Dipolar energy

Dipolar energy (often called magnetostatic or demagnetising energy) is the resultant energy from the interaction of magnetic moments with each other. Two magnetic moments at positions $\ensuremath{\mathbf{r}}_i$ and $\ensuremath{\mathbf{r}}_j$ have the dipolar energy:

$$\mathcal{E}_{\mathrm{di}}^{i,j} = \mu_0 \left( {\mbox{\boldmath {$\mu$}}_i\cdot\mbox{\boldmath {$\m... ...math{\mathbf{r}}_{ij}) \over \vert\ensuremath{\mathbf{r}}_{ij}\vert^5 } \right)$$ (2.12)

$\mathcal{E}_{\mathrm{di}}^{i,j}$The dipolar energy between two magnetic moments $\mu$$_i$ and $\mu$$_j$

where

$$\ensuremath{\mathbf{r}}_{ij} = \ensuremath{\mathbf{r}}_j - \ensuremath{\mathbf{r}}_i$$ (2.13)

$\ensuremath{\mathbf{r}}_{ij}$The distance between two magnetic moments $\mu$$_i$ and $\mu$$_j$ at positions $\ensuremath{\mathbf{r}}_i$ and $\ensuremath{\mathbf{r}}_j$

For $N$ magnetic moments this becomes:

$$\mathcal{E}_{\mathrm{di}} = {1 \over 2} \sum_{i=1}^{N}\sum_{j\neq i} \mathcal{E}_{\mathrm{di}}^{i,j}$$ (2.14)

Computing the dipolar energy is the most expensive part of any micromagnetic simulation as the dipolar energy is a long-range interaction and therefore must consider the interaction of each magnetic moment $\mu$$_i$ with every other magnetic moment $\mu$$_j$.