1.1.5.1 Magnetostatic energy

Now we will provide the expression for the contribution of magnetostatic interactions to the free energy of the system. The derivation of such expression is quite straightforward if one assumes that the energy density [14] of magnetostatic field is given by:

$$U_ =\int_{\Omega_\infty} \frac{1}{2} \mu_0{\mathbf{H}_\text{m}}^2 dV \quad,$$ (1.47)

where $\Omega_\infty$ is the whole space. In fact, by expressing the magnetostatic field as

$${\mathbf{H}_\text{m}}=\frac{\textbf{B}_\text{m}}{\mu_0}-\textbf{M}\quad,$$ (1.48)

Eq. (1.47) becomes:

$$U_ =\int_{\Omega_\infty} \frac{1}{2} \mu_0{\mathbf{H}_\text{m}}\cdot \left(\frac{\textbf{B}_\text{m}}{\mu_0}-\textbf{M}\right) dV \quad.$$ (1.49)

The first term in Eq. (1.49) vanishes owing to the integral orthogonality of the solenoidal field $\textbf{B}_$m and the conservative field ${\mathbf{H}_\text{m}}$ over the whole space [14]. The remaining part, remembering that $\textbf{M}$ is nonzero only within the region $\Omega$, is the magnetostatic free energy:

$$F_ =-\int_{\Omega} \frac{1}{2} \mu_0\textbf{M}\cdot{\mathbf{H}_\text{m}} dV \quad.$$ (1.50)

We observe that magnetostatic energy expresses a nonlocal interaction, since the magnetostatic field functionally depends, through the boundary value problem (1.45), on the whole magnetization vector field, as we anticipated in the beginning of the section. The latter equation has the physical meaning of an interaction energy of an assigned continuous magnetic moments distribution, namely it can be obtained by computing the work, made against the magnetic field generated by the continuous distribution, to bring an elementary magnetic moment $\mu_0\textbf{M} dV$ from infinity to its actual position within the distribution [15]. Discussion on the choice of the magnetostatic field energy density can be found in Ref. [14] and references therein.