Analytic Expressions for Shielded Halbach Multipoles

Abstract

We employ the method of images to derive analytic expressions for the magnetic field of Halbach multipoles that are enclosed in high-permeability shielding.

Figures (7)

• Figure 1: The geometry with the domain $\Omega$ containing permanent-magnet material with remanent field $\underline{B}_r$ and the field $\underline{B}$ that it creates at position $\hat{z}$.
• Figure 2: Dipole images of a plane a cylindrical surface.
• Figure 3: The image dipole at $z_4$ is constructed by first rotating the original dipole $\underline{B}_r$ from $z$ to the imaginary axis at $z_2$, followed by scaling its magnitude with $(R/r)^2$, complex conjugating it, and moving it to $z_3$. Finally the dipole is rotated back and arrives at $z_4$. See the text for further explanations.
• Figure 4: The original dipoles are visualized by the arrows on the inner dotted black circle. The rotate twice as one goes along the circle, consistent with a tumbling factor of $k = 2$. The shielding is shown as the red circle whereas the image dipoles, calculated from Equation \ref{['eq:imdip']} are located further outside and are also given by black arrows. The blue arrows on the shielding is calculated by adding the field from all dipoles, weighted by the Green's function $G(\hat{z},z)$, on the shielding.
• Figure 5: Two segments of a segmented multipole that extends radially from $r_i$ to $r_o$. Each segment has an azimuthal width, given by $2\alpha$. Note the orientation of the easy axis, determined by the tumbling factor $k$, changes from segment to segment.