# What is a minimal set of quantities fully describing the source of a magnetic field?

Assume I would like to compare different magnetic fields without knowing what generates them. What is the minimal set of physical properties describing a field that would let me calculate all the other properties of this field?

I would definitely need to be able to find vector potential at every point ($\mathbf A(x,y,z)$).

Currently I am thinking that magnetic moment alone is enough, is this the case?

However, for example this wikipedia article comparing Earths magnetic field to a dipole talks about three values of vector $\mathbf B$ (radial, asimuthal, and the magnitude).

So what is the minimal set of quantities that is enough to derive the rest?

• Not sure if this is related, but there is a standard approach in the antenna community where they define a surface that encloses all sources and then they place hypothetical magnetic charges on this surface such that they would produce the magnetic field. This is refered to as the surface equivalence theorem if I remember correctly. – flippiefanus Dec 24 '17 at 13:51

As a simple example, consider the magnetic field produced by a dipolar surface current distribution confined to the surface of a sphere of radius $a$, given by $\mathbf K(\theta,\phi) = K_0 \cos(\theta) \hat{\boldsymbol \phi}$: this will produce a purely dipolar magnetic field, $$\mathbf B(\mathbf r) = \frac{\mu_0}{4\pi}\frac{m}{r^3}\left[2\cos(\theta)\hat{\boldsymbol r}+\sin(\theta)\hat{\boldsymbol \theta}\right],$$ whose amplitude only depends on the magnetic dipole moment $m\propto K_0 a^4$. That looks innocent, but it means that if you shrink the sphere to some smaller radius $b<a$, and you increase the current density by an equal amount, then there will be absolutely no trace of the change in the magnetic field for at positions outside the original sphere.
This is a generic feature, in that you cannot tell what the sources of a magnetic field are if you know its behaviour on a limited bit of space, and you therefore can't infer what's happening with the field in the regions of space you don't have explicit information about. (In the examples above, the magnetic field is uniform inside the sphere - but a sphere of what size? you can't tell with data from $r>a$.)
Thus, to get that kind of information, you need $\mathbf B(\mathbf r)$ at all positions $\mathbf r$, and that information then encodes the current sources themselves through Ampère's law, $$\mathbf{J} = \frac{1}{\mu_0}\mathbf{\nabla} \times \mathbf{B}(\mathbf r).$$