# Maxwell Stress Tensor -> Force between magnets and perfect iron

1. Apr 9, 2014

### 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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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:

$Bm = Br / (1+Br*g(h_m*u0*Hc))$

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

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

Now integrate over volume:

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

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

Thank you.

2. Apr 9, 2014

### Meir Achuz

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

3. Apr 9, 2014

### SunnyBoyNY

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

$B_{iron} = \mu_{r} \cdot \mu_{0} \cdot H_{iron}$

$B_{mag} = B_{R} \cdot (1-H_{mag}/H_{c})$

4. Apr 10, 2014

### 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.

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