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

Halbach Arrays –Background –Physics –Current and Future Uses

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.

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.

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

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

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

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.

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).

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.

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

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).

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

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