What Produces a Magnetic Field? — Question 6

Question 6

A small imbalance of aligned microscopic moments can create a macroscopic magnetic source

A simplified material contains $N=10^{20}$ intrinsic magnetic moments, each of magnitude $\mu_a=9.27\times10^{-24}$ ampere square metre. A fraction f=.600 points along +z and the rest along -z. Treat the net moment as their vector sum.

1. Find the net moment and the fraction of the fully aligned value.

2. Find the change if f increases to .650 while N remains fixed; explain why no macroscopic transport current is required.

Question 6 - Solution

A small imbalance of aligned microscopic moments can create a macroscopic magnetic source

(a) Method and calculation

$$\vec\mu_{\mathrm{net}}=[fN-(1-f)N]\mu_a\hat z=N\mu_a(2f-1)\hat z.$$$$\mu_{\mathrm{net}}=1.854\times10^{-4}\,\mathrm{A\,m^2},\quad\mu_{\mathrm{net}}/(N\mu_a)=.200.$$

(b) Interpretation and further calculation

$$\Delta\vec\mu=2N\mu_a\Delta f\hat z=9.27\times10^{-5}\hat z\,\mathrm{A\,m^2}.$$$$I_{\mathrm{transport}}=0\quad\text{is compatible with}\quad\vec\mu_{\mathrm{net}}\ne0.$$Intrinsic moments do not require electrons to be literal rotating charged spheres.

Physical check

Permanent magnetism can arise from aligned microscopic moments even when the specimen is electrically neutral and carries no transport current.