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James Creel Physics Department Seminar 4117 Supervisor: Dr. Truman Black

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Halbach Arrays –Background –Physics –Current and Future Uses

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Concerning the physics of magnetic levitation (maglev), particle accelerators, and laser applications, Halbach arrays have played an incidental role. Discovered by Mallinson in 1973 as a “magnetic curiosity” but years later independently discovered by Klaus Halbach, these magnetic arrays had the ability to created a one- sided flux of magnetization. Originally, these arrays were recognized to make significant improvements in magnetic tape technology, but later were seen by Halbach as a novel way to enhance the effects of particle accelerators through guiding technology. Throughout this discussion, Halbach arrays will be explored for their uniqueness and applications current and future.

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What is a Halbach array? –Created by Klaus Halbach in late 70’s but discovered before this by Mallinson in 1973 –Uses magnets in configurations yielding magnetic fields stronger than individual couterparts.

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Representation of Magnetic Field Lines in for different views of the Halbach array

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Alternative Designs Axially symmetric array with linear field on the inside.

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Method: In order to find the field associated with a Halbach array, we need to think of a planar structure, of thickness d, lying in the x, z plane. x z We can take the magnetization to be two sinusoids in quadrature: m x = m 0 sin(kx) and m y = m 0 cos(kx) and m z =0. -d -y

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Next, we proceed with solving the boundary value problems for the scalar potentials above and below the sheet. –Potentials within the sheet obey Poisson’s equation: ▼ 2 Φ inside = m 0 kcos(kx) –As well as Laplace’s equation above and below ▼ 2 Φ below = 0 and ▼ 2 Φ above = 0.

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Since we know the particular solutions to Poisson’s Equation is –(m 0 /k)cos(kx), we can get the general solutions: Φ above = {Ae -ky +Be +ky }cos(kx) and Φ inside = {Ce -ky +De +ky -m o /k}cos(kx) and Φ below = {Ee -ky +Fe +ky }cos(kx).

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Matching the boundary conditions for the potential on either side of the sheet: –The fields and potentials must go to zero as y becomes infinite: Φ above = Φ below = 0. –The tangential fields must match on the sheet upper and lower surfaces: Φ above = Φ inside when y = 0 and Φ below = Φ inside when y = -d.

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–Also, the normal flux density must be continuous on the upper and lower surfaces by: and

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Rewriting our original solutions to match these boundary conditions gives: Φ above = 0 and Φ inside = (m 0 /k)(e ky -1)cos(kx) and Φ below = (m 0 /k)(1-e kd )e ky cos(kx).

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NOW YOU NOW HOW THEY WORK, LET’S SEE HOW THEY’RE USED

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Scientific American ArticleSoft Stop® Magnetic Brakes Study On Development Of An Atomic Force Microscope (Afm) Using A Magnetically Levitated Stage For Long Range Measurement

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Scientific American Article

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Inductrack Passive Magnetic Levitation

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