calculator:art_magnetic_force_on_layer_semi-numerical
Magnetic force on a thin layer due to a cuboid magnet
 | Stan Zurek, Magnetic force on a thin layer due to a cuboid magnet, Encyclopedia Magnetica, https://www.e-magnetica.pl/doku.php/calculator/art_magnetic_force_on_layer_semi-numerical, {accessed: 2025-03-14} |
The main equation is an analytical formula for calculating magnetic field strength at any point in space of a rectangular solenoid, which in reality approximates a permanent magnet.
The volume of a thin weakly-magnetic active layer is subdivided into M × M subregions for which the forces are calculated separately and then summed up. It is necessary to specify fixed value of magnetic susceptibility for the thin layer.
Therefore, this calculator operates as a hybrid between the analytical calculations from the main equation, but repeated for each sub-region, thus making it similar to numerical calculations.
| Magnetic field of a rectangular magnet (or solenoid) | ||
|---|---|---|
| Source: [1] S. Hampton, R. A. Lane, R. M. Hedlof, R. E. Phillips, C. A. Ordonez; Closed-form expressions for the magnetic fields of rectangular and circular finite-length solenoids and current loops. AIP Advances 1 June 2020; 10 (6): 065320, https://doi.org/10.1063/5.0010982 | ||
| [1], eq. (1) | $$ H_x = \frac{B_r}{8·π·μ_0}·\sum_{i=0}^{1} \sum_{j=0}^{1} \sum_{k=0}^{1} \left\{ (-1)^{i+j+k} · ln \left( \frac { r_{ijk} - [y+a_y · (-1)^{j+1}] } { r_{ijk} + [y+a_y · (-1)^{j+1}] } \right) \right\} $$ | (A/m) |
| [1], eq. (2) | $$ H_y = \frac{B_r}{8·π·μ_0}·\sum_{i=0}^{1} \sum_{j=0}^{1} \sum_{k=0}^{1} \left\{ (-1)^{i+j+k} · ln \left( \frac { r_{ijk} - [x+a_x · (-1)^{i+1}] } { r_{ijk} + [x+a_x · (-1)^{i+1}] } \right) \right\} $$ | |
| [1], eq. (3) | $$ H_z = \frac{B_r}{4·π·μ_0}·\sum_{i=0}^{1} \sum_{j=0}^{1} \sum_{k=0}^{1} \left\{ (-1)^{i+j+k+1} · \left[ tan^{-1} \left( \frac { [x+a_x · (-1)^{i+1}] · [z+a_z · (-1)^{k+1}] } { r_{ijk} · [y+a_y · (-1)^{j+1}] } \right) + \\ + tan^{-1} \left( \frac { [y+a_y · (-1)^{j+1}] · [z+a_z · (-1)^{k+1}] } { r_{ijk} · [x+a_x · (-1)^{i+1}] } \right) \right] \right\} $$ | |
| where: $B_r$ - remanence (T) of the permanent magnet, $μ_0$ - magnetic permeability of vacuum (H/m), $L$ - length of the solenoid along its axis (m), and $r_{ijk}$ in (m) is calculated as: $ r_{ijk} = \sqrt{ [x+a_x · (-1)^{i+1}]^2 + [y+a_y · (-1)^{j+1}]^2 + [z+a_z · (-1)^{k+1}]^2 } $ | ||
Illustration for the interactive calculator: Active layer volume is subdivided into M subregions for which the magnetic forces are calculated separately and then summed up over the active region
Geometry of rectangular solenoid as defined for the equations above (in the original paper [1])
by S. Hampton, R. A. Lane, R. M. Hedlof, R. E. Phillips, C. A. Ordonez, published in AIP Advances, CC-BY-4.0
by S. Hampton, R. A. Lane, R. M. Hedlof, R. E. Phillips, C. A. Ordonez, published in AIP Advances, CC-BY-4.0
| Notes: The number of subdivisions is set to M = 100. The calculator is based on equations for a rectangular solenoid, so for the magnet its magnetic permeability is assumed to be that of vacuum, and therefore the output values can differ significantly from real magnets. However, reasonable order-of-magnitude forces can be estimated with this approach, if the layer is kept “thin” so that t < L, and such that its magnetic susceptibility can be assumed constant. |
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calculator/art_magnetic_force_on_layer_semi-numerical.txt · Last modified: 2025/01/19 18:58 by stan_zurek