Dipolar energy

  The dipolar energy can be represented continuously by:

$$\mathcal{E}_{\mathrm{di}} = -\mu_0 \int_V \ensuremath{\mathbf{H}}_{\mathrm{de}}(\ensuremath{\mathbf{r}})\cdot\ensuremath{\mathbf{M}}(\ensuremath{\mathbf{r}}) \mathrm{d}^3r$$ (2.29)

$\mathcal{E}_{\mathrm{di}}$The dipolar energy of a system

where $\ensuremath{\mathbf{H}}_{\mathrm{de}}(\ensuremath{\mathbf{r}})$ is the demagnetising field with components contributed from the divergence of magnetisation within the volume and surface poles (O'Handley, 1999):

$$\ensuremath{\mathbf{H}}_{\mathrm{de}}(\ensuremath{\mathbf{r}}) = ... ...\over \vert\ensuremath{\mathbf{r}} - \ensuremath{\mathbf{r}}' \vert^3} \right )$$ (2.30)

$\ensuremath{\mathbf{H}}_{\mathrm{de}}$The demagnetising field in a system $\ensuremath{\mathbf{n}}$In magnetostatics, the vector normal to the surface of a sample

and $\ensuremath{\mathbf{n}}$ is the surface normal.

A complete derivation of $\ensuremath{\mathbf{H}}_{\mathrm{de}}$ is given in Brown (1963), Aharoni (2000) and Blundell (2001).