Halbach array - Knowledge

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their magnetization matches that of a Halbach cylinder, as shown for a 6-rod design. This design greatly increases access to the region of uniform field, at the expense of the volume of uniform field being smaller than in the cylindrical designs (although this area can be made larger by increasing the number of component rods). Rotating the rods relative to each other results in many possibilities, including a dynamically variable field and various dipolar configurations. It can be seen that the designs shown in (A) and (B) are closely related to the

4065: 6624:{\displaystyle \mathbf {H} =\nabla \varphi ={\begin{cases}M_{0}\ln \left({\frac {r_{\mathrm {o} }}{r_{\mathrm {i} }}}\right)\cos \theta \,{\hat {\mathbf {r} }}-M_{0}\ln \left({\frac {r_{\mathrm {o} }}{r_{\mathrm {i} }}}\right)\sin \theta \,{\hat {\boldsymbol {\theta }}}&r<r_{\mathrm {i} },\\M_{0}\cos \theta \left\,{\hat {\mathbf {r} }}-M_{0}\ln \left({\frac {r_{\mathrm {o} }}{r}}\right)\sin \theta \,{\hat {\boldsymbol {\theta }}}&r_{\mathrm {i} }<r<r_{\mathrm {o} },\\0&r>r_{\mathrm {o} },\end{cases}}} 3617: 3319: 1090: 655: 7043: 768: 776: 3698: 624: 297: 33: 6175: 7024: 663: 797: 3343: 144: 8315: 3047: 4279: 1230: 41: 4060:{\displaystyle {\begin{aligned}{\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial \varphi _{\mathrm {p} }}{\partial r}}\right)&={\frac {1}{r}}{\frac {\partial }{\partial r}}\left\\&={\frac {\cos \theta }{r}}{\frac {\partial }{\partial r}}\left\\&={\frac {\cos \theta }{r}}(\ln r+1+1)\\&={\frac {\ln r\cos \theta }{r}}+{\frac {2\cos \theta }{r}}\end{aligned}}} 5603: 5928: 6777: 5061: 682:. 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. 4782: 3612:{\displaystyle {\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial \varphi }{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}\varphi }{\partial \theta ^{2}}}={\begin{cases}0&r<r_{\mathrm {i} },\\-{\frac {2M_{0}\cos \theta }{r}}&r_{\mathrm {i} }<r<r_{\mathrm {o} },\\0&r>r_{\mathrm {0} }.\end{cases}}} 3314:{\displaystyle {\begin{aligned}\nabla \cdot \mathbf {M} _{\mathrm {cyl} }&={\frac {1}{r}}{\frac {\partial }{\partial r}}(rM_{0}\cos \theta )+{\frac {1}{r}}{\frac {\partial }{\partial \theta }}(M_{0}\sin \theta )\\&={\frac {M_{0}\cos \theta }{r}}+{\frac {M_{0}\cos \theta }{r}}\\&={\frac {2M_{0}\cos \theta }{r}}.\end{aligned}}} 7056:

finite-length cylinder design. The magnitude of the uniform field for a sphere also increases to 4/3 the amount for the ideal cylindrical design with the same inner and outer radii. However, for a spherical struction, access to the region of uniform field is usually restricted to a narrow hole at the top and bottom of the design.

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The direction of magnetization of the wedges in (A) can be calculated using a set of rules given by Abele and allows for great freedom in the shape of the cavity. Another class of design is the magnetic mangle (B), proposed by Coey and Cugat, in which uniformly magnetized rods are arranged such that

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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 is subject to

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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.

6170:{\displaystyle \varphi ={\begin{cases}M_{0}\ln \left({\frac {r_{\mathrm {o} }}{r_{\mathrm {i} }}}\right)r\cos \theta &r<r_{\mathrm {i} },\\M_{0}\ln \left({\frac {r_{\mathrm {o} }}{r}}\right)r\cos \theta &r_{\mathrm {i} }<r<r_{\mathrm {o} },\\0&r>r_{\mathrm {o} }.\end{cases}}} 1068:

flux magnetization patterns, as this configuration is the only system for infinitely long cylinders that produces a torque. 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

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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

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a strong torque from its neighboring rods, and therefore requires mechanical stabilization. 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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which is independent of position. Similarly, outside the cylinder the magnetisation also vanishes, and since the magnetic field vanishes there, the flux density does too. So indeed the field is uniform inside and zero outside the ideal Halbach cylinder, with a magnitude depending on its physical

7019:{\displaystyle B_{\mathrm {bore} }=\left|\mathbf {B} _{\mathrm {bore} }\right|={\sqrt {\cos ^{2}\theta +\sin ^{2}\theta }}{\frac {M_{0}}{\mu _{0}}}\ln \left({\frac {r_{\mathrm {o} }}{r_{\mathrm {i} }}}\right)={\frac {M_{0}}{\mu _{0}}}\ln \left({\frac {r_{\mathrm {o} }}{r_{\mathrm {i} }}}\right),} 1237:

This cylindrical design is only one class of designs that produce a uniform field inside a cavity within an array of permanent magnets. Other classes of design include wedge designs, proposed by Abele and Jensen, in which wedges of magnetized material are arranged to provide uniform field within

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Ideally, these structures would be created from an infinite-length cylinder of magnetic material with the direction of magnetization continuously varying. The magnetic flux produced by this ideal design would be perfectly uniform and be entirely confined to either the bore of the cylinder or the

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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

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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

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Halbach cylinders give a static field. However, cylinders can be nested, and by rotating one cylinder relative to the other, cancellation of the field and adjustment of the direction can be achieved. As the outside field of a cylinder is quite low, the relative rotation does not require strong

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If the two-dimensional magnetic distribution patterns of the Halbach cylinder are extended to three dimensions, the result is the Halbach sphere. These designs have an extremely uniform field within the interior of the design, as they are not affected by the "end effects" prevalent in the

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The rotating pattern of permanent magnets (on the front face; on the left, up, right, down) can be continued indefinitely and have the same effect. The effect of this arrangement is roughly similar to many horseshoe magnets placed adjacent to each other, with similar poles touching.

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in 1998, and this was increased further to 5 T in 2002, although over a smaller working volume of 0.05 mm. As hard materials are temperature-dependent, refrigeration of the entire magnet array can increase the field within the working area further, as shown by Kumada

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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.

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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.

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are arbitrary constants. The desired solution will be the sum of the particular and homogeneous solutions that satisfies the boundary conditions. Again with the benefit of hindsight, let us set most of the constants to zero immediately and assert that the solution is

4274:{\displaystyle {\begin{aligned}{\frac {1}{r^{2}}}{\frac {\partial ^{2}\varphi _{\mathrm {p} }}{\partial \theta ^{2}}}&={\frac {1}{r^{2}}}{\frac {\partial ^{2}}{\partial \theta ^{2}}}(r\ln r\cos \theta )\\&=-{\frac {\ln r\cos \theta }{r}}.\end{aligned}}} 2874: 6769: 2515: 1056: = 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. 642:

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

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across the boundary, but since the physical quantities we are interested in depend on gradients of this potential, we can arbitrarily set the constant to zero for convenience.) To obtain a second set of conditions, integrate Equation

1188: 5920: 2691: 5598:{\displaystyle (\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} }} 2525: 5614: 1896: 5056:{\displaystyle \varphi ={\begin{cases}\alpha r\cos \theta &r<r_{\mathrm {i} },\\\beta r\cos \theta -M_{0}r\ln r\cos \theta &r_{\mathrm {i} }<r<r_{\mathrm {o} },\\0&r>r_{\mathrm {o} },\end{cases}}} 4345: 5338: 2433:. The sign difference is due to the relative orientation of the magnetisation and the surface normal to the part of the integration volume inside the cylinder walls being opposite at the inner and outer boundaries. 72:

This magnetic orientation process replicates that applied by a magnetic recording tape head to the magnetic tape coating during the recording process. The principle was further described by James (Jim) M. Winey of

3703: 4430: 1823: 4777:{\displaystyle \varphi _{\mathrm {h} }=A_{0}+B_{0}\theta +C_{0}\ln r+\sum _{n=1}^{\infty }\left(A_{n}r^{n}+B_{n}r^{-n}\right)\cos n\theta +\sum _{n=1}^{\infty }\left(C_{n}r^{n}+D_{n}r^{-n}\right)\sin n\theta ,} 2312: 3052: 6686: 1733: 2025: 8178:

Bloch, F. and Cugat, O. and Meunier, G. and Toussaint, J.C. (1998). "Innovating approaches to the generation of intense magnetic fields: design and optimization of a 4 Tesla permanent magnet flux source".

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These cylindrical structures are used in devices such as brushless AC motors, magnetic couplings and high-field cylinders. Both brushless motors and coupling devices use multipole field arrangements:

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in 1973, and these "one-sided flux" structures were initially described by him as a "curiosity", although at the time he recognized from this discovery the potential for significant improvements in

4484: 2864: 1080:, a small fringe field, and no electrical power requirements or heat dissipation needs. The reduced stray fields also enhance safety and minimize interference with surrounding electronic devices. 1040:, which introduce non-uniformities in the field. The difficulty of manufacturing a cylinder with a continuously varying magnetization also usually leads to the design being broken into segments. 5240: 5845: 65:

that augments the magnetic field on one side of the array while cancelling the field to near zero on the other side. This is achieved by having a spatially rotating pattern of magnetisation.

1361: 4860: 2152: 3012:{\displaystyle \nabla ^{2}f={\frac {1}{r}}{\frac {\partial }{\partial r}}\left(r{\frac {\partial f}{\partial r}}\right)+{\frac {1}{r^{2}}}{\frac {\partial ^{2}f}{\partial \theta ^{2}}}.} 1770: 388: 805: 746: 1626: 1591: 1332: 5367: 2431: 2396: 2118: 2083: 5093: 561: 291: 1012: 2178: 1970: 1948: 1679: 1654: 472: 1301: 6691: 2446: 2218: 2198: 1916: 1257:

Minispec) standard plate geometries (C) of permanent magnets, they, as explained above, offer a huge bore diameter, while still having a reasonably homogeneous field.

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Raich, H.; Blümler, P. (21 October 2004). "Design and construction of a dipolar Halbach array with a homogeneous field from identical bar magnets: NMR Mandhalas".

2789:{\displaystyle \nabla f={\frac {\partial f}{\partial r}}{\hat {\mathbf {r} }}+{\frac {1}{r}}{\frac {\partial f}{\partial \theta }}{\hat {\boldsymbol {\theta }}}.} 7065: 1107: 7764: 5850: 2681: 2361: 523: 2633:{\displaystyle \nabla \cdot \mathbf {F} ={\frac {1}{r}}{\frac {\partial }{\partial r}}(rF_{r})+{\frac {1}{r}}{\frac {\partial F_{\theta }}{\partial \theta }}.} 2341: 1246: = 2 Halbach cylinder. Other very simple designs for a uniform field include separated magnets with soft iron return paths, as shown in figure (C). 44:

A Halbach array, showing the orientation of each piece's magnetic field. This array would give a strong field underneath, while the field above would cancel.

7436: 7537: 5774:{\displaystyle 0-(\beta \cos \theta -M_{0}\ln r_{\mathrm {0} }\cos \theta -M_{0}\cos \theta )=M_{0}\cos \theta \implies \beta -M_{0}\ln r_{\mathrm {o} }=0} 7215: 1848: 1036:

outside of the cylinder. Of course, the ideal case of infinite length is not realizable, and in practice the finite length of the cylinders produces

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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

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The method used to find the field created by the cylinder is mathematically very similar to that used to investigate a uniformly magnetised sphere.

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produced (in the ideal case) on the opposite side. This helps with field confinement, usually a problem in the design of magnetic structures.

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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

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The magnetic field around an infinite Halbach array of cube magnets. The field does not cancel perfectly due to the discrete magnets used.

155:, the reason for this flux distribution can be intuitively visualised using Mallinson's original diagram (note that it uses the negative 2241: 8135: 1052:

Brushless motors or alternators typically use cylindrical designs in which all the flux is confined to the centre of the bore (such as

6637: 1684: 1548:{\displaystyle \mathbf {M} _{\mathrm {cyl} }=M_{0}(\cos \theta \,{\hat {\mathbf {r} }}+\sin \theta \,{\hat {\boldsymbol {\theta }}}),} 1980: 7039:

forces. In the ideal case of infinitely long cylinders, no force would be required to rotate a cylinder with respect to the other.

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The direction of magnetization within the ferromagnetic material, in plane perpendicular to the axis of the cylinder, is given by

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Because of the symmetry of the arrangement along the cylinder's axis, the problem can be treated as two-dimensional. Work in

638:) refrigerator magnets are created with a Halbach magnetization pattern for a stronger holding force when attached to a flat 96: 5095:

are constants to be determined. If we can choose the constants such that the boundary conditions are satisfied, then by the

7932: 8161: 4444: 2818: 246:; the components of the magnetization vector can therefore be any Hilbert-transform pair (the simplest of which is simply 620:

and Inductrack rocket-launch system utilize the Halbach array to lift the train by repelling loops of wire in the track.

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A. R. Insinga; C. R. H. Bahl; R. Bjørk; A. Smith (2016). "Performance of Halbach magnet arrays with finite coercivity".

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T. R. Ni Mhiochain; D. Weaire; S. M. McMurry; J. M. D. Coey (1999). "Analysis of torque in nested magnetic cylinders".

7406:"Magnetically levitated train takes flight | US Department of Energy Science News | EurekAlert! Science News" 151:

Although this magnetic flux distribution seems somewhat counter-intuitive to those familiar with simple bar magnets or

5799: 8114: 1337: 7571:"Final progress report for the NASA Inductrack model rocket launcher at the Lawrence Livermore National Laboratory" 4790: 17: 2123: 7440: 1741: 310: 7544: 8319: 8136:"Tip Magazine: Magnets, Markets, and Magic Cylinders The Industrial Physicist by Michael Coey and Denis Weaire" 77:

in 1970, for the ideal case of continuously rotating magnetization, induced by a one-sided stripe-shaped coil.

7824: 7262: 8245: 700: 8350: 1736: 1596: 1561: 1306: 8110: 1842: 7353:"Electromagnetic Transducer, James Winey, Figure 29; United States Patent 3,674,946, filed Dec. 23, 1970" 5343: 2401: 2366: 2088: 2053: 759:

of the magnetic sheet given by the spacing between magnetic blocks with the same magnetization vector.

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The field on the non-cancelling side of the ideal, continuously varying, infinite array is of the form

8247: 7327: 5069: 1032: − 1 gives an internal magnetic field, and a negative one gives an external magnetic field. 8360: 8355: 7577: 7413: 1919: 1216: 532: 7609: 6212: 5943: 4884: 3467: 249: 8335: 830: 159:

component, unlike the diagram in Mallinson's article). The diagram shows the field from a strip of

57: 7352: 6764:{\displaystyle \mathbf {B} _{\mathrm {bore} }={\frac {1}{\mu _{0}}}\mathbf {H} _{\mathrm {bore} }} 2161: 1953: 1931: 1662: 1637: 434: 4491: 2510:{\displaystyle \mathbf {F} =F_{r}{\hat {\mathbf {r} }}+F_{\theta }{\hat {\boldsymbol {\theta }}}} 1274: 1269: 1076:

Scanners. They offer the potential for a relatively lightweight, low to mid-field system with no

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during the 1980s, independently invented the Halbach array to focus particle accelerator beams.

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J. M. D. Coey; T.R. Ní Mhíocháin (2003). "Permanent Magnets". In F. Herlach; N. Miura (eds.).

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The field is twice as large on the side on which the flux is confined (in the idealized case).

8290: 8208: 7596: 4497: 4487: 2044: 1657: 648: 627: 599: 8142: 399: 217: 189: 8262: 8188: 8177: 8000: 7957: 7789: 7729: 7684: 7639: 7492: 7379: 7303: 7230: 7139:{\displaystyle B={\frac {4}{3}}B_{0}\ln \left({\frac {R_{\text{o}}}{R_{\text{i}}}}\right).} 6634:

while the magnetic flux density can then be found everywhere using the previous definition

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Since the problem is static there are no free currents and all time derivatives vanish, so

1420: 1183:{\displaystyle H=M_{r}\ln \left({\frac {R_{\text{o}}}{R_{\text{i}}}}\right){\widehat {y}},} 603: 479: 7892:"Optimal Halbach permanent magnet designs for maximally pulling and pushing nanoparticles" 7478:"Optimal Halbach permanent magnet designs for maximally pulling and pushing nanoparticles" 5915:{\displaystyle \alpha =M_{0}\ln \left({\frac {r_{\mathrm {o} }}{r_{\mathrm {i} }}}\right)} 8: 7765:"Analysis of the magnetic field, force, and torque for two-dimensional Halbach cylinders" 3636: 1922:(up to a sign under some definitions). Substituting this back into the previous Equation 679: 607: 587: 179:

directions. The effect of superimposing both of these structures is shown in the figure.

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Three designs (A) (B) (C) producing uniform magnetic fields within their central air gap

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This group also reported development of a 5.16 T Halbach dipole cylinder in 2003.

2666: 2346: 2232: 2155: 508: 4494:, it can be shown that the general homogeneous solution whose gradient is periodic in 2320: 123: 8246:

Kumada, M. and Antokhin, E.I. and Iwashita, Y. and Aoki, M. and Sugiyama, E. (2004).

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O. Cugat; F. Bloch; J.C. Toussaint (1998). "4-Tesla Permanent Magnetic Flux Source".

8039: 8020: 7749: 7674: 7518: 7319: 7246: 7242: 243: 81: 8282: 8095: 7977: 7809: 7258: 8270: 8196: 8083: 8008: 7965: 7911: 7903: 7870: 7862: 7851:"Design of Sparse Halbach Magnet Arrays for Portable MRI Using a Genetic Algorithm" 7797: 7737: 7692: 7655: 7647: 7508: 7500: 7462: 7387: 7311: 7238: 7177: 7172: 2813: 7046:

Magnetic levitation using a planar Halbach array and concentric structure windings

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A ferromagnetic cylinder showing various magnetization patterns and magnetic field

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at the outer boundary. The potential gradient has non-vanishing radial component

1891:{\displaystyle \nabla \times \mathbf {H} =0\implies \mathbf {H} =\nabla \varphi } 8228: 7710:

R. Bjørk (2011). "The ideal dimensions of a Halbach cylinder of finite length".

7216:"Design of permanent multipole magnets with oriented rare earth cobalt material" 3639:

of this equation in the cylinder walls. With the benefit of hindsight, consider

8158: 8012: 7969: 7907: 7801: 7651: 7504: 7193: 1632: 671: 186:. In fact, any magnetization pattern where the components of magnetization are 8061:

Proc. 15th International Workshop on Rare Earth Magnets and Their Applications

7866: 7391: 7289:"Applications of Permanent Magnets in Accelerators and Electron Storage Rings" 8344: 8274: 7323: 7250: 4340:{\displaystyle \nabla ^{2}\varphi _{\mathrm {p} }={\frac {2\cos \theta }{r}}} 1414: 817: 639: 92: 85: 7933:"The efficiency and the demagnetization field of a general Halbach cylinder" 7522: 7151: 7150:

Using this method, as well as soft pole pieces within the design, 4.5

5333:{\displaystyle \beta \cos \theta -M_{0}\ln r\cos \theta -M_{0}\cos \theta } 2661: 2441: 1249:

In recent years, these Halbach dipoles have been used to conduct low-field

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while the magnetisation vanishes outside the walls, that is for the bore

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in the bore, and so the conditions on the potential derivative become

4514:(such that all the physical quantities are single-valued) is given by 4425:{\displaystyle -M_{0}\varphi _{\mathrm {p} }=-M_{0}r\ln r\cos \theta } 586:

Thus they have a surprising number of applications, ranging from flat

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Concepts in Magnetic Resonance Part B: Magnetic Resonance Engineering

7696: 7568: 7315: 1818:{\displaystyle \nabla \cdot \mathbf {H} =-\nabla \cdot \mathbf {M} .} 1222: 1025: 644: 171:

direction (top right). Note that the field above the plane is in the

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is continuous across the boundary. (More properly this implies that

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Consider now the boundary conditions at the cylinder-air interfaces

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Mallinson J.C. (1973). "One-Sided Fluxes — A Magnetic Curiosity?".

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Schematic of a Halbach array consisting of a series magnetized rods

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direction for both structures, but the field below the plane is in

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at the outer boundary. Note that these are identical to Equations

2307:{\displaystyle \left=\pm \mathbf {M} \cdot {\hat {\mathbf {r} }},} 690:

components of the magnetic fields from the sheets cancel, and the

6688:. In the bore, where the magnetisation vanishes, this reduces to 147:

Cancellation of magnetic components resulting in a one-sided flux

8248:"Super Strong Permanent Magnet Quadrupole for a Linear Collider" 7475: 6681:{\displaystyle \mathbf {B} =\mu _{0}(\mathbf {H} +\mathbf {M} )} 1728:{\displaystyle \mathbf {B} =\mu _{0}(\mathbf {H} +\mathbf {M} )} 143: 8314: 7538:"Toward More Efficient Transport: The Inductrack Maglev System" 7190:

uses Halbach arrays to generate strong fields for maglev trains

2020:{\displaystyle \nabla ^{2}\varphi =-\nabla \cdot \mathbf {M} ,} 1254: 62: 8058: 8033: 132:

Orientation of strong and weak sides in a linear Halbach array

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across the boundary, and in our case the sign is negative at

8038:. Vol. 1. World Scientific Publishing. pp. 25–47. 3330:, which is that we want to solve, becomes by using Equation 2231:

across a small volume straddling the boundary and apply the

598:, magnetic drug targeting to high-tech applications such as 571:

The advantages of one-sided flux distributions are twofold:

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cancel below the plane and reinforce itself above the plane

40: 7569:

L. S. Tung; R. F. Post; J. Martinez-Frias (27 June 2001).

3685:{\displaystyle \varphi _{\mathrm {p} }=r\ln r\cos \theta } 1101:= 2, the field inside the bore is uniform and is given by 503:

is the magnitude of the field at the surface of the array,

7622: 1406:{\displaystyle r_{\mathrm {i} }<r<r_{\mathrm {o} }} 1250: 1073: 7543:. Lawrence Livermore National Laboratory. Archived from 7154:

in a working volume of 20 mm was achieved by Bloch

5157:{\displaystyle \alpha =\beta -M_{0}\ln r_{\mathrm {i} }} 3041:, the magnetisation divergence in the cylinder walls is 2154:

over a small loop straddling the boundary and applying

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Equal-gearing arrangement for a variable Halbach array

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Scaling up this design and adding a top sheet gives a

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through industrial applications such as the brushless

7068: 6780: 6694: 6640: 6189: 5931: 5853: 5802: 5617: 5378: 5346: 5264: 5191: 5113: 5072: 4872: 4793: 4523: 4500: 4447: 4359: 4290: 4079: 3701: 3645: 3346: 3050: 2877: 2821: 2694: 2669: 2528: 2449: 2404: 2369: 2349: 2323: 2244: 2206: 2186: 2164: 2126: 2091: 2056: 1983: 1956: 1934: 1904: 1851: 1783: 1744: 1687: 1665: 1640: 1599: 1564: 1453: 1423: 1369: 1340: 1309: 1277: 1110: 833: 703: 535: 511: 482: 437: 402: 313: 252: 220: 192: 8073: 5796:, so indeed the guess was consistent. Hence we have 4479:{\displaystyle \nabla ^{2}\varphi _{\mathrm {h} }=0} 2859:{\displaystyle \nabla ^{2}f=\nabla \cdot (\nabla f)} 694:

components reinforce, so that the field is given by

6771:. Hence the magnitude of the flux density there is 4435:Now consider the homogeneous equation for Equation 7138: 7059:The equation for the field in a Halbach sphere is 7018: 6763: 6680: 6623: 6169: 5914: 5839: 5773: 5597: 5361: 5332: 5235:{\displaystyle \beta -M_{0}\ln r_{\mathrm {o} }=0} 5234: 5156: 5087: 5055: 4854: 4776: 4506: 4478: 4424: 4339: 4273: 4059: 3684: 3611: 3313: 3011: 2858: 2788: 2675: 2632: 2509: 2425: 2390: 2355: 2335: 2306: 2212: 2192: 2172: 2146: 2112: 2077: 2019: 1964: 1942: 1910: 1890: 1817: 1764: 1727: 1673: 1648: 1620: 1585: 1547: 1436: 1405: 1355: 1326: 1295: 1182: 1006: 740: 630:showing a flat refrigerator magnet's magnetization 555: 517: 495: 466: 423: 382: 285: 234: 206: 1253:experiments. Compared to commercially available ( 8342: 8159:Permanent magnet based sources of magnetic field 7889: 5840:{\displaystyle \beta =M_{0}\ln r_{\mathrm {o} }} 1221:, the flux inside the bore actually exceeds the 1072:Cylindrical Halbach Arrays are used in Portable 658:Flux distribution for a flat refrigerator magnet 7535: 1260: 8255:IEEE Transactions on Applied Superconductivity 7369: 1356:{\displaystyle {\hat {\boldsymbol {\theta }}}} 8229:"Record-breaking magnet has five-tesla field" 7848: 7668: 7476:A. Sarwar; A. Nemirovski; B. Shapiro (2012). 7286: 7213: 6180:Consequently, the magnetic field is given by 4855:{\displaystyle A_{i},\ B_{i},\ C_{i},\ D_{i}} 1193:where the inner and outer cylinder radii are 8295:: CS1 maint: multiple names: authors list ( 8213:: CS1 maint: multiple names: authors list ( 8036:High Magnetic Fields: Science and Technology 2147:{\displaystyle \nabla \times \mathbf {H} =0} 8108: 7993:Journal of Magnetism and Magnetic Materials 7984: 7940:Journal of Magnetism and Magnetic Materials 7896:Journal of Magnetism and Magnetic Materials 7825:"Halbach Magnets: History, Types, and Uses" 7772:Journal of Magnetism and Magnetic Materials 7632:Journal of Magnetism and Magnetic Materials 7485:Journal of Magnetism and Magnetic Materials 1765:{\displaystyle \nabla \cdot \mathbf {B} =0} 1417:in the cylinder walls, which has magnitude 383:{\displaystyle F(x,y)=F_{0}e^{ikx}e^{-ky},} 8239: 7931:R. Bjørk; A. Smith; C. R. H. Bahl (2015). 7924: 7763:R. Bjørk; A. Smith; C. R. H. Bahl (2010). 7756: 7703: 7616: 7437:"Concerning the Physics of Halbach Arrays" 7280: 7207: 5730: 5726: 5554: 5550: 5503: 5499: 2180:is continuous. This in turn requires that 1873: 1869: 1363:, and let the cylinder have radial extent 1060:coaxially nested cylinders with opposite + 666:Schematic diagram of a free-electron laser 8336:magnet levitated vehicle switching system 8171: 8052: 7951: 7915: 7874: 7783: 7723: 7659: 7512: 7469: 6533: 6463: 6355: 6276: 5097:uniqueness theorem for Poisson's equation 1526: 1499: 762: 8027: 7709: 7562: 7041: 4432:is the appropriate particular solution. 2158:requires that the parallel component of 1228: 1088: 803: 795: 774: 766: 661: 653: 622: 295: 182:The crucial point is that the flux will 142: 61:) is a special arrangement of permanent 39: 31: 27:Special arrangement of permanent magnets 7822: 6537: 6359: 2774: 2498: 1530: 1344: 14: 8343: 7849:Cooley, C. Z.; Haskell, M. W. (2018). 741:{\displaystyle H_{y}\approx \cos(kx),} 163:with alternating magnetization in the 8109:Fitzpatrick, Richard (27 June 2014). 7434: 1621:{\displaystyle r>r_{\mathrm {o} }} 1586:{\displaystyle r<r_{\mathrm {i} }} 1327:{\displaystyle {\hat {\mathbf {r} }}} 1238:cavities inside the design as shown. 1093:Uniform field inside Halbach cylinder 816:is a magnetized cylinder composed of 97:Lawrence Berkeley National Laboratory 56: 7625:"An adjustable linear Halbach array" 7623:J. E. Hilton; S. M. McMurry (2012). 7033: 5182: 5104: 3337: 2868: 2685: 2519: 1974: 1774: 1656:is related to the magnetisation and 1069:state results in a restoring torque. 7890:Sarwar, A.; Nemirovski, A. (2012). 7536:Richard F. Post (10 October 2005). 5362:{\displaystyle \alpha \cos \theta } 5099:, we must have found the solution. 2812:Combining these two relations, the 36:The flux diagram of a Halbach array 24: 7463:"High efficiency voice coil motor" 7196:can give very even magnetic fields 7001: 6989: 6932: 6920: 6826: 6823: 6820: 6817: 6796: 6793: 6790: 6787: 6755: 6752: 6749: 6746: 6712: 6709: 6706: 6703: 6605: 6575: 6554: 6509: 6437: 6382: 6334: 6322: 6255: 6243: 6198: 6151: 6121: 6100: 6065: 6024: 5986: 5974: 5900: 5888: 5831: 5759: 5589: 5532: 5419: 5220: 5148: 5037: 5007: 4986: 4916: 4698: 4606: 4530: 4464: 4449: 4379: 4307: 4292: 4181: 4171: 4128: 4120: 4105: 3894: 3890: 3819: 3815: 3796: 3792: 3759: 3751: 3742: 3722: 3718: 3652: 3563: 3542: 3487: 3443: 3429: 3391: 3383: 3363: 3359: 3160: 3156: 3104: 3100: 3075: 3072: 3069: 3055: 2990: 2976: 2938: 2930: 2910: 2906: 2879: 2847: 2838: 2823: 2762: 2754: 2715: 2707: 2695: 2618: 2603: 2559: 2555: 2529: 2426:{\displaystyle r=r_{\mathrm {o} }} 2417: 2391:{\displaystyle r=r_{\mathrm {i} }} 2382: 2260: 2252: 2127: 2113:{\displaystyle r=r_{\mathrm {o} }} 2104: 2078:{\displaystyle r=r_{\mathrm {i} }} 2069: 2003: 1985: 1882: 1852: 1801: 1784: 1745: 1612: 1577: 1468: 1465: 1462: 1397: 1376: 293:, as shown in the diagram above). 80:The effect was also discovered by 25: 8372: 8307: 8115:The University of Texas at Austin 1084: 1043: 8313: 7576:. UCRL-ID-144455. Archived from 6811: 6740: 6697: 6671: 6663: 6642: 6468: 6281: 6191: 5088:{\displaystyle \alpha ,\ \beta } 3063: 2728: 2536: 2472: 2451: 2436:In plane-polar coordinates, the 2291: 2280: 2166: 2134: 2010: 1972:, we find that we need to solve 1958: 1936: 1875: 1859: 1808: 1791: 1752: 1718: 1710: 1689: 1667: 1642: 1504: 1456: 1314: 167:direction (top left) and in the 138: 122: 111: 102: 8221: 8152: 8128: 8102: 8067: 7883: 7842: 7816: 7223:Nuclear Instruments and Methods 5102:The continuity conditions give 4347:, and comparison with Equation 2343:denotes a jump in the quantity 566: 556:{\displaystyle 2\pi /\lambda .} 8181:IEEE Transactions on Magnetics 7855:IEEE Transactions on Magnetics 7823:Marchio, Cathy (23 May 2024). 7529: 7455: 7428: 7398: 7372:IEEE Transactions on Magnetics 7363: 7345: 6675: 6659: 6540: 6472: 6362: 6285: 5727: 5701: 5624: 5551: 5500: 5456: 5379: 4221: 4197: 3989: 3965: 3929: 3911: 3852: 3828: 3191: 3169: 3138: 3113: 2853: 2844: 2777: 2732: 2584: 2568: 2501: 2476: 2330: 2324: 2295: 1870: 1722: 1706: 1539: 1533: 1508: 1487: 1347: 1318: 1290: 1278: 950: 938: 878: 866: 732: 723: 529:(i.e., the spatial frequency) 418: 406: 329: 317: 286:{\displaystyle \sin(x)\cos(y)} 280: 274: 265: 259: 13: 1: 8331:Electric model aircraft motor 8326:Passive levitation of a shaft 8111:"Uniformly Magnetized Sphere" 7200: 1631:By definition, the auxiliary 1303:with associated unit vectors 1007:{\displaystyle M=M_{r}\left,} 686:design is chosen so that the 7243:10.1016/0029-554X(80)90094-4 2173:{\displaystyle \mathbf {H} } 1965:{\displaystyle \mathbf {M} } 1943:{\displaystyle \mathbf {H} } 1674:{\displaystyle \mathbf {B} } 1649:{\displaystyle \mathbf {H} } 1261:Derivation in the ideal case 467:{\displaystyle F_{x}+iF_{y}} 7: 7166: 5792: 5786: 5248: 5170: 4437: 4349: 3625: 3332: 3326: 3037: 3025: 2802: 2646: 2227: 2033: 1924: 1831: 1296:{\displaystyle (r,\theta )} 1028:(A/m). A positive value of 791: 10: 8377: 8013:10.1016/j.jmmm.2016.01.076 7970:10.1016/j.jmmm.2015.02.034 7908:10.1016/j.jmmm.2011.09.008 7802:10.1016/j.jmmm.2009.08.044 7712:Journal of Applied Physics 7677:Journal of Applied Physics 7652:10.1016/j.jmmm.2012.02.014 7505:10.1016/j.jmmm.2011.09.008 7296:Journal of Applied Physics 5608:at the inner boundary and 5340:in the cylinder walls and 5180:at the inner boundary and 7867:10.1109/TMAG.2017.2751001 7392:10.1109/TMAG.1973.1067714 7050: 1920:magnetic scalar potential 431:is the field in the form 8275:10.1109/TASC.2004.830555 7435:Creel, James R. (2006). 4490:. Through the method of 2213:{\displaystyle \varphi } 2193:{\displaystyle \varphi } 1911:{\displaystyle \varphi } 1097:For the special case of 4507:{\displaystyle \theta } 4492:separation of variables 4486:. This has the form of 3692:, because then we have 1772:, this is equivalently 1633:magnetic field strength 1270:plane-polar coordinates 7718:(1): 013915–013915–6. 7604:Cite journal requires 7287:Klaus Halbach (1985). 7214:Klaus Halbach (1980). 7140: 7047: 7020: 6765: 6682: 6625: 6171: 5922:, giving the solution 5916: 5841: 5775: 5599: 5363: 5334: 5236: 5158: 5089: 5057: 4856: 4778: 4702: 4610: 4508: 4480: 4426: 4341: 4275: 4061: 3686: 3613: 3315: 3013: 2860: 2790: 2677: 2634: 2511: 2427: 2392: 2357: 2337: 2308: 2214: 2194: 2174: 2148: 2114: 2079: 2043:which has the form of 2021: 1966: 1944: 1912: 1892: 1845:additionally requires 1819: 1766: 1729: 1675: 1650: 1622: 1587: 1549: 1444:, rotates smoothly as 1438: 1407: 1357: 1328: 1297: 1234: 1184: 1094: 1008: 821:shown in the figures. 809: 808:Cylinder magnetization 801: 780: 772: 763:Variable linear arrays 742: 667: 659: 631: 557: 519: 497: 468: 425: 424:{\displaystyle F(x,y)} 384: 301: 287: 236: 235:{\displaystyle \pi /2} 208: 207:{\displaystyle \pi /2} 161:ferromagnetic material 148: 45: 37: 8164:24 April 2012 at the 7141: 7045: 7021: 6766: 6683: 6626: 6172: 5917: 5842: 5776: 5600: 5364: 5335: 5237: 5159: 5090: 5058: 4857: 4779: 4682: 4590: 4509: 4481: 4427: 4342: 4276: 4062: 3687: 3614: 3316: 3014: 2861: 2791: 2678: 2635: 2512: 2428: 2393: 2358: 2338: 2309: 2215: 2195: 2175: 2149: 2115: 2080: 2022: 1967: 1945: 1913: 1893: 1820: 1767: 1730: 1676: 1658:magnetic flux density 1651: 1623: 1588: 1550: 1439: 1437:{\displaystyle M_{0}} 1408: 1358: 1329: 1298: 1232: 1185: 1092: 1024:is the ferromagnetic 1009: 807: 799: 778: 770: 743: 665: 657: 649:magnetic viewing film 628:Magnetic viewing film 626: 604:particle accelerators 558: 520: 498: 496:{\displaystyle F_{0}} 469: 426: 385: 299: 288: 237: 209: 146: 43: 35: 8322:at Wikimedia Commons 7066: 6778: 6692: 6638: 6187: 5929: 5851: 5800: 5615: 5376: 5344: 5262: 5189: 5111: 5070: 4870: 4791: 4521: 4498: 4445: 4357: 4288: 4077: 3699: 3643: 3344: 3048: 2875: 2819: 2692: 2667: 2526: 2447: 2402: 2367: 2347: 2321: 2242: 2204: 2184: 2162: 2124: 2089: 2054: 1981: 1954: 1932: 1902: 1849: 1781: 1742: 1685: 1663: 1638: 1597: 1562: 1451: 1421: 1367: 1338: 1307: 1275: 1108: 831: 701: 680:free-electron lasers 636:hard ceramic ferrite 608:free-electron lasers 588:refrigerator magnets 533: 509: 480: 435: 400: 311: 250: 218: 190: 8351:Magnetic levitation 8267:2004ITAS...14.1287K 8193:1998ITM....34.2465B 8088:10.1002/cmr.b.20018 8005:2016JMMM..407..369I 7962:2015JMMM..384..128B 7794:2010JMMM..322..133B 7734:2011JAP...109a3915B 7689:1999JAP....86.6412N 7644:2012JMMM..324.2051H 7497:2012JMMM..324..742S 7416:on 12 February 2019 7384:1973ITM.....9..678M 7308:1985JAP....57.3605H 7268:on 31 December 2018 7235:1980NucIM.169....1H 3637:particular solution 2317:where the notation 634:Flat flexible (not 7410:www.eurekalert.org 7136: 7048: 7016: 6761: 6678: 6621: 6616: 6167: 6162: 5912: 5837: 5771: 5595: 5359: 5330: 5232: 5154: 5085: 5053: 5048: 4852: 4774: 4504: 4488:Laplace's equation 4476: 4422: 4337: 4271: 4269: 4057: 4055: 3682: 3609: 3604: 3311: 3309: 3009: 2856: 2786: 2673: 2630: 2507: 2423: 2388: 2353: 2333: 2304: 2233:divergence theorem 2210: 2190: 2170: 2144: 2110: 2075: 2045:Poisson's equation 2017: 1962: 1940: 1908: 1888: 1815: 1762: 1725: 1671: 1646: 1618: 1583: 1545: 1434: 1403: 1353: 1324: 1293: 1235: 1180: 1095: 1004: 810: 802: 781: 773: 738: 668: 660: 632: 553: 515: 493: 464: 421: 380: 302: 283: 232: 204: 149: 46: 38: 8318:Media related to 8201:10.1109/20.717567 8148:on 28 March 2006. 8045:978-981-02-4964-9 7742:10.1063/1.3525646 7683:(11): 6412–6424. 7638:(13): 2051–2056. 7357:www.espacenet.com 7127: 7124: 7114: 7083: 7034:Varying the field 7007: 6969: 6938: 6900: 6876: 6736: 6543: 6518: 6475: 6446: 6365: 6340: 6288: 6261: 6074: 5992: 5906: 5256: 5255: 5178: 5177: 5081: 4841: 4825: 4809: 4335: 4262: 4195: 4166: 4142: 4099: 4051: 4027: 3963: 3901: 3886: 3826: 3803: 3788: 3766: 3729: 3714: 3633: 3632: 3532: 3457: 3423: 3398: 3370: 3355: 3302: 3261: 3230: 3167: 3152: 3111: 3096: 3033: 3032: 3004: 2970: 2945: 2917: 2902: 2810: 2809: 2780: 2769: 2749: 2735: 2722: 2676:{\displaystyle f} 2654: 2653: 2625: 2598: 2566: 2551: 2504: 2479: 2356:{\displaystyle f} 2298: 2267: 2220:must differ by a 2041: 2040: 1839: 1838: 1593:and surroundings 1536: 1511: 1350: 1321: 1174: 1159: 1156: 1146: 993: 972: 921: 900: 518:{\displaystyle k} 244:Hilbert transform 82:John C. Mallinson 58:[ˈhalbax] 16:(Redirected from 8368: 8361:Types of magnets 8356:Magnetic devices 8317: 8301: 8300: 8294: 8286: 8261:(2): 1287–1289. 8252: 8243: 8237: 8236: 8235:. 22 March 2002. 8225: 8219: 8218: 8212: 8204: 8187:(5): 2465–2468. 8175: 8169: 8156: 8150: 8149: 8147: 8141:. Archived from 8140: 8132: 8126: 8125: 8123: 8121: 8106: 8100: 8099: 8071: 8065: 8064: 8056: 8050: 8049: 8031: 8025: 8024: 7988: 7982: 7981: 7955: 7937: 7928: 7922: 7921: 7919: 7887: 7881: 7880: 7878: 7846: 7840: 7839: 7837: 7835: 7829:Stanford Magnets 7820: 7814: 7813: 7787: 7769: 7760: 7754: 7753: 7727: 7707: 7701: 7700: 7697:10.1063/1.371705 7672: 7666: 7665: 7663: 7629: 7620: 7614: 7613: 7607: 7602: 7600: 7592: 7590: 7588: 7582: 7575: 7566: 7560: 7559: 7557: 7555: 7549: 7542: 7533: 7527: 7526: 7516: 7482: 7473: 7467: 7466: 7459: 7453: 7452: 7450: 7448: 7439:. Archived from 7432: 7426: 7425: 7423: 7421: 7412:. Archived from 7402: 7396: 7395: 7367: 7361: 7360: 7349: 7343: 7342: 7340: 7338: 7332: 7326:. Archived from 7316:10.1063/1.335021 7302:(1): 3605–3608. 7293: 7284: 7278: 7277: 7275: 7273: 7267: 7261:. Archived from 7220: 7211: 7178:Permanent magnet 7173:Neodymium magnet 7145: 7143: 7142: 7137: 7132: 7128: 7126: 7125: 7122: 7116: 7115: 7112: 7106: 7094: 7093: 7084: 7076: 7025: 7023: 7022: 7017: 7012: 7008: 7006: 7005: 7004: 6994: 6993: 6992: 6982: 6970: 6968: 6967: 6958: 6957: 6948: 6943: 6939: 6937: 6936: 6935: 6925: 6924: 6923: 6913: 6901: 6899: 6898: 6889: 6888: 6879: 6877: 6869: 6868: 6850: 6849: 6840: 6835: 6831: 6830: 6829: 6814: 6801: 6800: 6799: 6770: 6768: 6767: 6762: 6760: 6759: 6758: 6743: 6737: 6735: 6734: 6722: 6717: 6716: 6715: 6700: 6687: 6685: 6684: 6679: 6674: 6666: 6658: 6657: 6645: 6630: 6628: 6627: 6622: 6620: 6619: 6610: 6609: 6608: 6580: 6579: 6578: 6559: 6558: 6557: 6545: 6544: 6536: 6523: 6519: 6514: 6513: 6512: 6502: 6490: 6489: 6477: 6476: 6471: 6466: 6462: 6458: 6451: 6447: 6442: 6441: 6440: 6430: 6404: 6403: 6387: 6386: 6385: 6367: 6366: 6358: 6345: 6341: 6339: 6338: 6337: 6327: 6326: 6325: 6315: 6303: 6302: 6290: 6289: 6284: 6279: 6266: 6262: 6260: 6259: 6258: 6248: 6247: 6246: 6236: 6224: 6223: 6194: 6176: 6174: 6173: 6168: 6166: 6165: 6156: 6155: 6154: 6126: 6125: 6124: 6105: 6104: 6103: 6079: 6075: 6070: 6069: 6068: 6058: 6046: 6045: 6029: 6028: 6027: 5997: 5993: 5991: 5990: 5989: 5979: 5978: 5977: 5967: 5955: 5954: 5921: 5919: 5918: 5913: 5911: 5907: 5905: 5904: 5903: 5893: 5892: 5891: 5881: 5869: 5868: 5846: 5844: 5843: 5838: 5836: 5835: 5834: 5818: 5817: 5780: 5778: 5777: 5772: 5764: 5763: 5762: 5746: 5745: 5716: 5715: 5691: 5690: 5669: 5668: 5667: 5651: 5650: 5604: 5602: 5601: 5596: 5594: 5593: 5592: 5576: 5575: 5537: 5536: 5535: 5519: 5518: 5489: 5488: 5446: 5445: 5424: 5423: 5422: 5406: 5405: 5368: 5366: 5365: 5360: 5339: 5337: 5336: 5331: 5320: 5319: 5289: 5288: 5250: 5241: 5239: 5238: 5233: 5225: 5224: 5223: 5207: 5206: 5183: 5172: 5163: 5161: 5160: 5155: 5153: 5152: 5151: 5135: 5134: 5105: 5094: 5092: 5091: 5086: 5079: 5062: 5060: 5059: 5054: 5052: 5051: 5042: 5041: 5040: 5012: 5011: 5010: 4991: 4990: 4989: 4956: 4955: 4921: 4920: 4919: 4861: 4859: 4858: 4853: 4851: 4850: 4839: 4835: 4834: 4823: 4819: 4818: 4807: 4803: 4802: 4783: 4781: 4780: 4775: 4758: 4754: 4753: 4752: 4740: 4739: 4727: 4726: 4717: 4716: 4701: 4696: 4666: 4662: 4661: 4660: 4648: 4647: 4635: 4634: 4625: 4624: 4609: 4604: 4577: 4576: 4561: 4560: 4548: 4547: 4535: 4534: 4533: 4513: 4511: 4510: 4505: 4485: 4483: 4482: 4477: 4469: 4468: 4467: 4457: 4456: 4431: 4429: 4428: 4423: 4400: 4399: 4384: 4383: 4382: 4372: 4371: 4346: 4344: 4343: 4338: 4336: 4331: 4317: 4312: 4311: 4310: 4300: 4299: 4280: 4278: 4277: 4272: 4270: 4263: 4258: 4238: 4227: 4196: 4194: 4193: 4192: 4179: 4178: 4169: 4167: 4165: 4164: 4152: 4143: 4141: 4140: 4139: 4126: 4125: 4124: 4123: 4113: 4112: 4102: 4100: 4098: 4097: 4085: 4066: 4064: 4063: 4058: 4056: 4052: 4047: 4033: 4028: 4023: 4003: 3995: 3964: 3959: 3948: 3940: 3936: 3932: 3902: 3900: 3889: 3887: 3882: 3871: 3863: 3859: 3855: 3827: 3825: 3814: 3804: 3802: 3791: 3789: 3781: 3772: 3768: 3767: 3765: 3757: 3756: 3755: 3754: 3740: 3730: 3728: 3717: 3715: 3707: 3691: 3689: 3688: 3683: 3657: 3656: 3655: 3627: 3618: 3616: 3615: 3610: 3608: 3607: 3598: 3597: 3596: 3568: 3567: 3566: 3547: 3546: 3545: 3533: 3528: 3518: 3517: 3504: 3492: 3491: 3490: 3458: 3456: 3455: 3454: 3441: 3437: 3436: 3426: 3424: 3422: 3421: 3409: 3404: 3400: 3399: 3397: 3389: 3381: 3371: 3369: 3358: 3356: 3348: 3338: 3320: 3318: 3317: 3312: 3310: 3303: 3298: 3288: 3287: 3274: 3266: 3262: 3257: 3247: 3246: 3236: 3231: 3226: 3216: 3215: 3205: 3197: 3181: 3180: 3168: 3166: 3155: 3153: 3145: 3128: 3127: 3112: 3110: 3099: 3097: 3089: 3080: 3079: 3078: 3066: 3027: 3018: 3016: 3015: 3010: 3005: 3003: 3002: 3001: 2988: 2984: 2983: 2973: 2971: 2969: 2968: 2956: 2951: 2947: 2946: 2944: 2936: 2928: 2918: 2916: 2905: 2903: 2895: 2887: 2886: 2869: 2865: 2863: 2862: 2857: 2831: 2830: 2804: 2795: 2793: 2792: 2787: 2782: 2781: 2773: 2770: 2768: 2760: 2752: 2750: 2742: 2737: 2736: 2731: 2726: 2723: 2721: 2713: 2705: 2686: 2682: 2680: 2679: 2674: 2648: 2639: 2637: 2636: 2631: 2626: 2624: 2616: 2615: 2614: 2601: 2599: 2591: 2583: 2582: 2567: 2565: 2554: 2552: 2544: 2539: 2520: 2516: 2514: 2513: 2508: 2506: 2505: 2497: 2494: 2493: 2481: 2480: 2475: 2470: 2467: 2466: 2454: 2432: 2430: 2429: 2424: 2422: 2421: 2420: 2398:and positive at 2397: 2395: 2394: 2389: 2387: 2386: 2385: 2362: 2360: 2359: 2354: 2342: 2340: 2339: 2336:{\displaystyle } 2334: 2313: 2311: 2310: 2305: 2300: 2299: 2294: 2289: 2283: 2272: 2268: 2266: 2258: 2250: 2219: 2217: 2216: 2211: 2199: 2197: 2196: 2191: 2179: 2177: 2176: 2171: 2169: 2153: 2151: 2150: 2145: 2137: 2119: 2117: 2116: 2111: 2109: 2108: 2107: 2084: 2082: 2081: 2076: 2074: 2073: 2072: 2035: 2026: 2024: 2023: 2018: 2013: 1993: 1992: 1975: 1971: 1969: 1968: 1963: 1961: 1949: 1947: 1946: 1941: 1939: 1917: 1915: 1914: 1909: 1897: 1895: 1894: 1889: 1878: 1862: 1833: 1824: 1822: 1821: 1816: 1811: 1794: 1775: 1771: 1769: 1768: 1763: 1755: 1734: 1732: 1731: 1726: 1721: 1713: 1705: 1704: 1692: 1680: 1678: 1677: 1672: 1670: 1655: 1653: 1652: 1647: 1645: 1627: 1625: 1624: 1619: 1617: 1616: 1615: 1592: 1590: 1589: 1584: 1582: 1581: 1580: 1554: 1552: 1551: 1546: 1538: 1537: 1529: 1513: 1512: 1507: 1502: 1486: 1485: 1473: 1472: 1471: 1459: 1443: 1441: 1440: 1435: 1433: 1432: 1412: 1410: 1409: 1404: 1402: 1401: 1400: 1381: 1380: 1379: 1362: 1360: 1359: 1354: 1352: 1351: 1343: 1333: 1331: 1330: 1325: 1323: 1322: 1317: 1312: 1302: 1300: 1299: 1294: 1189: 1187: 1186: 1181: 1176: 1175: 1167: 1164: 1160: 1158: 1157: 1154: 1148: 1147: 1144: 1138: 1126: 1125: 1013: 1011: 1010: 1005: 1000: 996: 995: 994: 986: 983: 979: 978: 974: 973: 965: 923: 922: 914: 911: 907: 906: 902: 901: 893: 849: 848: 814:Halbach cylinder 747: 745: 744: 739: 713: 712: 602:magnets used in 562: 560: 559: 554: 546: 524: 522: 521: 516: 502: 500: 499: 494: 492: 491: 473: 471: 470: 465: 463: 462: 447: 446: 430: 428: 427: 422: 389: 387: 386: 381: 376: 375: 360: 359: 344: 343: 292: 290: 289: 284: 241: 239: 238: 233: 228: 213: 211: 210: 205: 200: 126: 115: 60: 55: 21: 18:Halbach cylinder 8376: 8375: 8371: 8370: 8369: 8367: 8366: 8365: 8341: 8340: 8310: 8305: 8304: 8288: 8287: 8250: 8244: 8240: 8227: 8226: 8222: 8206: 8205: 8176: 8172: 8166:Wayback Machine 8157: 8153: 8145: 8138: 8134: 8133: 8129: 8119: 8117: 8107: 8103: 8072: 8068: 8057: 8053: 8046: 8032: 8028: 7989: 7985: 7935: 7929: 7925: 7888: 7884: 7847: 7843: 7833: 7831: 7821: 7817: 7767: 7761: 7757: 7708: 7704: 7673: 7669: 7627: 7621: 7617: 7605: 7603: 7594: 7593: 7586: 7584: 7583:on 5 March 2016 7580: 7573: 7567: 7563: 7553: 7551: 7550:on 4 April 2023 7547: 7540: 7534: 7530: 7480: 7474: 7470: 7461: 7460: 7456: 7446: 7444: 7433: 7429: 7419: 7417: 7404: 7403: 7399: 7368: 7364: 7351: 7350: 7346: 7336: 7334: 7333:on 6 April 2012 7330: 7291: 7285: 7281: 7271: 7269: 7265: 7218: 7212: 7208: 7203: 7183:Strong focusing 7169: 7121: 7117: 7111: 7107: 7105: 7101: 7089: 7085: 7075: 7067: 7064: 7063: 7053: 7036: 7000: 6999: 6995: 6988: 6987: 6983: 6981: 6977: 6963: 6959: 6953: 6949: 6947: 6931: 6930: 6926: 6919: 6918: 6914: 6912: 6908: 6894: 6890: 6884: 6880: 6878: 6864: 6860: 6845: 6841: 6839: 6816: 6815: 6810: 6809: 6805: 6786: 6785: 6781: 6779: 6776: 6775: 6745: 6744: 6739: 6738: 6730: 6726: 6721: 6702: 6701: 6696: 6695: 6693: 6690: 6689: 6670: 6662: 6653: 6649: 6641: 6639: 6636: 6635: 6615: 6614: 6604: 6603: 6599: 6591: 6585: 6584: 6574: 6573: 6569: 6553: 6552: 6548: 6546: 6535: 6534: 6508: 6507: 6503: 6501: 6497: 6485: 6481: 6467: 6465: 6464: 6436: 6435: 6431: 6429: 6425: 6418: 6414: 6399: 6395: 6392: 6391: 6381: 6380: 6376: 6368: 6357: 6356: 6333: 6332: 6328: 6321: 6320: 6316: 6314: 6310: 6298: 6294: 6280: 6278: 6277: 6254: 6253: 6249: 6242: 6241: 6237: 6235: 6231: 6219: 6215: 6208: 6207: 6190: 6188: 6185: 6184: 6161: 6160: 6150: 6149: 6145: 6137: 6131: 6130: 6120: 6119: 6115: 6099: 6098: 6094: 6092: 6064: 6063: 6059: 6057: 6053: 6041: 6037: 6034: 6033: 6023: 6022: 6018: 6010: 5985: 5984: 5980: 5973: 5972: 5968: 5966: 5962: 5950: 5946: 5939: 5938: 5930: 5927: 5926: 5899: 5898: 5894: 5887: 5886: 5882: 5880: 5876: 5864: 5860: 5852: 5849: 5848: 5830: 5829: 5825: 5813: 5809: 5801: 5798: 5797: 5758: 5757: 5753: 5741: 5737: 5711: 5707: 5686: 5682: 5663: 5662: 5658: 5646: 5642: 5616: 5613: 5612: 5588: 5587: 5583: 5571: 5567: 5531: 5530: 5526: 5514: 5510: 5484: 5480: 5441: 5437: 5418: 5417: 5413: 5401: 5397: 5377: 5374: 5373: 5345: 5342: 5341: 5315: 5311: 5284: 5280: 5263: 5260: 5259: 5219: 5218: 5214: 5202: 5198: 5190: 5187: 5186: 5147: 5146: 5142: 5130: 5126: 5112: 5109: 5108: 5071: 5068: 5067: 5047: 5046: 5036: 5035: 5031: 5023: 5017: 5016: 5006: 5005: 5001: 4985: 4984: 4980: 4978: 4951: 4947: 4926: 4925: 4915: 4914: 4910: 4902: 4880: 4879: 4871: 4868: 4867: 4846: 4842: 4830: 4826: 4814: 4810: 4798: 4794: 4792: 4789: 4788: 4745: 4741: 4735: 4731: 4722: 4718: 4712: 4708: 4707: 4703: 4697: 4686: 4653: 4649: 4643: 4639: 4630: 4626: 4620: 4616: 4615: 4611: 4605: 4594: 4572: 4568: 4556: 4552: 4543: 4539: 4529: 4528: 4524: 4522: 4519: 4518: 4499: 4496: 4495: 4463: 4462: 4458: 4452: 4448: 4446: 4443: 4442: 4395: 4391: 4378: 4377: 4373: 4367: 4363: 4358: 4355: 4354: 4318: 4316: 4306: 4305: 4301: 4295: 4291: 4289: 4286: 4285: 4268: 4267: 4239: 4237: 4225: 4224: 4188: 4184: 4180: 4174: 4170: 4168: 4160: 4156: 4151: 4144: 4135: 4131: 4127: 4119: 4118: 4114: 4108: 4104: 4103: 4101: 4093: 4089: 4084: 4080: 4078: 4075: 4074: 4054: 4053: 4034: 4032: 4004: 4002: 3993: 3992: 3949: 3947: 3938: 3937: 3907: 3903: 3893: 3888: 3872: 3870: 3861: 3860: 3818: 3813: 3809: 3805: 3795: 3790: 3780: 3773: 3758: 3750: 3749: 3745: 3741: 3739: 3735: 3731: 3721: 3716: 3706: 3702: 3700: 3697: 3696: 3651: 3650: 3646: 3644: 3641: 3640: 3603: 3602: 3592: 3591: 3587: 3579: 3573: 3572: 3562: 3561: 3557: 3541: 3540: 3536: 3534: 3513: 3509: 3505: 3503: 3497: 3496: 3486: 3485: 3481: 3473: 3463: 3462: 3450: 3446: 3442: 3432: 3428: 3427: 3425: 3417: 3413: 3408: 3390: 3382: 3380: 3376: 3372: 3362: 3357: 3347: 3345: 3342: 3341: 3324:Hence Equation 3308: 3307: 3283: 3279: 3275: 3273: 3264: 3263: 3242: 3238: 3237: 3235: 3211: 3207: 3206: 3204: 3195: 3194: 3176: 3172: 3159: 3154: 3144: 3123: 3119: 3103: 3098: 3088: 3081: 3068: 3067: 3062: 3061: 3051: 3049: 3046: 3045: 3035:Using Equation 2997: 2993: 2989: 2979: 2975: 2974: 2972: 2964: 2960: 2955: 2937: 2929: 2927: 2923: 2919: 2909: 2904: 2894: 2882: 2878: 2876: 2873: 2872: 2826: 2822: 2820: 2817: 2816: 2772: 2771: 2761: 2753: 2751: 2741: 2727: 2725: 2724: 2714: 2706: 2704: 2693: 2690: 2689: 2668: 2665: 2664: 2656:Similarly, the 2617: 2610: 2606: 2602: 2600: 2590: 2578: 2574: 2558: 2553: 2543: 2535: 2527: 2524: 2523: 2496: 2495: 2489: 2485: 2471: 2469: 2468: 2462: 2458: 2450: 2448: 2445: 2444: 2416: 2415: 2411: 2403: 2400: 2399: 2381: 2380: 2376: 2368: 2365: 2364: 2348: 2345: 2344: 2322: 2319: 2318: 2290: 2288: 2287: 2279: 2259: 2251: 2249: 2245: 2243: 2240: 2239: 2205: 2202: 2201: 2185: 2182: 2181: 2165: 2163: 2160: 2159: 2156:Stokes' theorem 2133: 2125: 2122: 2121: 2103: 2102: 2098: 2090: 2087: 2086: 2068: 2067: 2063: 2055: 2052: 2051: 2009: 1988: 1984: 1982: 1979: 1978: 1957: 1955: 1952: 1951: 1935: 1933: 1930: 1929: 1903: 1900: 1899: 1874: 1858: 1850: 1847: 1846: 1807: 1790: 1782: 1779: 1778: 1751: 1743: 1740: 1739: 1717: 1709: 1700: 1696: 1688: 1686: 1683: 1682: 1666: 1664: 1661: 1660: 1641: 1639: 1636: 1635: 1611: 1610: 1606: 1598: 1595: 1594: 1576: 1575: 1571: 1563: 1560: 1559: 1528: 1527: 1503: 1501: 1500: 1481: 1477: 1461: 1460: 1455: 1454: 1452: 1449: 1448: 1428: 1424: 1422: 1419: 1418: 1396: 1395: 1391: 1375: 1374: 1370: 1368: 1365: 1364: 1342: 1341: 1339: 1336: 1335: 1313: 1311: 1310: 1308: 1305: 1304: 1276: 1273: 1272: 1263: 1206: 1199: 1166: 1165: 1153: 1149: 1143: 1139: 1137: 1133: 1121: 1117: 1109: 1106: 1105: 1087: 1046: 1022: 985: 984: 964: 957: 953: 937: 933: 913: 912: 892: 885: 881: 865: 861: 854: 850: 844: 840: 832: 829: 828: 794: 765: 708: 704: 702: 699: 698: 569: 542: 534: 531: 530: 510: 507: 506: 487: 483: 481: 478: 477: 458: 454: 442: 438: 436: 433: 432: 401: 398: 397: 365: 361: 349: 345: 339: 335: 312: 309: 308: 251: 248: 247: 224: 219: 216: 215: 196: 191: 188: 187: 141: 136: 135: 134: 133: 129: 128: 127: 118: 117: 116: 105: 95:, while at the 53: 28: 23: 22: 15: 12: 11: 5: 8374: 8364: 8363: 8358: 8353: 8339: 8338: 8333: 8328: 8323: 8309: 8308:External links 8306: 8303: 8302: 8238: 8220: 8170: 8151: 8127: 8101: 8066: 8051: 8044: 8026: 7983: 7923: 7902:(5): 742–754. 7882: 7841: 7815: 7778:(1): 133–141. 7755: 7702: 7667: 7615: 7606:|journal= 7561: 7528: 7491:(5): 742–754. 7468: 7454: 7443:on 4 June 2011 7427: 7397: 7378:(4): 678–682. 7362: 7344: 7279: 7205: 7204: 7202: 7199: 7198: 7197: 7194:Helmholtz coil 7191: 7185: 7180: 7175: 7168: 7165: 7147: 7146: 7135: 7131: 7120: 7110: 7104: 7100: 7097: 7092: 7088: 7082: 7079: 7074: 7071: 7052: 7049: 7035: 7032: 7027: 7026: 7015: 7011: 7003: 6998: 6991: 6986: 6980: 6976: 6973: 6966: 6962: 6956: 6952: 6946: 6942: 6934: 6929: 6922: 6917: 6911: 6907: 6904: 6897: 6893: 6887: 6883: 6875: 6872: 6867: 6863: 6859: 6856: 6853: 6848: 6844: 6838: 6834: 6828: 6825: 6822: 6819: 6813: 6808: 6804: 6798: 6795: 6792: 6789: 6784: 6757: 6754: 6751: 6748: 6742: 6733: 6729: 6725: 6720: 6714: 6711: 6708: 6705: 6699: 6677: 6673: 6669: 6665: 6661: 6656: 6652: 6648: 6644: 6632: 6631: 6618: 6613: 6607: 6602: 6598: 6595: 6592: 6590: 6587: 6586: 6583: 6577: 6572: 6568: 6565: 6562: 6556: 6551: 6547: 6542: 6539: 6532: 6529: 6526: 6522: 6517: 6511: 6506: 6500: 6496: 6493: 6488: 6484: 6480: 6474: 6470: 6461: 6457: 6454: 6450: 6445: 6439: 6434: 6428: 6424: 6421: 6417: 6413: 6410: 6407: 6402: 6398: 6394: 6393: 6390: 6384: 6379: 6375: 6372: 6369: 6364: 6361: 6354: 6351: 6348: 6344: 6336: 6331: 6324: 6319: 6313: 6309: 6306: 6301: 6297: 6293: 6287: 6283: 6275: 6272: 6269: 6265: 6257: 6252: 6245: 6240: 6234: 6230: 6227: 6222: 6218: 6214: 6213: 6211: 6206: 6203: 6200: 6197: 6193: 6178: 6177: 6164: 6159: 6153: 6148: 6144: 6141: 6138: 6136: 6133: 6132: 6129: 6123: 6118: 6114: 6111: 6108: 6102: 6097: 6093: 6091: 6088: 6085: 6082: 6078: 6073: 6067: 6062: 6056: 6052: 6049: 6044: 6040: 6036: 6035: 6032: 6026: 6021: 6017: 6014: 6011: 6009: 6006: 6003: 6000: 5996: 5988: 5983: 5976: 5971: 5965: 5961: 5958: 5953: 5949: 5945: 5944: 5942: 5937: 5934: 5910: 5902: 5897: 5890: 5885: 5879: 5875: 5872: 5867: 5863: 5859: 5856: 5833: 5828: 5824: 5821: 5816: 5812: 5808: 5805: 5782: 5781: 5770: 5767: 5761: 5756: 5752: 5749: 5744: 5740: 5736: 5733: 5729: 5725: 5722: 5719: 5714: 5710: 5706: 5703: 5700: 5697: 5694: 5689: 5685: 5681: 5678: 5675: 5672: 5666: 5661: 5657: 5654: 5649: 5645: 5641: 5638: 5635: 5632: 5629: 5626: 5623: 5620: 5606: 5605: 5591: 5586: 5582: 5579: 5574: 5570: 5566: 5563: 5560: 5557: 5553: 5549: 5546: 5543: 5540: 5534: 5529: 5525: 5522: 5517: 5513: 5509: 5506: 5502: 5498: 5495: 5492: 5487: 5483: 5479: 5476: 5473: 5470: 5467: 5464: 5461: 5458: 5455: 5452: 5449: 5444: 5440: 5436: 5433: 5430: 5427: 5421: 5416: 5412: 5409: 5404: 5400: 5396: 5393: 5390: 5387: 5384: 5381: 5358: 5355: 5352: 5349: 5329: 5326: 5323: 5318: 5314: 5310: 5307: 5304: 5301: 5298: 5295: 5292: 5287: 5283: 5279: 5276: 5273: 5270: 5267: 5254: 5253: 5244: 5242: 5231: 5228: 5222: 5217: 5213: 5210: 5205: 5201: 5197: 5194: 5176: 5175: 5166: 5164: 5150: 5145: 5141: 5138: 5133: 5129: 5125: 5122: 5119: 5116: 5084: 5078: 5075: 5064: 5063: 5050: 5045: 5039: 5034: 5030: 5027: 5024: 5022: 5019: 5018: 5015: 5009: 5004: 5000: 4997: 4994: 4988: 4983: 4979: 4977: 4974: 4971: 4968: 4965: 4962: 4959: 4954: 4950: 4946: 4943: 4940: 4937: 4934: 4931: 4928: 4927: 4924: 4918: 4913: 4909: 4906: 4903: 4901: 4898: 4895: 4892: 4889: 4886: 4885: 4883: 4878: 4875: 4849: 4845: 4838: 4833: 4829: 4822: 4817: 4813: 4806: 4801: 4797: 4785: 4784: 4773: 4770: 4767: 4764: 4761: 4757: 4751: 4748: 4744: 4738: 4734: 4730: 4725: 4721: 4715: 4711: 4706: 4700: 4695: 4692: 4689: 4685: 4681: 4678: 4675: 4672: 4669: 4665: 4659: 4656: 4652: 4646: 4642: 4638: 4633: 4629: 4623: 4619: 4614: 4608: 4603: 4600: 4597: 4593: 4589: 4586: 4583: 4580: 4575: 4571: 4567: 4564: 4559: 4555: 4551: 4546: 4542: 4538: 4532: 4527: 4503: 4475: 4472: 4466: 4461: 4455: 4451: 4421: 4418: 4415: 4412: 4409: 4406: 4403: 4398: 4394: 4390: 4387: 4381: 4376: 4370: 4366: 4362: 4334: 4330: 4327: 4324: 4321: 4315: 4309: 4304: 4298: 4294: 4282: 4281: 4266: 4261: 4257: 4254: 4251: 4248: 4245: 4242: 4236: 4233: 4230: 4228: 4226: 4223: 4220: 4217: 4214: 4211: 4208: 4205: 4202: 4199: 4191: 4187: 4183: 4177: 4173: 4163: 4159: 4155: 4150: 4147: 4145: 4138: 4134: 4130: 4122: 4117: 4111: 4107: 4096: 4092: 4088: 4083: 4082: 4068: 4067: 4050: 4046: 4043: 4040: 4037: 4031: 4026: 4022: 4019: 4016: 4013: 4010: 4007: 4001: 3998: 3996: 3994: 3991: 3988: 3985: 3982: 3979: 3976: 3973: 3970: 3967: 3962: 3958: 3955: 3952: 3946: 3943: 3941: 3939: 3935: 3931: 3928: 3925: 3922: 3919: 3916: 3913: 3910: 3906: 3899: 3896: 3892: 3885: 3881: 3878: 3875: 3869: 3866: 3864: 3862: 3858: 3854: 3851: 3848: 3845: 3842: 3839: 3836: 3833: 3830: 3824: 3821: 3817: 3812: 3808: 3801: 3798: 3794: 3787: 3784: 3779: 3776: 3774: 3771: 3764: 3761: 3753: 3748: 3744: 3738: 3734: 3727: 3724: 3720: 3713: 3710: 3705: 3704: 3681: 3678: 3675: 3672: 3669: 3666: 3663: 3660: 3654: 3649: 3631: 3630: 3621: 3619: 3606: 3601: 3595: 3590: 3586: 3583: 3580: 3578: 3575: 3574: 3571: 3565: 3560: 3556: 3553: 3550: 3544: 3539: 3535: 3531: 3527: 3524: 3521: 3516: 3512: 3508: 3502: 3499: 3498: 3495: 3489: 3484: 3480: 3477: 3474: 3472: 3469: 3468: 3466: 3461: 3453: 3449: 3445: 3440: 3435: 3431: 3420: 3416: 3412: 3407: 3403: 3396: 3393: 3388: 3385: 3379: 3375: 3368: 3365: 3361: 3354: 3351: 3322: 3321: 3306: 3301: 3297: 3294: 3291: 3286: 3282: 3278: 3272: 3269: 3267: 3265: 3260: 3256: 3253: 3250: 3245: 3241: 3234: 3229: 3225: 3222: 3219: 3214: 3210: 3203: 3200: 3198: 3196: 3193: 3190: 3187: 3184: 3179: 3175: 3171: 3165: 3162: 3158: 3151: 3148: 3143: 3140: 3137: 3134: 3131: 3126: 3122: 3118: 3115: 3109: 3106: 3102: 3095: 3092: 3087: 3084: 3082: 3077: 3074: 3071: 3065: 3060: 3057: 3054: 3053: 3031: 3030: 3021: 3019: 3008: 3000: 2996: 2992: 2987: 2982: 2978: 2967: 2963: 2959: 2954: 2950: 2943: 2940: 2935: 2932: 2926: 2922: 2915: 2912: 2908: 2901: 2898: 2893: 2890: 2885: 2881: 2855: 2852: 2849: 2846: 2843: 2840: 2837: 2834: 2829: 2825: 2808: 2807: 2798: 2796: 2785: 2779: 2776: 2767: 2764: 2759: 2756: 2748: 2745: 2740: 2734: 2730: 2720: 2717: 2712: 2709: 2703: 2700: 2697: 2672: 2652: 2651: 2642: 2640: 2629: 2623: 2620: 2613: 2609: 2605: 2597: 2594: 2589: 2586: 2581: 2577: 2573: 2570: 2564: 2561: 2557: 2550: 2547: 2542: 2538: 2534: 2531: 2503: 2500: 2492: 2488: 2484: 2478: 2474: 2465: 2461: 2457: 2453: 2419: 2414: 2410: 2407: 2384: 2379: 2375: 2372: 2352: 2332: 2329: 2326: 2315: 2314: 2303: 2297: 2293: 2286: 2282: 2278: 2275: 2271: 2265: 2262: 2257: 2254: 2248: 2209: 2189: 2168: 2143: 2140: 2136: 2132: 2129: 2120:. Integrating 2106: 2101: 2097: 2094: 2071: 2066: 2062: 2059: 2039: 2038: 2029: 2027: 2016: 2012: 2008: 2005: 2002: 1999: 1996: 1991: 1987: 1960: 1938: 1907: 1887: 1884: 1881: 1877: 1872: 1868: 1865: 1861: 1857: 1854: 1837: 1836: 1827: 1825: 1814: 1810: 1806: 1803: 1800: 1797: 1793: 1789: 1786: 1761: 1758: 1754: 1750: 1747: 1724: 1720: 1716: 1712: 1708: 1703: 1699: 1695: 1691: 1669: 1644: 1614: 1609: 1605: 1602: 1579: 1574: 1570: 1567: 1556: 1555: 1544: 1541: 1535: 1532: 1525: 1522: 1519: 1516: 1510: 1506: 1498: 1495: 1492: 1489: 1484: 1480: 1476: 1470: 1467: 1464: 1458: 1431: 1427: 1399: 1394: 1390: 1387: 1384: 1378: 1373: 1349: 1346: 1320: 1316: 1292: 1289: 1286: 1283: 1280: 1262: 1259: 1207:respectively. 1204: 1197: 1191: 1190: 1179: 1173: 1170: 1163: 1152: 1142: 1136: 1132: 1129: 1124: 1120: 1116: 1113: 1086: 1085:Uniform fields 1083: 1082: 1081: 1070: 1057: 1045: 1042: 1020: 1015: 1014: 1003: 999: 992: 989: 982: 977: 971: 968: 963: 960: 956: 952: 949: 946: 943: 940: 936: 932: 929: 926: 920: 917: 910: 905: 899: 896: 891: 888: 884: 880: 877: 874: 871: 868: 864: 860: 857: 853: 847: 843: 839: 836: 793: 790: 764: 761: 749: 748: 737: 734: 731: 728: 725: 722: 719: 716: 711: 707: 672:wiggler magnet 584: 583: 576: 568: 565: 564: 563: 552: 549: 545: 541: 538: 514: 504: 490: 486: 475: 461: 457: 453: 450: 445: 441: 420: 417: 414: 411: 408: 405: 391: 390: 379: 374: 371: 368: 364: 358: 355: 352: 348: 342: 338: 334: 331: 328: 325: 322: 319: 316: 282: 279: 276: 273: 270: 267: 264: 261: 258: 255: 231: 227: 223: 203: 199: 195: 140: 137: 131: 130: 121: 120: 119: 110: 109: 108: 107: 106: 104: 101: 26: 9: 6: 4: 3: 2: 8373: 8362: 8359: 8357: 8354: 8352: 8349: 8348: 8346: 8337: 8334: 8332: 8329: 8327: 8324: 8321: 8320:Halbach array 8316: 8312: 8311: 8298: 8292: 8284: 8280: 8276: 8272: 8268: 8264: 8260: 8256: 8249: 8242: 8234: 8230: 8224: 8216: 8210: 8202: 8198: 8194: 8190: 8186: 8182: 8174: 8167: 8163: 8160: 8155: 8144: 8137: 8131: 8116: 8112: 8105: 8097: 8093: 8089: 8085: 8081: 8077: 8070: 8062: 8055: 8047: 8041: 8037: 8030: 8022: 8018: 8014: 8010: 8006: 8002: 7998: 7994: 7987: 7979: 7975: 7971: 7967: 7963: 7959: 7954: 7949: 7945: 7941: 7934: 7927: 7918: 7913: 7909: 7905: 7901: 7897: 7893: 7886: 7877: 7872: 7868: 7864: 7860: 7856: 7852: 7845: 7830: 7826: 7819: 7811: 7807: 7803: 7799: 7795: 7791: 7786: 7781: 7777: 7773: 7766: 7759: 7751: 7747: 7743: 7739: 7735: 7731: 7726: 7721: 7717: 7713: 7706: 7698: 7694: 7690: 7686: 7682: 7678: 7671: 7662: 7657: 7653: 7649: 7645: 7641: 7637: 7633: 7626: 7619: 7611: 7598: 7579: 7572: 7565: 7546: 7539: 7532: 7524: 7520: 7515: 7510: 7506: 7502: 7498: 7494: 7490: 7486: 7479: 7472: 7464: 7458: 7442: 7438: 7431: 7415: 7411: 7407: 7401: 7393: 7389: 7385: 7381: 7377: 7373: 7366: 7358: 7354: 7348: 7329: 7325: 7321: 7317: 7313: 7309: 7305: 7301: 7297: 7290: 7283: 7264: 7260: 7256: 7252: 7248: 7244: 7240: 7236: 7232: 7228: 7224: 7217: 7210: 7206: 7195: 7192: 7189: 7186: 7184: 7181: 7179: 7176: 7174: 7171: 7170: 7164: 7162: 7157: 7153: 7133: 7129: 7118: 7108: 7102: 7098: 7095: 7090: 7086: 7080: 7077: 7072: 7069: 7062: 7061: 7060: 7057: 7044: 7040: 7031: 7013: 7009: 6996: 6984: 6978: 6974: 6971: 6964: 6960: 6954: 6950: 6944: 6940: 6927: 6915: 6909: 6905: 6902: 6895: 6891: 6885: 6881: 6873: 6870: 6865: 6861: 6857: 6854: 6851: 6846: 6842: 6836: 6832: 6806: 6802: 6782: 6774: 6773: 6772: 6731: 6727: 6723: 6718: 6667: 6654: 6650: 6646: 6611: 6600: 6596: 6593: 6588: 6581: 6570: 6566: 6563: 6560: 6549: 6530: 6527: 6524: 6520: 6515: 6504: 6498: 6494: 6491: 6486: 6482: 6478: 6459: 6455: 6452: 6448: 6443: 6432: 6426: 6422: 6419: 6415: 6411: 6408: 6405: 6400: 6396: 6388: 6377: 6373: 6370: 6352: 6349: 6346: 6342: 6329: 6317: 6311: 6307: 6304: 6299: 6295: 6291: 6273: 6270: 6267: 6263: 6250: 6238: 6232: 6228: 6225: 6220: 6216: 6209: 6204: 6201: 6195: 6183: 6182: 6181: 6157: 6146: 6142: 6139: 6134: 6127: 6116: 6112: 6109: 6106: 6095: 6089: 6086: 6083: 6080: 6076: 6071: 6060: 6054: 6050: 6047: 6042: 6038: 6030: 6019: 6015: 6012: 6007: 6004: 6001: 5998: 5994: 5981: 5969: 5963: 5959: 5956: 5951: 5947: 5940: 5935: 5932: 5925: 5924: 5923: 5908: 5895: 5883: 5877: 5873: 5870: 5865: 5861: 5857: 5854: 5826: 5822: 5819: 5814: 5810: 5806: 5803: 5795: 5794: 5789: 5788: 5768: 5765: 5754: 5750: 5747: 5742: 5738: 5734: 5731: 5723: 5720: 5717: 5712: 5708: 5704: 5698: 5695: 5692: 5687: 5683: 5679: 5676: 5673: 5670: 5664: 5659: 5655: 5652: 5647: 5643: 5639: 5636: 5633: 5630: 5627: 5621: 5618: 5611: 5610: 5609: 5584: 5580: 5577: 5572: 5568: 5564: 5561: 5558: 5555: 5547: 5544: 5541: 5538: 5527: 5523: 5520: 5515: 5511: 5507: 5504: 5496: 5493: 5490: 5485: 5481: 5477: 5474: 5471: 5468: 5465: 5462: 5459: 5453: 5450: 5447: 5442: 5438: 5434: 5431: 5428: 5425: 5414: 5410: 5407: 5402: 5398: 5394: 5391: 5388: 5385: 5382: 5372: 5371: 5370: 5356: 5353: 5350: 5347: 5327: 5324: 5321: 5316: 5312: 5308: 5305: 5302: 5299: 5296: 5293: 5290: 5285: 5281: 5277: 5274: 5271: 5268: 5265: 5252: 5245: 5243: 5229: 5226: 5215: 5211: 5208: 5203: 5199: 5195: 5192: 5185: 5184: 5181: 5174: 5167: 5165: 5143: 5139: 5136: 5131: 5127: 5123: 5120: 5117: 5114: 5107: 5106: 5103: 5100: 5098: 5082: 5076: 5073: 5043: 5032: 5028: 5025: 5020: 5013: 5002: 4998: 4995: 4992: 4981: 4975: 4972: 4969: 4966: 4963: 4960: 4957: 4952: 4948: 4944: 4941: 4938: 4935: 4932: 4929: 4922: 4911: 4907: 4904: 4899: 4896: 4893: 4890: 4887: 4881: 4876: 4873: 4866: 4865: 4864: 4847: 4843: 4836: 4831: 4827: 4820: 4815: 4811: 4804: 4799: 4795: 4771: 4768: 4765: 4762: 4759: 4755: 4749: 4746: 4742: 4736: 4732: 4728: 4723: 4719: 4713: 4709: 4704: 4693: 4690: 4687: 4683: 4679: 4676: 4673: 4670: 4667: 4663: 4657: 4654: 4650: 4644: 4640: 4636: 4631: 4627: 4621: 4617: 4612: 4601: 4598: 4595: 4591: 4587: 4584: 4581: 4578: 4573: 4569: 4565: 4562: 4557: 4553: 4549: 4544: 4540: 4536: 4525: 4517: 4516: 4515: 4501: 4493: 4489: 4473: 4470: 4459: 4453: 4440: 4439: 4433: 4419: 4416: 4413: 4410: 4407: 4404: 4401: 4396: 4392: 4388: 4385: 4374: 4368: 4364: 4360: 4352: 4351: 4332: 4328: 4325: 4322: 4319: 4313: 4302: 4296: 4264: 4259: 4255: 4252: 4249: 4246: 4243: 4240: 4234: 4231: 4229: 4218: 4215: 4212: 4209: 4206: 4203: 4200: 4189: 4185: 4175: 4161: 4157: 4153: 4148: 4146: 4136: 4132: 4115: 4109: 4094: 4090: 4086: 4073: 4072: 4071: 4048: 4044: 4041: 4038: 4035: 4029: 4024: 4020: 4017: 4014: 4011: 4008: 4005: 3999: 3997: 3986: 3983: 3980: 3977: 3974: 3971: 3968: 3960: 3956: 3953: 3950: 3944: 3942: 3933: 3926: 3923: 3920: 3917: 3914: 3908: 3904: 3897: 3883: 3879: 3876: 3873: 3867: 3865: 3856: 3849: 3846: 3843: 3840: 3837: 3834: 3831: 3822: 3810: 3806: 3799: 3785: 3782: 3777: 3775: 3769: 3762: 3746: 3736: 3732: 3725: 3711: 3708: 3695: 3694: 3693: 3679: 3676: 3673: 3670: 3667: 3664: 3661: 3658: 3647: 3638: 3629: 3622: 3620: 3599: 3593: 3588: 3584: 3581: 3576: 3569: 3558: 3554: 3551: 3548: 3537: 3529: 3525: 3522: 3519: 3514: 3510: 3506: 3500: 3493: 3482: 3478: 3475: 3470: 3464: 3459: 3451: 3447: 3438: 3433: 3418: 3414: 3410: 3405: 3401: 3394: 3386: 3377: 3373: 3366: 3352: 3349: 3340: 3339: 3336: 3335: 3334: 3329: 3328: 3304: 3299: 3295: 3292: 3289: 3284: 3280: 3276: 3270: 3268: 3258: 3254: 3251: 3248: 3243: 3239: 3232: 3227: 3223: 3220: 3217: 3212: 3208: 3201: 3199: 3188: 3185: 3182: 3177: 3173: 3163: 3149: 3146: 3141: 3135: 3132: 3129: 3124: 3120: 3116: 3107: 3093: 3090: 3085: 3083: 3058: 3044: 3043: 3042: 3040: 3039: 3029: 3022: 3020: 3006: 2998: 2994: 2985: 2980: 2965: 2961: 2957: 2952: 2948: 2941: 2933: 2924: 2920: 2913: 2899: 2896: 2891: 2888: 2883: 2871: 2870: 2867: 2850: 2841: 2835: 2832: 2827: 2815: 2806: 2799: 2797: 2783: 2765: 2757: 2746: 2743: 2738: 2718: 2710: 2701: 2698: 2688: 2687: 2684: 2670: 2663: 2659: 2650: 2643: 2641: 2627: 2621: 2611: 2607: 2595: 2592: 2587: 2579: 2575: 2571: 2562: 2548: 2545: 2540: 2532: 2522: 2521: 2518: 2490: 2486: 2482: 2463: 2459: 2455: 2443: 2439: 2434: 2412: 2408: 2405: 2377: 2373: 2370: 2350: 2327: 2301: 2284: 2276: 2273: 2269: 2263: 2255: 2246: 2238: 2237: 2236: 2234: 2230: 2229: 2223: 2207: 2187: 2157: 2141: 2138: 2130: 2099: 2095: 2092: 2064: 2060: 2057: 2048: 2046: 2037: 2030: 2028: 2014: 2006: 2000: 1997: 1994: 1989: 1977: 1976: 1973: 1927: 1926: 1921: 1905: 1885: 1879: 1866: 1863: 1855: 1844: 1835: 1828: 1826: 1812: 1804: 1798: 1795: 1787: 1777: 1776: 1773: 1759: 1756: 1748: 1738: 1714: 1701: 1697: 1693: 1659: 1634: 1629: 1607: 1603: 1600: 1572: 1568: 1565: 1542: 1523: 1520: 1517: 1514: 1496: 1493: 1490: 1482: 1478: 1474: 1447: 1446: 1445: 1429: 1425: 1416: 1415:magnetisation 1392: 1388: 1385: 1382: 1371: 1287: 1284: 1281: 1271: 1266: 1258: 1256: 1252: 1247: 1245: 1239: 1231: 1227: 1224: 1220: 1219: 1214: 1210: 1203: 1196: 1177: 1171: 1168: 1161: 1150: 1140: 1134: 1130: 1127: 1122: 1118: 1114: 1111: 1104: 1103: 1102: 1100: 1091: 1079: 1075: 1071: 1067: 1063: 1058: 1055: 1051: 1050: 1049: 1041: 1039: 1033: 1031: 1027: 1023: 1001: 997: 990: 987: 980: 975: 969: 966: 961: 958: 954: 947: 944: 941: 934: 930: 927: 924: 918: 915: 908: 903: 897: 894: 889: 886: 882: 875: 872: 869: 862: 858: 855: 851: 845: 841: 837: 834: 827: 826: 825: 822: 819: 818:ferromagnetic 815: 806: 798: 789: 785: 777: 769: 760: 758: 754: 735: 729: 726: 720: 717: 714: 709: 705: 697: 696: 695: 693: 689: 683: 681: 677: 673: 664: 656: 652: 650: 646: 641: 640:ferromagnetic 637: 629: 625: 621: 619: 616: 611: 609: 605: 601: 597: 593: 589: 581: 577: 574: 573: 572: 550: 547: 543: 539: 536: 528: 512: 505: 488: 484: 476: 459: 455: 451: 448: 443: 439: 415: 412: 409: 403: 396: 395: 394: 377: 372: 369: 366: 362: 356: 353: 350: 346: 340: 336: 332: 326: 323: 320: 314: 307: 306: 305: 298: 294: 277: 271: 268: 262: 256: 253: 245: 229: 225: 221: 201: 197: 193: 185: 180: 178: 174: 170: 166: 162: 158: 154: 145: 139:Magnetization 125: 114: 103:Linear arrays 100: 98: 94: 93:Klaus Halbach 89: 87: 86:magnetic tape 83: 78: 76: 70: 66: 64: 59: 51: 50:Halbach array 42: 34: 30: 19: 8291:cite journal 8258: 8254: 8241: 8233:CERN Courier 8232: 8223: 8209:cite journal 8184: 8180: 8173: 8154: 8143:the original 8130: 8118:. Retrieved 8104: 8079: 8075: 8069: 8060: 8054: 8035: 8029: 7996: 7992: 7986: 7943: 7939: 7926: 7899: 7895: 7885: 7858: 7854: 7844: 7832:. Retrieved 7828: 7818: 7775: 7771: 7758: 7715: 7711: 7705: 7680: 7676: 7670: 7635: 7631: 7618: 7597:cite journal 7585:. Retrieved 7578:the original 7564: 7552:. Retrieved 7545:the original 7531: 7488: 7484: 7471: 7457: 7445:. Retrieved 7441:the original 7430: 7418:. Retrieved 7414:the original 7409: 7400: 7375: 7371: 7365: 7356: 7347: 7335:. Retrieved 7328:the original 7299: 7295: 7282: 7270:. Retrieved 7263:the original 7226: 7222: 7209: 7160: 7155: 7148: 7058: 7054: 7037: 7030:dimensions. 7028: 6633: 6179: 5791: 5785: 5783: 5607: 5257: 5246: 5179: 5168: 5101: 5065: 4786: 4436: 4434: 4348: 4283: 4069: 3634: 3623: 3331: 3325: 3323: 3036: 3034: 3023: 2811: 2800: 2683:is given by 2662:scalar field 2655: 2644: 2517:is given by 2442:vector field 2435: 2316: 2226: 2221: 2049: 2042: 2031: 1923: 1843:Ampère's law 1840: 1829: 1630: 1557: 1267: 1264: 1248: 1243: 1240: 1236: 1217: 1212: 1208: 1201: 1194: 1192: 1098: 1096: 1065: 1061: 1053: 1047: 1044:Applications 1037: 1034: 1029: 1018: 1016: 823: 813: 811: 786: 782: 752: 750: 691: 687: 684: 676:synchrotrons 669: 633: 618:Maglev train 612: 585: 578:There is no 570: 567:Applications 392: 303: 242:is called a 183: 181: 176: 172: 168: 164: 156: 150: 90: 88:technology. 79: 71: 67: 49: 47: 29: 7999:: 369–376. 7946:: 128–132. 7861:(1): 1–12. 7420:11 February 7229:(1): 1–10. 4353:shows that 3635:Look for a 1413:. Then the 1038:end effects 596:voice coils 580:stray field 8345:Categories 7953:1502.06700 7661:2262/63909 7587:12 January 7554:1 December 7272:12 January 7201:References 7188:Inductrack 5066:where now 4787:where the 2438:divergence 1928:governing 1737:Gauss' law 1211:is in the 1078:cryogenics 757:wavenumber 674:, used in 615:Inductrack 527:wavenumber 91:Physicist 8082:: 16–25. 8021:124300587 7785:1409.1712 7750:119168717 7725:1410.0496 7447:31 August 7324:0029-554X 7251:0029-554X 7099:⁡ 6975:⁡ 6961:μ 6906:⁡ 6892:μ 6874:θ 6871:⁡ 6855:θ 6852:⁡ 6728:μ 6651:μ 6541:^ 6538:θ 6531:θ 6528:⁡ 6495:⁡ 6479:− 6473:^ 6453:− 6423:⁡ 6412:θ 6409:⁡ 6363:^ 6360:θ 6353:θ 6350:⁡ 6308:⁡ 6292:− 6286:^ 6274:θ 6271:⁡ 6229:⁡ 6202:φ 6199:∇ 6090:θ 6087:⁡ 6051:⁡ 6008:θ 6005:⁡ 5960:⁡ 5933:φ 5874:⁡ 5855:α 5823:⁡ 5804:β 5751:⁡ 5735:− 5732:β 5728:⟹ 5724:θ 5721:⁡ 5699:θ 5696:⁡ 5680:− 5677:θ 5674:⁡ 5656:⁡ 5640:− 5637:θ 5634:⁡ 5628:β 5622:− 5581:⁡ 5565:− 5562:β 5556:α 5552:⟹ 5542:α 5539:− 5524:⁡ 5508:− 5505:β 5501:⟹ 5497:θ 5494:⁡ 5478:− 5472:θ 5469:⁡ 5463:α 5460:− 5454:θ 5451:⁡ 5435:− 5432:θ 5429:⁡ 5411:⁡ 5395:− 5392:θ 5389:⁡ 5383:β 5357:θ 5354:⁡ 5348:α 5328:θ 5325:⁡ 5309:− 5306:θ 5303:⁡ 5294:⁡ 5278:− 5275:θ 5272:⁡ 5266:β 5212:⁡ 5196:− 5193:β 5140:⁡ 5124:− 5121:β 5115:α 5083:β 5074:α 4976:θ 4973:⁡ 4964:⁡ 4945:− 4942:θ 4939:⁡ 4930:β 4900:θ 4897:⁡ 4888:α 4874:φ 4769:θ 4763:⁡ 4747:− 4699:∞ 4684:∑ 4677:θ 4671:⁡ 4655:− 4607:∞ 4592:∑ 4582:⁡ 4563:θ 4526:φ 4502:θ 4460:φ 4450:∇ 4441:, namely 4420:θ 4417:⁡ 4408:⁡ 4389:− 4375:φ 4361:− 4329:θ 4326:⁡ 4303:φ 4293:∇ 4256:θ 4253:⁡ 4244:⁡ 4235:− 4219:θ 4216:⁡ 4207:⁡ 4186:θ 4182:∂ 4172:∂ 4133:θ 4129:∂ 4116:φ 4106:∂ 4070:and also 4045:θ 4042:⁡ 4021:θ 4018:⁡ 4009:⁡ 3972:⁡ 3957:θ 3954:⁡ 3918:⁡ 3895:∂ 3891:∂ 3880:θ 3877:⁡ 3850:θ 3847:⁡ 3838:⁡ 3820:∂ 3816:∂ 3797:∂ 3793:∂ 3760:∂ 3747:φ 3743:∂ 3723:∂ 3719:∂ 3680:θ 3677:⁡ 3668:⁡ 3648:φ 3526:θ 3523:⁡ 3501:− 3448:θ 3444:∂ 3439:φ 3430:∂ 3392:∂ 3387:φ 3384:∂ 3364:∂ 3360:∂ 3296:θ 3293:⁡ 3255:θ 3252:⁡ 3224:θ 3221:⁡ 3189:θ 3186:⁡ 3164:θ 3161:∂ 3157:∂ 3136:θ 3133:⁡ 3105:∂ 3101:∂ 3059:⋅ 3056:∇ 2995:θ 2991:∂ 2977:∂ 2939:∂ 2931:∂ 2911:∂ 2907:∂ 2880:∇ 2848:∇ 2842:⋅ 2839:∇ 2824:∇ 2814:Laplacian 2778:^ 2775:θ 2766:θ 2763:∂ 2755:∂ 2733:^ 2716:∂ 2708:∂ 2696:∇ 2622:θ 2619:∂ 2612:θ 2604:∂ 2560:∂ 2556:∂ 2533:⋅ 2530:∇ 2502:^ 2499:θ 2491:θ 2477:^ 2296:^ 2285:⋅ 2277:± 2261:∂ 2256:φ 2253:∂ 2208:φ 2188:φ 2131:× 2128:∇ 2007:⋅ 2004:∇ 2001:− 1995:φ 1986:∇ 1906:φ 1886:φ 1883:∇ 1871:⟹ 1856:× 1853:∇ 1805:⋅ 1802:∇ 1799:− 1788:⋅ 1785:∇ 1749:⋅ 1746:∇ 1698:μ 1534:^ 1531:θ 1524:θ 1521:⁡ 1509:^ 1497:θ 1494:⁡ 1348:^ 1345:θ 1319:^ 1288:θ 1223:remanence 1172:^ 1131:⁡ 1026:remanence 991:^ 988:φ 967:π 962:− 959:φ 945:− 931:⁡ 919:^ 916:ρ 895:π 890:− 887:φ 873:− 859:⁡ 721:⁡ 715:≈ 548:λ 540:π 367:− 272:⁡ 257:⁡ 222:π 194:π 153:solenoids 8283:23698444 8162:Archived 8120:28 March 8096:58309210 7978:54826296 7810:56325133 7523:23335834 7259:53486791 7167:See also 2866:becomes 2658:gradient 2235:to find 2222:constant 1898:, where 1735:. Using 792:Cylinder 645:extruded 592:DC motor 177:opposite 75:Magnepan 8263:Bibcode 8189:Bibcode 8001:Bibcode 7958:Bibcode 7917:3547684 7876:5937527 7834:31 July 7790:Bibcode 7730:Bibcode 7685:Bibcode 7640:Bibcode 7514:3547684 7493:Bibcode 7380:Bibcode 7337:13 June 7304:Bibcode 7231:Bibcode 1918:is the 755:is the 600:wiggler 525:is the 63:magnets 54:German: 8281: 8094: 8063:: 807. 8042: 8019: 7976: 7914: 7873: 7808: 7748: 7521: 7511: 7322: 7257: 7249: 7161:et al. 7156:et al. 7051:Sphere 5080: 4840: 4824: 4808: 4284:Hence 1255:Bruker 1017:where 751:where 393:where 8279:S2CID 8251:(PDF) 8146:(PDF) 8139:(PDF) 8092:S2CID 8017:S2CID 7974:S2CID 7948:arXiv 7936:(PDF) 7806:S2CID 7780:arXiv 7768:(PDF) 7746:S2CID 7720:arXiv 7628:(PDF) 7581:(PDF) 7574:(PDF) 7548:(PDF) 7541:(PDF) 7481:(PDF) 7331:(PDF) 7292:(PDF) 7266:(PDF) 7255:S2CID 7219:(PDF) 2660:of a 2440:of a 1064:and − 8297:link 8215:link 8122:2022 8040:ISBN 7836:2024 7610:help 7589:2016 7556:2017 7519:PMID 7449:2008 7422:2019 7339:2017 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2460:F 2456:= 2452:F 2418:o 2413:r 2409:= 2406:r 2383:i 2378:r 2374:= 2371:r 2351:f 2331:] 2328:f 2325:[ 2302:, 2292:r 2281:M 2274:= 2270:] 2264:r 2247:[ 2228:1 2167:H 2142:0 2139:= 2135:H 2105:o 2100:r 2096:= 2093:r 2070:i 2065:r 2061:= 2058:r 2036:) 2034:2 2032:( 2015:, 2011:M 1998:= 1990:2 1959:M 1937:H 1925:1 1880:= 1876:H 1867:0 1864:= 1860:H 1834:) 1832:1 1830:( 1813:. 1809:M 1796:= 1792:H 1760:0 1757:= 1753:B 1723:) 1719:M 1715:+ 1711:H 1707:( 1702:0 1694:= 1690:B 1668:B 1643:H 1613:o 1608:r 1601:r 1578:i 1573:r 1566:r 1543:, 1540:) 1515:+ 1505:r 1488:( 1483:0 1479:M 1475:= 1469:l 1466:y 1463:c 1457:M 1430:0 1426:M 1398:o 1393:r 1386:r 1377:i 1372:r 1315:r 1291:) 1285:, 1282:r 1279:( 1244:k 1218:e 1213:y 1209:H 1205:o 1202:R 1198:i 1195:R 1178:, 1169:y 1162:) 1155:i 1151:R 1145:o 1141:R 1135:( 1123:r 1119:M 1115:= 1112:H 1099:k 1066:k 1062:k 1054:k 1030:k 1021:r 1019:M 1002:, 998:] 981:) 976:) 970:2 955:( 951:) 948:1 942:k 939:( 935:( 925:+ 909:) 904:) 898:2 883:( 879:) 876:1 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