Magnetic Materials — Question 4

Question 4

A highly permeable sphere still has a finite apparent susceptibility

A linear sphere has N=1/3 and intrinsic susceptibility chi. Apply a uniform field Ha. Assume the linear model remains valid as chi is made large.

1. Find internal H and B in terms of Ha and chi.

2. Find the large-chi limits of M/Ha and B/(mu0Ha), and explain why internal H can vanish while B remains finite.

Question 4 - Solution

A highly permeable sphere still has a finite apparent susceptibility

(a) Method and calculation

$$H=H_a-M/3,\quad M=\chi H\Rightarrow\boxed{H=\frac{H_a}{1+\chi/3}}.$$$$\boxed{B=\mu_0H_a\frac{1+\chi}{1+\chi/3}}.$$

(b) Interpretation and further calculation

$$\lim_{\chi\to\infty}\frac{M}{H_a}=3,\qquad\lim_{\chi\to\infty}\frac{B}{\mu_0H_a}=3.$$$$\boxed{H\to0,\quad M\to3H_a,\quad B\to\mu_0M=3\mu_0H_a}.$$

Physical check

This is a shape-limited linear-model result; a real ferromagnet may saturate before this limit is reached.