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A small linearly magnetizable sample has a magnetic dipole moment induced by its applied field. In a fixed principal direction, write $m={\alpha }_B{B}_0$; this convention gives ${\alpha }_B$ units $\mathrm{A}{\mathrm{m}}^2{\mathrm{T}}^{-1}$. A uniformly magnetized ellipsoid of volume $V$ has ${\alpha }_B=V\chi /[{\mu }_0(1+N_d\chi )]$. The effective interaction potential energy at fixed source current is $-{\alpha }_B{B}_0^2/2$. The factor one half arises because the moment must be induced as the field is raised; differentiating $-m(B_0)B_0$ as though it described a permanent moment would count the response twice. The response model excludes remanence, nonlinear saturation and hysteresis.

The printed formula has a physical problem for the specified geometry. Let $\chi =\mu -1$ be the linear magnetic susceptibility, where $\mu$ is the source's relative magnetic permeability. The constitutive equation is $M=\chi H_{\mathrm{in}}$, not in general $\chi H_0/\mu$.

For a small uniformly magnetized ellipsoid with its principal axis along the applied field, include its demagnetizing field through the SI demagnetizing factor $N_d$:Solving givesA slender rod aligned along the field has $N_d\to 0$ as its aspect ratio grows. Its ideal slender-rod limit at fixed $\mu$ is thereforeprovided $N_d|\mu -1|\ll 1$. A finite cylinder generally has nonuniform magnetization, so an ellipsoidal shape factor is an approximation, and its aspect ratio is needed for a more precise correction. Small sample length compared with the field-variation scale makes $H_0$ approximately uniform across it; that condition does not set the demagnetizing factor to one.

The source's displayed $M=(\mu -1)H_0/\mu$ instead follows from $N_d=1$, the thin-plate limit with field normal to the plate. It can also be obtained by imposing $B_{\mathrm{in}}=B_0$, but that identification is not justified for a thin rod parallel to the field. Consequently the requested extra $1/\mu$ cannot honestly be derived from the stated rod geometry. The next part gives both the shape-consistent result and the force conditional on using the printed magnetisation.