Maxwell Stress Tensor -> Force between magnets and perfect iron

[SunnyBoyNY]

(this is not a hw)

Assume you have a magnet of dimensions x_m, h_m, remanent flux density Br, and coercive field density Hc. The magnet is placed in a magnetic "C" structure (perfect iron) such that it is connected on one side but there is an airgap on the other side.

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xx... xx
xx...gg
xx...mm
xx...mm
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Stack length is 1 m for simplicity.

I know how to calculate gap flux density as a function of airgap length. I am struggling, however, with using the Maxwell Stress tensor to calculate the force between the magnet and the structure through the airgap.

This is what I tried:

[itex]
Bm = Br / (1+Br*g(h_m*u0*Hc))
[/itex]

[itex]
\nabla\cdot\sigma_{xyz}= \frac{1}{\mu_{0}} \begin{pmatrix}
\frac{\partial 0.5B_{x}^{2}}{\partial x} & \frac{\partial B_{x}B_{y}}{\partial x} & \frac{\partial B_{x}B_{z}}{\partial x}\\

\frac{\partial B_{y}B_{x}}{\partial y} & \frac{\partial 0.5B_{y}^{2}}{\partial y} & \frac{\partial B_{y}B_{z}}{\partial y}\\

\frac{\partial B_{z}B_{x}}{\partial z} & \frac{\partial B_{z}B_{y}}{\partial z}& \frac{\partial 0.5B_{z}^{2}}{\partial z} \\
\end{pmatrix}

[/itex]

care only about one direction, which simplifies the equation to 1/2u0 * dBm^2/dx.

Now integrate over volume:

[itex]
F = \int_{V} \mathbf{f} \mathrm{d} V = \int_{V} (\nabla\cdot\sigma)\mathrm{d} V = \oint_{S} (\mathbf{f}\cdot\mathbf{n})\mathrm{d} A
[/itex]

But the numbers consistently come out wrong - with respect to a FEA simulation. Do I use the MST incorrectly?

Thank you.

 

[Meir Achuz]

The derivation of the MST requires a linear material (constant mu). Therefor, the MST cannot be applied to ferromagnets.

 

[SunnyBoyNY]

The derivation of the MST requires a linear material (constant mu). Therefor, the MST cannot be applied to ferromagnets.

Thank you. I forgot to mention that the iron has ideal (linear) magnetic properties such that:

[itex]
B_{iron} = \mu_{r} \cdot \mu_{0} \cdot H_{iron}
[/itex]

Also, the permanent magnet has a linear loading curve:

[itex]
B_{mag} = B_{R} \cdot (1-H_{mag}/H_{c})
[/itex]

 

[SunnyBoyNY]

Found the problem: I used the divergence theorem incorrectly.

Volume integral of field divergence is equal to the closed surface integral of the field itself, not its divergence.

[itex]
F = \int_{V} \mathbf{f} \mathrm{d} V = \int_{V} (\nabla\cdot\sigma)\mathrm{d} V = \oint_{S} (\sigma\cdot\mathbf{n})\mathrm{d} A = \oint_{S} \sigma \cdot\mathrm{d} \vec{A}
[/itex]

 

[sardar]

I appreciate your efforts in trying to use the Maxwell Stress Tensor to calculate the force between the magnets and the perfect iron structure. However, there are a few issues with your approach that may be causing the incorrect results.

Firstly, the Maxwell Stress Tensor is typically used to calculate the force between two electromagnetic fields, not between a magnet and a perfect iron structure. So, it may not be the most appropriate method for this scenario.

Secondly, the equation you have used for the Maxwell Stress Tensor assumes a continuous and uniform magnetic field, which may not be the case in this setup. The airgap between the magnet and the structure can cause variations in the magnetic field, making the equation inaccurate.

Lastly, the integration over volume may also not be accurate as it assumes a constant magnetic field throughout the volume, which may not be the case in this setup.

I would suggest using a different approach, such as using magnetic circuit analysis or FEA simulation, to accurately calculate the force between the magnets and the structure. These methods take into account the non-uniformity of the magnetic field and can provide more accurate results. Additionally, performing experimental tests can also help validate the results obtained from these methods.

I hope this helps. Keep up the good work in trying to understand and apply scientific concepts. Best of luck in your research.