - \alpha = 0 \implies \alpha = \beta - M_0 \ln r_\mathrm{i}</math>
	at the inner boundary and		at the inner boundary and
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	[[File:Halbach_planar_array.svg\|thumb\|Magnetic levitation using a planar Halbach array and concentric structure windings\|220x220px]]		[[File:Halbach_planar_array.svg\|thumb\|Magnetic levitation using a planar Halbach array and concentric structure windings\|220x220px]]


	==Sphere==		==Sphere==
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	: <math>B = \frac{4}{3} B_0 \ln\left(\frac{R_\text{o}}{R_\text{i}}\right).</math>		: <math>B = \frac{4}{3} B_0 \ln\left(\frac{R_\text{o}}{R_\text{i}}\right).</math>

	Higher fields are possible by optimising the spherical design to take account of the fact that it is composed of point dipoles (and not line dipoles). This results in the stretching of the sphere to an elliptical shape and having a non-uniform distribution of magnetization over the component parts. Using this method, as well as soft pole pieces within the design, 4.5 [[~~Tesla~~ (unit)\|T]] in a working volume of 20 mm<sup>3</sup> was achieved,<ref name="Bloch1998"/> and this was increased further to 5 T in 2002,<ref>{{Cite web \| url=http://cerncourier.com/cws/article/cern/28598 \| title=Record-breaking magnet has five-tesla field \|website=CERN Courier\| date=22 March 2002 }}</ref> although over a smaller working volume of 0.05 mm<sup>3</sup>. As hard materials are temperature-dependent, refrigeration of the entire magnet array can increase the field within the working area further. This group also reported development of a 5.16 T Halbach dipole cylinder in 2003.<ref name="Kumada2004"/>		Higher fields are possible by optimising the spherical design to take account of the fact that it is composed of point dipoles (and not line dipoles). This results in the stretching of the sphere to an elliptical shape and having a non-uniform distribution of magnetization over the component parts. Using this method, as well as soft pole pieces within the design, 4.5 [[tesla (unit)\|T]] in a working volume of 20 mm<sup>3</sup> was achieved,<ref name="Bloch1998"/> and this was increased further to 5 T in 2002,<ref>{{Cite web \| url=http://cerncourier.com/cws/article/cern/28598 \| title=Record-breaking magnet has five-tesla field \|website=CERN Courier\| date=22 March 2002 }}</ref> although over a smaller working volume of 0.05 mm<sup>3</sup>. As hard materials are temperature-dependent, refrigeration of the entire magnet array can increase the field within the working area further. This group also reported development of a 5.16 T Halbach dipole cylinder in 2003.<ref name="Kumada2004"/>

	== See also ==		== See also ==
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	<ref name="Bloch1998">{{cite journal		<ref name="Bloch1998">{{cite journal
	\| author = Bloch, F. and Cugat, O. and Meunier, G. and Toussaint, J.C.		\| author = Bloch, F. and Cugat, O. and Meunier, G. and Toussaint, J.C.
	\| title = Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4 ~~Tesla~~ permanent magnet flux source		\| title = Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4-tesla permanent magnet flux source
	\| journal = IEEE Transactions on Magnetics		\| journal = IEEE Transactions on Magnetics
	\| volume = 34		\| volume = 34
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	\| author2 = F. Bloch		\| author2 = F. Bloch
	\| author3 = J.C. Toussaint		\| author3 = J.C. Toussaint
	\| title = 4-~~Tesla~~ Permanent Magnetic Flux Source		\| title = 4-tesla Permanent Magnetic Flux Source
	\| journal = Proc. 15th International Workshop on Rare Earth Magnets and Their Applications		\| journal = Proc. 15th International Workshop on Rare Earth Magnets and Their Applications
	\| page = 807		\| page = 807

imported>L3erdnik: /* Variable linear arrays */ Adjusted phrasing for the fact that in the Halbach orientation the rods (except the edge cases) experience 0 torque and it is only a problem mid-rotation

2025-05-16T17:42:20Z

Variable linear arrays: Adjusted phrasing for the fact that in the Halbach orientation the rods (except the edge cases) experience 0 torque and it is only a problem mid-rotation

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Halbach array - Revision history

imported>Poocat at 14:24, 28 December 2025

imported>L3erdnik: /* Variable linear arrays */ Adjusted phrasing for the fact that in the Halbach orientation the rods (except the edge cases) experience 0 torque and it is only a problem mid-rotation