Auxiliary field

The auxiliary field H\mathbf{H} is a vector field that combines the effect of both free currents added externally and bound currents induced by magnetization. It is

H=1μ0B−M[Am2]\mathbf{H}=\frac{1}{\mu_{0}}\mathbf{B}-\mathbf{M}\qquad\left[ \frac{\text{A}}{\text{m}^{2}} \right]

where B\mathbf{B} is the magnetic field due to free currents and M\mathbf{M} is the magnetization.

Derivation

Consider any magnetized object upon which is set a free current Jf\mathbf{J}_{f}. The total current then is the sum of the free one and the bound one Jb\mathbf{J}_{b}:

J=Jb+Jf\mathbf{J}=\mathbf{J}_{b}+\mathbf{J}_{f}

From Ampere's law we get

1μ0(∇×B)=J=Jb+Jf=Jf+(∇×M)\frac{1}{\mu_{0}}(\nabla\times\mathbf{B})=\mathbf{J}=\mathbf{J}_{b}+\mathbf{J}_{f}=\mathbf{J}_{f}+(\nabla\times\mathbf{M})

if we collect the curls we get

∇×(1μ0B−M)=Jf\nabla \times\left( \frac{1}{\mu_{0}}\mathbf{B}-\mathbf{M} \right)=\mathbf{J}_{f}

The quantity in parenthesis is defined as the auxiliary field H\mathbf{H}.

Boundary conditions

The auxiliary field inherits B\mathbf{B}'s discontinuities over a surface current density. We know from the definition that the Divergence of H\mathbf{H} is ∇⋅H=−∇⋅M\nabla\cdot\mathbf{H}=-\nabla\cdot\mathbf{M}, so

Habove⊥−Hbelow⊥=−(Mabove⊥−Mbelow⊥)H_\text{above}^{\perp}-H_\text{below}^{\perp}=-(M_\text{above}^{\perp}-M_\text{below}^{\perp})

whereas from the curl of H\mathbf{H} derived from Ampere's law, ∇×H=Jf\nabla\times\mathbf{H}=\mathbf{J}_{f}, we get

Habove∥−Hbelow∥=Kf×n^\mathbf{H}^{\parallel}_\text{above}-\mathbf{H}^{\parallel}_\text{below}=\mathbf{K}_{f}\times \hat{\mathbf{n}}

which may be more useful than the boundary conditions on B\mathbf{B} when dealing with magnetized materials.