Editing Halbach array (section)

==Sphere==

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 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.

The equation for the field in a Halbach sphere is<ref>[http://www.magnetocaloric.com/Code/pub01.htm Permanent magnet based sources of magnetic field] {{Webarchive|url=https://web.archive.org/web/20120424225940/http://www.magnetocaloric.com/Code/pub01.htm |date=24 April 2012 }}.</ref>

: <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&nbsp;[[Tesla (unit)|T]] in a working volume of 20&nbsp;mm<sup>3</sup> was achieved,<ref name="Bloch1998"/> and this was increased further to 5&nbsp;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&nbsp;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&nbsp;T Halbach dipole cylinder in 2003.<ref name="Kumada2004"/>