Demagnetizing field (${\mathbf{H}}_d$)

The demagnetizing field is the contribution to magnetic field intensity produced by the sample's own magnetization. Inside a uniformly magnetized ellipsoid, it is uniform and has the form ${\mathbf{H}}_d=-\mathsf{N}\mathbf{M}$; the demagnetizing factor along a principal axis gives its longitudinal component. These conditions make the scalar correction in field conventions for magnetic susceptibility useful.

A cylinder generally has spatially varying magnetization and demagnetizing field. A volume-sensitive instrument and a short central pickup coil consequently need different effective corrections: one averages over the sample, while the other samples central flux. A shape factor copied from an ellipsoid is not an exact correction for every cylinder, especially with nonlinear response or hysteresis. ^[ref]

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.

For an ellipsoidal sample in a uniform applied magnetic field intensity $H_0$ along a principal axis, its uniform demagnetizing field is $-N_{d}M$. The SI demagnetizing factor obeys $0\le N_d\le 1$. Linear magnetic susceptibility $\chi ={\mu }_r-1$ gives $M=\chi (H_0-N_{d}M)$, hence $M=\chi H_0/(1+N_d\chi )$. An axially slender prolate limit has $N_d\to 0$, a sphere has $N_d=1/3$, and a thin plate with field normal to it has $N_d\to 1$. 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.

A demagnetizing factor relates a component of demagnetizing field to magnetization. The three principal factors of a uniformly magnetized ellipsoid sum to one in the International System of Units. Factors quoted in a different unit convention need conversion; state the convention and measurement geometry before applying them.