The Wikipedia page for magnetization states that the energy density associated with sphere $j$ due to the magnetic field $\mathbf{B}_i$ is given by $-\mathbf{M}_j\cdot\mathbf{B}_i$. Where does this come from? Why does the magnetization $\mathbf{M}$ belong to $j$ while the $\mathbf{B}$-field belongs to $i$?
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$\begingroup$ You are right to ask, and they assume similar thing already in (7). Mathematically, you can take these energy formulae as postulates, and the paper shows these postulates lead to the usual formulae for forces between magnets, both for point dipoles and magnetized spheres, and these formulae turn out to be the same (not surprising, since the energy formulae are very similar). The paper is lacking in motivating these exercises. The interesting thing here is to show the dipole model and the Ampere current model of spheres predict the same energy formulae and forces, but they did not do that. $\endgroup$– Ján LalinskýCommented Sep 16 at 13:15
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$\begingroup$ The energy formulae for permanent magnets like $-\mathbf m_1\cdot \mathbf B_2$ are actually derived from other knowledge that is closer to experimental basis and basic laws. One can derive it for the dipole model of magnet, from the Coulomb law (you can find this in EM textbooks for energy of electric dipole in external field, but it works the same for magnetic dipole in external magnetic field). One can also derive it for the Ampere model with fixed current, from macroscopic magnetic force acting on the magnetization current $\mathbf F = \int \mathbf j_M\times\mathbf B_{ext}~dV$. $\endgroup$– Ján LalinskýCommented Sep 16 at 13:26
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The energy density due to a single PERMANENTLY magnetized sphere of magnetization per unit volume $\bf{M}$ at a location in an external magnetic field where, absent the sphere, the value of the flux density is $\bf{B}$ is equal to $\bf{M} \cdot \bf{B}$. This derivation can be found in Jackson's "Classical Electrodynamics," Griifiths' "Introduction to Electrodynamics," and probably other places. I prefer the expression for total energy $\bf{m} \cdot \bf{B}$ where $\bf m$ is the volume integral of $\bf M$, but it's basically the same.
The energy due to a soft magnetic sphere (constant $\mu$) with INDUCED magnetization $\bf m$ due to an external field $\bf B$ is $\frac{1}{2} (\bf{m} \cdot \bf{B})$. This is also mentioned in Jackson (at least it is in my older edition), although you have to dig a little for it. Note that some texts only use $\bf m$ when referring to permanent magnetization. But the definition of $\bf m$ as the volume integral of $\bf M$ applies in both cases.
Technically the definition of PE also includes a negative sign which I have disregarded.
As noted in textbooks, neither of the above expressions for energy include the energy associated with maintaining the external field, so can't necessarily be called "total" energy. But more often than not, they are the quantities of interest.
