# On the definition of magnetic field intensity

In magnetostatics theory, the magnetic field of any magnetic body with magnetization $$\mathbf M$$ is equivalent to that of a volume current density $$\mathbf J_{m,v}$$ and a surface current density $$\mathbf J_{m,s}$$ with the following relations:

$$\mathbf J_{m,v} = \nabla \times \mathbf M \\ \mathbf J_{m,s} = \mathbf M \times \mathbf n$$

In defining the magnetic field intensity (aka, auxiliary magnetic field) $$\mathbf H$$ it is assumed that only the volume magnetization current density $$\mathbf J_{m,v}$$ contributes to the free current $$\mathbf J_f$$: (See D.K Cheng, Field and Wave Electromagnetics)

$$\nabla \times \frac{\mathbf B}{\mu_0} = \mathbf J_f + \nabla \times \mathbf M \\ \nabla \times \left(\frac{\mathbf B}{\mu_0} - \mathbf M \right) = \mathbf J_f$$

However all the three sources $$\mathbf J_{f}$$, $$\mathbf J_{m,v}$$ and $$\mathbf J_{m,s}$$ should contribute to the magnetic field. I cannot see how the surface current is included.

The surface magnetization current is accounted for by the magnetization current term $$\nabla \times \mathbf M$$. On a boundary between the medium and vacuum, the term $$\nabla \times \mathbf M$$ is singular; when integrated over an arbitrarily small line normal to the boundary passing through the boundary, we get value $$\mathbf M\times \mathbf n$$, which is thus the surface current per unit length on the boundary.