= Uniformly magnetized ellipsoid
{title2=$M=\chi H_0/(1+N_d\chi)$}
For an ellipsoidal sample in a uniform applied <magnetic field intensity> $H_0$ along a principal axis, its uniform <demagnetizing field> is $-N_dM$. The SI <demagnetizing factor> obeys $0\le N_d\le1$. Linear <magnetic susceptibility> $\chi=\mu_r-1$ gives $M=\chi(H_0-N_dM)$, hence $M=\chi H_0/(1+N_d\chi)$. An axially slender prolate limit has $N_d\to0$, a sphere has $N_d=1/3$, and a thin plate with field normal to it has $N_d\to1$. Thus $M=(\mu_r-1)H_0/\mu_r$ belongs to the plate limit, not the slender axial-rod limit. A finite cylinder generally has nonuniform magnetization; an effective ellipsoidal factor is an approximation rather than an exact universal correction.
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