Magnets — Question 8

Question 8

Equal axial and equatorial readings occur at unequal radii

A small magnet produces far fields $B_a=\mu_0m/(2\pi r_a^3)$ on its axis and $B_e=\mu_0m/(4\pi r_e^3)$ on its equator. Two probes read equal magnitudes Bstar.

1. Find ra/re and infer m from Bstar and re.

2. A third probe lies on the opposite axial side at radius ra. Compare all three field directions for a moment along +z.

Question 8 - Solution

Equal axial and equatorial readings occur at unequal radii

(a) Method and calculation

$$\frac{2}{r_a^3}=\frac{1}{r_e^3}\Rightarrow\boxed{r_a/r_e=2^{1/3}}.$$$$\boxed{m=4\pi r_e^3B_*/\mu_0}.$$

(b) Interpretation and further calculation

$$\mathbf B_{+z}=B_*\hat z,\qquad\mathbf B_{-z}=B_*\hat z.$$$$\boxed{\mathbf B_{\mathrm{equator}}=-B_*\hat z}.$$

Physical check

On the negative axial side the field points toward the dipole, but its vector still points along +z.