calculator:art_layer_force_magnet_cuboid

#### Calculator of force on thin layer from cuboid magnet

This section is an interactive calculator. |
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* This page is being edited and may be incomplete or incorrect.

Magnetic force F acting on the thin magnetic layer |
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expressed by permeability | expressed by susceptibility | |

$$ F = (μ_r - 1) · H_{avg} ⋅ t ⋅ A ⋅ µ_0 ⋅ \frac{dH}{dx} $$ | $$ F = χ · H_{avg} ⋅ t ⋅ A ⋅ µ_0 ⋅ \frac{dH}{dx} $$ | (N) |

where: H(x) - magnetic field strength H generated along the axis x of the cuboid magnet (see also: Calculator of H along axis of rectangular magnet) | $ H(x) = \frac{B_r}{π·µ_0}· \left( \text{atan} \frac{2·(x+L)·\sqrt{a^2+b^2+4·(x+L)^2}}{a·b} - \text{atan} \frac{2·x·\sqrt{a^2+b^2+4·x^2}}{a·b} \right) $ | (A/m) |

H - average _{avg}H in the sample (average of H on both surfaces of the sample, equivalent to H at the centre of the sample) | $$ H_{avg} = \frac{H(x_0)+H(x_0+t)}{2} \approx H(x_0 + \frac{t}{2}) $$ | (A/m) |

dH/dx - gradient of H across the thickness of the sample (similar for half of the thickness) | $$ \frac{dH}{dx} = \frac{H(x_0)-H(x_0+t)}{t} \approx \frac{H(x_0)-H(x_0+\frac{t}{2})}{\frac{t}{2}} $$ | (A/m)/(m) ≡ (A/m ^{2}) |

and: $B_r$ - remanence (flux density) of the permanent magnet (T), $L$ - axial length (m) of the cylindrical magnet, $a$ and $b$ - side lengths (m) of the of the magnet, $x$ - location (m) from the surface of the magnet (magnet surface is synonymous with x = 0), μ_{0} - magnetic permeability of vacuum (H/m), μ - relative magnetic permeability of the layer (unitless), _{r}χ = μ - 1 - magnetic susceptibility of the layer (unitless, based on SI units), _{r}t - thickness of the sample (m), A - active area of the layer (m^{2}) equivalent to the cross-section of the magnet |

Magnetic field along the axis of a cuboid (rectangular) magnet

^{S. Zurek, E-Magnetica.pl, CC-BY-4.0}

*Notes: This equation is gives only an approximation of expected forces, and it is applicable only for thin layers positioned close to the magnet surface. The magnetic permeability or susceptibility needs to be provided for the value of magnetic field acting at the centre of the sample. More details and some experimental data are available in (Szozda, 2023). ^{1)} *

calculator/art_layer_force_magnet_cuboid.txt · Last modified: 2023/10/03 22:18 by stan_zurek