Gauss' Law for Magnetic Fields — Question 7

Question 7

A radial inverse-square field passes a local test but fails a global magnetic test

For r greater than zero, a proposed field is $\mathbf B=K\hat r/r^2$. Assume it is intended to describe an ordinary magnetic field in all space, with no magnetic monopoles.

1. Compute its divergence away from the origin and its flux through a sphere.

2. Determine what K must be if every closed surface must have zero magnetic flux.

Question 7 - Solution

A radial inverse-square field passes a local test but fails a global magnetic test

(a) Method and calculation

$$\nabla\cdot\mathbf B=\frac1{r^2}\frac d{dr}(r^2B_r)=0\quad(r>0).$$$$\boxed{\Phi_{\mathrm{sphere}}=4\pi r^2(K/r^2)=4\pi K}.$$

(b) Interpretation and further calculation

$$\oint\mathbf B\cdot d\mathbf A=0\Rightarrow4\pi K=0.$$$$\boxed{K=0\quad\text{for the stated all-space model}}.$$

Physical check

Excluding a singular point from a differential calculation can hide a nonzero global flux.