What Produces a Magnetic Field? — Question 7

Question 7

Random microscopic moments have a nonzero root-mean-square sum even when their mean vanishes

N independent moments have isotropic random orientations and fixed magnitude mu. Assume $\langle\vec\mu_i\rangle=0$ and $\langle\vec\mu_i\cdot\vec\mu_j\rangle=0$ for distinct i and j. Let $\vec M=\sum_i\vec\mu_i$.

1. Find the mean vector and root-mean-square magnitude of M.

2. Compare the rms magnitude with the fully aligned moment for $N=10^{20}$. State why a zero average does not mean each microscopic source disappears.

Question 7 - Solution

Random microscopic moments have a nonzero root-mean-square sum even when their mean vanishes

(a) Method and calculation

$$\langle\vec M\rangle=\sum_i\langle\vec\mu_i\rangle=0.$$$$\langle M^2\rangle=\sum_i\langle\mu_i^2\rangle+2\sum_{i<j}\langle\vec\mu_i\cdot\vec\mu_j\rangle=N\mu^2.$$

(b) Interpretation and further calculation

$$M_{\mathrm{rms}}=\sqrt N\mu,\quad M_{\mathrm{aligned}}=N\mu.$$$$M_{\mathrm{rms}}/M_{\mathrm{aligned}}=N^{-1/2}=10^{-10}.$$Individual moments remain present; their directions largely cancel.

Physical check

The square-root scaling relies on independent, unbiased orientations; correlated domains need a different model.