Magnetic Field Question (saturating an iron plate)

[lcvoth23]

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?

Relevant 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]

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?

 

[Charles Link]

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.