A three-dimensional space with particle P at the center of a Cartesian coordinate system. P includes a proton-core located at the origin and a photon somewhere in the vicinity. |

Consider a three-dimensional space where a proton-core provides a frame of reference for describing positions noted by $\overline{r}$. Let P be a centered particle that includes the proton-core and some collection of dynamic quarks. We vaguely represent these dynamic quarks using photons but do not assign a postion to a photon unless it is absorbed by a material particle that has a well-defined position. Instead we define some characteristics of electromagnetic fields that *do* depend on postion, and use them to describe how a photon $\gamma$ would interact with some hypothetical test-particle located at position $\overline{r}$. Let P be characterized by a magnetic moment noted as $\, \overline{\mu}$. Outside of P we define the **magnetic vector potential** as

$\begin{align} \overline{A} \equiv \frac{ \mu_{\sf{o}} }{ 4 \pi } \frac{ \overline{\mu} \times \overline{r} }{ r^{3} } \end{align}$

where $\mu_{\sf{o}}$ is a constant and $\times$ notes a cross-product. In the International System units for $\overline{A}$ are the same as momentum per unit charge, and abbreviated as (V·s·m^{−1}). Definition: the

**magnetic flux density**is

$\begin{align} \overline{B} \equiv \nabla \times \overline{A} \end{align}$

where $\nabla \times$ notes a mathematical vector operator called the curl. The units used to measure flux density are called tesla and abbreviated by (T). The magnetic potentials and flux densities around multiple magnetic moments are obtained from vector sums over the contributions of individual particles.$\begin{align} \overline{B} = \frac{ \mu_{\sf{o}} }{ 4 \pi } \left( \frac{ 3 \overline{r} \left( \overline{\mu} \cdot \overline{r} \right) }{ r^{5} } - \frac{ \overline{\mu} }{ r^{3} } \right) \end{align}$