= Magnetization of a permeable sphere
{title2=$\mathbf M=\chi\mathbf H_0/(1+\chi/3)$}
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 $-H_ir\cos\theta$ inside and $-H_0r\cos\theta+C\cos\theta/r^2$ 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.
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