Magnetic Materials — Question 10

Question 10

Uniform magnetization becomes a surface current rather than a volume current

An infinitely long cylinder is uniformly magnetized $\mathbf M=M_0\hat z$ with no free currents. Use $\mathbf J_b=\nabla\times\mathbf M$ and $\mathbf K_b=\mathbf M\times\hat n$. Take fields to vanish far outside.

1. Find the bound volume and surface currents.

2. Use the infinite-solenoid equivalence to find B inside and outside, then determine H inside.

Question 10 - Solution

Uniform magnetization becomes a surface current rather than a volume current

(a) Method and calculation

$$\boxed{\mathbf J_b=0},\qquad\hat n=\hat r,\quad\boxed{\mathbf K_b=M_0\hat\phi}.$$$$K_b\text{ has units A/m and circulates around the cylinder.}$$

(b) Interpretation and further calculation

$$\boxed{\mathbf B_{\mathrm{in}}=\mu_0M_0\hat z,\quad\mathbf B_{\mathrm{out}}=0}.$$$$\boxed{\mathbf H_{\mathrm{in}}=\mathbf B_{\mathrm{in}}/\mu_0-\mathbf M=0}.$$

Physical check

Zero free current and zero H do not imply zero B in a permanently magnetized material.