Calculator of magnetic force between cylindrical magnet and thick ferromagnetic plate, with air gap
 | Stan Zurek, Calculator of magnetic force between cylindrical magnet and thick ferromagnetic plate, with air gap, Encyclopedia Magnetica, https://www.e-magnetica.pl/doku.php/calculator/art_force_cylinder_magnet_and_plate_with_gap, {accessed 2025-03-14} |
This calculator assumes that the magnetic plate has infinitely high magnetic permeability and it is thick enough so that magnetic saturation does not take place, which would be typically true for typical steel plates and small size permanent magnets.
The magnetic field strength of a cylindrical magnet along its axis can be calculated from the equation as given below.
It is further assumed that the size of the non-magnetic gap is negligible as compared to the length of the magnet, and that the magnetic plate acts like a “magnetic mirror”, which produces a magnetic field of a magnet which has twice the length of the original magnet. The equation is derived for x = 0 meaning the surface of the cylinder, so with magnetic mirroring it requires calculation at the centre of the apparent structure, such that x = -L.
However, further correction is required for the presence of the gap because the magnetic flux density in the gap is decreased due to flux fringing and flux fringing factor, for example as given by Hurley and Wölfle1)
| Magnetic field strength H' of a cylindrical permanent magnet along its axis, with “mirroring” in a large magnetic plate | |
|---|---|
| $$ H'(x) = \frac{B_r}{2·µ_0}· \left( \frac{x+2·L}{\sqrt{D^2 /4 +(x+2·L)^2}} - \frac{x}{\sqrt{D^2 /4 +x^2}} \right)$$ | (A/m) |
| $$ H'(centre) = \frac{B_r}{µ_0}· \left( \frac{L}{\sqrt{D^2 /4 + L^2}} \right)$$ | (A/m) | $$ B'(centre) = \frac{B_r·L}{\sqrt{D^2 /4 + L^2}} $$ | (T) |
| where: $H'$ - magnetic field strength (A/m) of the mirrored structure (so that the apparent magnet length is doubled), $B'$ - magnetic flux density (T) of the mirrored structure, $B_r$ - remanence (flux density) of the permanent magnet (T), $L$ - axial length (m) of the cylindrical magnet, $D$ - diameter (m) of the of the magnet, $x$ - location (m) from the surface of the magnet (surface is synonymous with x = 0), μ0 - magnetic permeability of vacuum (H/m) | |||
| Flux fringing factor k of a cylindrical air gap | |
|---|---|
| $$k = \frac{A_{eq}}{A} = \frac{(D + 2 · g )^2 }{D^2}$$ | (unitless) |
| where: $A_{eq}$ - equivalent cross-sectional area (m2) corrected for the flux fringing due to non-magnetic gap, $A$ - area of magnet pole (m2), $D$ - magnet diameter (m), $g$ - length of non-magnetic gap (m). | |
The magnetic flux density B'(centre) is then corrected with the flux fringing factor and the resulting force is calculated as in the equation below.
| Magnetic force F between a cylindrical magnet and a thick magnetic plate | |
|---|---|
| $$F = \left( \frac{B'(centre)}{k} \right)^2 · \frac{A}{2·μ_0} $$ | (N) |
| where all variables are as defined above. | |
S. Zurek, E-Magnetica.pl, CC-BY-4.0
Notes: This calculation is valid only for a uniformly magnetised cylindrical magnet, acting with the field along its axis, placed on a thick ferromagnetic plate.2) The non-magnetic spacing must be much smaller than magnet length (g ≪ L).
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