Magnetic monopole

Remarks

In this convention $q_m$ is measured in ampere metres; another common convention, in webers, puts $\mu_0$ in other places. With $\rho_m = 0$ and $J_m = 0$ the system is Maxwell's equations, and the second equation is Gauss's law for magnetism.

Example: the field above has $\nabla \cdot B = 0$ away from $r_0$, and its flux through any sphere around $r_0$ is $\frac{\mu_0 q_m}{4\pi} \cdot \frac{1}{R^2} \cdot 4 \pi R^2 = \mu_0 q_m$, as $\nabla \cdot B = \mu_0 q_m \delta(r - r_0)$ requires.

Non-example: a magnetic dipole, such as a small current loop, has zero net flux through every closed surface. So its magnetic charge is $0$ and it is not a monopole.

On $\R^3 \setminus \set{r_0}$ the monopole field has no vector potential that is smooth on the whole set: by Stokes' theorem the flux through a sphere around $r_0$ would be $0$. One can instead use two potentials, each smooth outside a half-line that starts at $r_0$, which differ by a gauge transformation where both are defined.