Magnetic Materials — Question 3

Question 3

Ellipsoidal demagnetization limits the measured response

A linear ellipsoid has intrinsic susceptibility chi and demagnetizing factor N along the applied field. Its uniform internal field satisfies $H=H_a-NM$ and $M=\chi H$.

1. Derive the apparent susceptibility M/Ha.

2. Invert the relation to obtain intrinsic chi from measured apparent susceptibility q. State the positive-chi consistency bound when N is positive.

Question 3 - Solution

Ellipsoidal demagnetization limits the measured response

(a) Method and calculation

$$M=\chi(H_a-NM)\Rightarrow M(1+N\chi)=\chi H_a.$$$$\boxed{q\equiv M/H_a=\frac{\chi}{1+N\chi}}.$$

(b) Interpretation and further calculation

$$q(1+N\chi)=\chi\Rightarrow\boxed{\chi=\frac{q}{1-Nq}}.$$$$\chi>0,\ N>0\Rightarrow\boxed{0<q<1/N}.$$

Physical check

As q approaches 1/N, a small measurement error can produce a large error in the inferred intrinsic susceptibility.