 Force on a small linearly magnetizable body

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.