Magnetization of a permeable sphere ($\mathbf{M}=\chi {\mathbf{H}}_0/(1+\chi /3)$)

For a homogeneous linear isotropic sphere in a uniform applied field, the internal field is ${\mathbf{H}}_i={\mathbf{H}}_0-\mathbf{M}/3$. Combining this geometric demagnetizing field with $\mathbf{M}=\chi {\mathbf{H}}_i$ gives the title expression. Matching the scalar magnetic potentials $-H_{i}r\cos \theta$ inside and $-H_{0}r\cos \theta +C\cos \theta /r^2$ outside provides an independent boundary-value derivation. Tangential $H$ is continuous without free surface current and normal $B$ is continuous. A negative susceptibility makes magnetization antiparallel, not parallel in sense, to the applied field.