Induced magnetic dipole ($\mathbf{m}={\alpha }_B{\mathbf{B}}_0,U_{\mathrm{eff}}=-{\alpha }_B{B}_0^2/2$)

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.

For constant scalar response $\mathbf{m}={\alpha }_B{\mathbf{B}}_0$ and a body small compared with the applied field's spatial variation scale, the gradient of the induced magnetic dipole energy gives $\mathbf{F}={\alpha }_B\nabla B_0^2/2$. Along an axis this is $F_z=md{B}_0/dz$. For an ellipsoidal body, put ${\chi }_{\mathrm{eff}}=\chi /(1+N_d\chi )$ to obtain $F_z={\mu }_{0}V{\chi }_{\mathrm{eff}}{H}_0{H}_0^{\prime }$. Positive effective magnetic susceptibility attracts the body towards stronger field; a negative response repels it. Reversing the source current leaves the force unchanged because both the induced moment and the field gradient reverse. Shape and orientation enter through the demagnetizing factor, not merely through volume.