calculator:art_layer_force_magnet_cuboid
Calculator of force on thin layer from cuboid magnet
 | Stan Zurek, Calculator of force on thin layer from cuboid magnet, Encyclopedia Magnetica, http://www.e-magnetica.pl/doku.php/calculator/art_layer_force_magnet_cuboid, {accessed: 2025-02-19} |
| Magnetic force F acting on the thin magnetic layer from a cuboid magnet | |||
|---|---|---|---|
| Source: [1] L.H. Bennett, C.H. Page, L.J. Swartzendruber, Comments on Units in Magnetism, Journal of Research of the National Bureau of Standards Vol. 83, No.1 , Jan-Feb 1978 | |||
| expressed by permeability | expressed by susceptibility | ||
| [1], eq. (4), p. 10 | $$ F = (μ_r - 1) · H_{avg} ⋅ t ⋅ A ⋅ µ_0 ⋅ \frac{dH}{dx} $$ | $$ F = χ · H_{avg} ⋅ t ⋅ A ⋅ µ_0 ⋅ \frac{dH}{dx} $$ | (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) |
| Havg - average 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/m2) |
| 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), μ0 - magnetic permeability of vacuum (H/m), μr - relative magnetic permeability of the layer (unitless), χ = μr - 1 - magnetic susceptibility of the layer (unitless, based on SI units), t - thickness of the sample (m), A - active area of the layer (m2) equivalent to the cross-section of the magnet | ||
Notes: This calculator gives only an approximation of an order of magnitude of expected force, and it is applicable only for thin layers positioned close to the magnet surface. The calculation significantly underestimates forces for short/flat magnets (L < a, b). The magnetic permeability or susceptibility needs to be provided for the value of magnetic field acting at the centre of the sample. More details are available on the summary Art page https://e-magnetica.pl/art and in (Szozda, 2023).1)
↑
| → → → Helpful page? Support us! → → → | PayPal | ← ← ← Help us with just $0.10 per month? Come on…  ← ← ← |
calculator/art_layer_force_magnet_cuboid.txt · Last modified: 2025/01/18 22:05 by stan_zurek