Diamagnetism — Question 8

Question 8

A perfectly diamagnetic sphere needs a demagnetizing-field correction

An ideal sphere excludes magnetic induction from its interior: B=0. In a uniform applied field Ha its internal field obeys $H=H_a-M/3$. Use $B=\mu_0(H+M)$. Ignore penetration depth.

1. Find internal H, magnetization M, and apparent susceptibility M/Ha.

2. For sphere radius a, find its total dipole moment and explain why apparent susceptibility differs from intrinsic M/H.

Question 8 - Solution

A perfectly diamagnetic sphere needs a demagnetizing-field correction

(a) Method and calculation

$$B=0\Rightarrow M=-H,\quad H=H_a+H/3.$$$$\boxed{H=\tfrac32H_a,\quad M=-\tfrac32H_a}.$$

(b) Interpretation and further calculation

$$\boxed{\chi_{\mathrm{app}}=M/H_a=-3/2,\quad\chi_{\mathrm{int}}=M/H=-1}.$$$$\boxed{m=\frac{4\pi a^3}{3}M=-2\pi a^3H_a}.$$

Physical check

This ideal complete-exclusion limit is distinct from the tiny negative susceptibility of ordinary weak diamagnets.