Ferromagnetism — Question 5

Question 5

A saturated core keeps the vacuum differential response

In a narrow reversible region deep in positive saturation, a core obeys $M(H)=M_s-a/H^2$ with a positive and H sufficiently large. Take $B=\mu_0(H+M)$.

1. Find differential permeability and its large-H limit.

2. For a long tightly wound solenoid of length ell, area A and N turns, derive the small-signal inductance about a bias field Hb. Compare it with the empty-coil inductance.

Question 5 - Solution

A saturated core keeps the vacuum differential response

(a) Method and calculation

$$\frac{dM}{dH}=\frac{2a}{H^3}\Rightarrow\boxed{\mu_{\mathrm{diff}}=\mu_0(1+2a/H^3)}.$$$$H\to\infty\Rightarrow\boxed{\mu_{\mathrm{diff}}\to\mu_0}.$$

(b) Interpretation and further calculation

$$H=NI/\ell,\quad\Lambda=NAB\Rightarrow L_{\mathrm{small}}=\left.\frac{d\Lambda}{dI}\right|_{H_b}=\frac{N^2A}{\ell}\mu_{\mathrm{diff}}(H_b).$$$$\boxed{L_{\mathrm{small}}/L_0=1+2a/H_b^3\to1}.$$

Physical check

Saturation removes most incremental magnetization response, but it does not make B or incremental inductance vanish.