Full analytical solution for the magnetic field of uniformly magnetized cylinder tiles

Exact and Computationally Robust Solutions for Cylindrical Magnets Systems with Programmable Magnetization

“…The advantage of our solution can be further appreciated by comparing Equation with the considerable number of corresponding expressions reported through multiple tables in ref. [52] where the use of a computer algebra system (to automatically integrate the relevant governing equations) implied to cope with singularities (potentially introduced through the involved integration constants) by considering multiple cases. (iv) Our complete solutions, for both field and gradient, can be computed by calling a single function, namely $$\mathcal {C}$$.…”

“…[ 48 ], which is limited to axial magnetization, and refs. [ 49 , 50 , 51 , 52 ], which did not determine the gradient solution. Indeed, no analytical solutions were previously achieved for the gradient when considering diametric magnetization.…”

“…[ 49 , 50 , 51 ] Furthermore, an analytical solution has been recently reported for uniformly magnetized cylindrical tiles, which, however, requires multiple representations to circumvent computational singularities, and which is nonetheless limited to magnetic field only. [ 52 ] Corresponding gaps in the expanding field of magnetic actuation remain. For instance, exact analytical solutions for force and torque between cylindrical magnets, which are of utmost utility for the design of related systems, are lacking: only the force between coaxial magnets with axial magnetization was disclosed (either by assuming the same radius for both magnets [ 53 ] or by allowing for different radii [ 54 , 55 , 56 ] ), whereas analytical expressions for force and torque between coaxial magnets with diametric magnetization have not been achieved so far.…”

Stray magnetic fields from elliptical-shaped and stadium-shaped ferromagnets

“…Computation of magnetic field from a ferromagnetic body, or more generally solving a Possion equation, has been an attractive problem in both mathematics and physics, [1][2][3][4][5][6][7][8][9][10][11][12] and still prompts publications recently. [13][14][15][16][17][18] Even after great development of numerical solvers, [19][20][21][22][23][24] deriving a solution of potential, or field, in terms of analytical functions or in forms of some integrals with simplified approximations, such as macrospin or rigid-vortex assumption, is highly demanded, particularly when many-body problems are of interest. [25][26][27][28][29] This is because it enables us to estimate various parameters, such as coercive and stray fields, with adequate calculation cost.…”

“…The past works have mainly focused on the internal magnetic field and derived the solutions of the demagnetization coefficients for various shapes of ferromagnets such as ellipsoid, cylinder, and cuboid, [1][2][3][4][5][6][8][9][10] while the stray magnetic field, originated from magnetostatic interaction, for vortex, cylinder, and so on has also been investigated. 7,[11][12][13][14][15][16][17][18] An interesting target nowadays related to this is to compare stray magnetic fields from two kinds of uniformly magnetized ferromagnets, namely elliptical-shaped and stadium-shaped ferromagnets, 30,31) which are schematically shown in Figs. 1(a) and 1(b), respectively.…”