How to measure the "distance" from magnetic dipoles?

Question

We can talk about scalar potentials $\Phi$ for static electric fields:

$${\bf v} = \nabla {\Phi}$$

where $\nabla$ is the differential operator:

$$\nabla = \left[\frac{\partial}{\partial x}, \frac{\partial}{\partial y},\frac{\partial}{\partial z}\right]^T$$

Similarly we can talk about vector potentials $\bf \Psi$:

$${\bf v} = \nabla \times \bf \Psi$$

Now to my question, can we use these somehow to measure the "distance" to a magnetic dipole (and if we are closer to the north or the south pole)?

Answer 1

Note that you started to talk about electric fields, than switched to magnetic ones. The distance is to the center of the dipole. In the multipole expansion you get fields from monopoles, dipoles, quadrupoles, etc., contribution. What it means that you have a distribution of charges (electric or "magnetic") in a region around origin, then you approximate this by a sum of fields originating at the origin for infinitesimally small extent charge distributions for which you know how to calculate the electric field. At large distances from the center, if one of the moments is finite, the higher orders can usually be neglected. At small distances, you need to add not just the dipole at the origin, but the quadrupole as well, maybe even higher order moments. What does large and small mean in this context? It means much larger than the size of the charge distribution.