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Engineering Magnetic Field Question (saturating an iron plate)

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Homework Statement
How strong a magnetic field is necessary to magnetize an iron plate to its saturation magnetization for a field applied normal to the plate surface? Assume that iron has a relative permeability 200 and a saturation magnetization of 1700 emu/cm^3. If the field is applied in the plane of the plate, how large a field is required?
Homework Equations
B = H + 4*pi*M
H (total) = Hd + H (applied)
I know that the field inside sample is a combination of the demagnetizing field and whatever applied field you may have. So these two fields together influence how big a field you need in order to magnetize the sample.
 

Charles Link

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With a plate geometry, the ## B ## field inside the material is approximately the applied field (## B=H_o ##) because the lines of flux for ## B ## are continuous. Even though ## H ## is effectively zero inside the material of the plate, (because of the demagnetizing field), I think in calculations like this one, you need to assume that the effective ## H ## in the material is the applied ## B ##, even though it really is a bit of hand waving to make such an assumption. It is a very interesting problem, but I think it also brings to light some discrepancies that arise when working with ferromagnetic materials, and the assumption that these materials behave linearly. Anyway, I think what I have proposed above is about the best you can do with such a problem. @vanhees71 Might you have an input here?
 
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Charles Link

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A follow-on: I put some more effort into this one, and I believe I got a result that makes sense.
For very large ## \chi ##, ## H_{total} \approx 0 ##, but more precisely ## H_{total}=H_o+H_D ##, where ## H_D=-4 \pi M ##. Since ## M=\chi H_{total}=\chi (H_o-4 \pi M) ##, we get ## M=\frac{\chi H_o}{1+4 \pi \chi} ##. ## \\ ## For large ## \chi ##, ## M \approx \frac{H_o}{4 \pi} ##.
Since ## B=H_{total}+4 \pi M ##, we see for large ## \chi ## that ## H_{total} \approx 0 ## , and ## B \approx H_o ##.
One other item, is they give you ## \mu=200=1+4 \pi \chi ##, so from this you can compute ## \chi ##. ## \\ ## It appears this one does not suffer from the linearity difficulty that I originally thought. ## \\ ## For this latest result, ## H_{total}=\frac{H_o}{1+4 \pi \chi} ##. This result is much different from what I proposed in post 2, and it suggests that the iron in the plate will be very slow to develop a magnetization in response to the applied field ## H_o ##. Whether this actually happens in practice, I'm not completely sure.
 
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