The magnetic scalar potential and demagnetization vector for a cylinder tile

TL;DR

This work derives closed-form analytical solutions for the magnetic scalar potential generated by uniformly magnetized cylindrical slices and full cylinders by solving Poisson's equation and expressing the potential as a dot product $\Psi_M(\mathbf{r})=\mathbb{N}_\Psi(\mathbf{r})\cdot\mathbf{M}$, where the demagnetization vector $\mathbb{N}_\Psi$ encodes all geometric information and is constructed from elliptic integrals. The authors provide explicit surface-term decompositions for the six faces of a cylinder slice, including careful treatment of angular, radial, and axial singularities, and show that the same approach yields the full-cylinder case with appropriate angular limits and complete elliptic integrals. Validation against finite element simulations demonstrates perfect agreement for both the slice and the full cylinder, confirming the correctness and numerical reliability of the analytic expressions. The resulting framework facilitates high-precision magnetic field calculations and can be advantageous for physics-informed machine learning and other applications requiring conservative scalar potentials.

Abstract

A closed-form solution for the magnetic scalar potential generated by a uniformly magnetized cylindrical slice and a full cylinder is determined by solving Poisson's equation analytically. The solution is given in terms of elliptic integrals of the first, second and third kind. We show that the magnetic scalar potential can be written as the dot product of a demagnetization vector, containing all the geometric information of the generating cylinder, and the magnetization. We validate the derived analytical expressions for the magnetic scalar potential by comparing with a finite element simulation and show that these agree perfectly for both the cylindrical slice and the full cylinder.

Figures (5)

• Figure 1: The cylinder slice and full cylinder considered. The origin is placed such that it is at the center of the cylinder, regardless of whether this is a slice or a full cylinder. The cylinder slice is defined by its extent in the radial, $R_i$ to $R_o$, and angular, $\phi_1$ to $\phi_2$, directions and by its length from $z_1=-h/2$ to $z_2=h/2$ where $h$ is the height of the slice. The full cylinder is defined by its radius, $R_o$ and length, $h$.
• Figure 2: The magnetic scalar potential, $\Psi_M$, and the magnetic field lines generated by a cylindrical slice with an inner radius of $R_i = 0.25$ m, an outer radius of $R_o=0.35$ m, a height of $h= 0.7$ m and an angular extension from $\phi_1 = \pi/7$ to $\phi_2 = 2\pi-\pi/3$ and magnetization of $M = [2,\, 3,\, 4]$ A/m in the plane $z=0$ m. The magnetization direction is indicated by the red arrow, and the sides of the cylindrical slice are shown with the red lines.
• Figure 3: The magnetic scalar potential, $\Psi_M$, generated by a cylindrical slice with dimensions as given in Fig. \ref{['fig:Slice_4_0.001_z_0.0']} along the line $[0,\, 0,\, 0]\rightarrow[1.5,\, 0.8,\, 0.75]$ m as given by Eq. \ref{['Eq.Final_slice']} and as computed using finite element modeling.
• Figure 4: The magnetic scalar potential, $\Psi_M$, and the magnetic field lines generated by a cylinder with radius $R_o = 0.25$ m and height $h= 0.7$ m and magnetization of $M = [-2,\, -3,\, 4]$ A/m in the plane $z=0.2$ m. The magnetization direction is indicated by the red arrow, and the side of the cylinder are shown with the red line.
• Figure 5: The magnetic scalar potential, $\Psi_M$, generated by a cylinder with radius $R_o = 0.25$ m and height $h= 0.7$ m and magnetization of $M = [-2,\, -3,\, 4]$ A/m along the line $[0,\, 0,\, 0]\rightarrow[-1.5,\, -0.8,\, 0.75]$ m as given by Eq. \ref{['Eq.final_full']} and as computed using finite element modeling.