Closed-form expressions for the magnetic field of permanent magnets in three dimensions

Abstract

We derive a closed-form expression of the magnetic field of a finite-size current sheet and use it to calculate the field of permanent magnets, which are modeled through their surface current densities. We illustrate the method by determining the multipoles and the effective length due to fringe fields of a finite-length dipole constructed of magnetic cubes.

Strap-on magnets: a framework for rapid prototyping of magnets and beam lines

“…Based on the analytic results for the magnetic fields from permanent magnets from [3] we explored the use of permanent-magnet cubes to create dipoles, quadrupoles, and solenoids. All examples are based on the frames for 10 mm cubes, shown in Figure 1, but are easily scalable to use cubes of a different size, to include more cubes, or produce other multipoles.…”

“…We then Fouriertransform the component of the field that is tangential to this circle and show it on the upper panel on the bottom-left in Figure 2. This algorithm, which is explained in [3], gives us the magnitude and angle of the multipole coefficients of the assembly shown on the middle panel. We see that the integrated field strength is 5.72 Tmm and the lower panel indicates that the relative magnitude of other multipoles is around 6 × 10 −3 with the decapole contribution (m = 5) being the largest.…”

“…In the frame for an eight-magnet quadrupole, shown on the right-hand image in Figure 1, subsequent magnets rotate by 3 × 360 o /8 as illustrated by the little notch in each of the eight holes for the cubes. In order to calculate the magnetic fields from the cubes in three dimensions we use the closed-form expressions from [3], which allow us to prepare MATLAB scripts to determine the relevant field quantities and the multipole contents of the magnets in parameterized form. This proves very convenient to adapt the design to different sized cubes, different geometries, and other multipolarities.…”

Strap-On Magnets: A Framework for Rapid Prototyping of Magnets and Beam Lines

“…In [3], we calculated the field generated by a rectangular current sheet, which proved useful to describe permanent-magnet cubes that can be modeled by four rectangular sheets. In the top-left image in Figure 2, we see that each of the eight magnets consists of four blue sheets with the green line indicating the direction of the current, such that we can visualize the cube as a square solenoid that generates a magnetic field in its inside that is represented by the yellow arrow, which coincides with the direction of the easy axis of the magnet.…”

“…In order to calculate the magnetic fields from the cubes in three dimensions, we used the closed-form expressions from [3], which allowed us to prepare MATLAB [4] scripts to determine the relevant field quantities and the multipole contents of the magnets in parameterized form. This proves very convenient to adapt the design to different sized cubes, different geometries, and other multipolarities.…”