# 2.7: Magnetic Field Intensity

Magnetic field intensity $${\bf H}$$ is an alternative description of the magnetic field in which the effect of material is factored out. For example, the magnetic flux density $${\bf B}$$ (reminder: Section 2.5) due to a point charge $$q$$ moving at velocity $${\bf v}$$ can be written in terms of the Biot-Savart Law:

${\bf B} = \mu ~ \frac{q {\bf v}}{4\pi R^2} \times \hat{\bf R} \label{m0012_eMatB}$

where $$\hat{\bf R}$$ is the unit vector pointing from the charged particle to the field point $${\bf r}$$, $$R$$ is this distance, “$$\times$$” is the cross product, and $$\mu$$ is the permeability of the material. We can rewrite Equation \ref{m0012_eMatB} as:

${\bf B} \triangleq \mu {\bf H} \label{m0012_eHDef}$

with:

${\bf H} = \frac{q {\bf v}}{4\pi R^2} \times \hat{\bf R} \label{m0012_eHHH}$

so $${\bf H}$$ in homogeneous media does not depend on $$\mu$$.

Dimensional analysis of Equation \ref{m0012_eHHH} reveals that the units for $${\bf H}$$ are amperes per meter (A/m). However, $${\bf H}$$ does not represent surface current density, as the units might suggest. While it is certainly true that a distribution of current (A) over some linear cross-section (m) can be described as a current density having units of A/m, $${\bf H}$$ is associated with the magnetic field and not a particular current distribution (the concept of current density is not essential to understand this section; however, a primer can be found in Section 6.2). Said differently, $${\bf H}$$ can be viewed as a description of the magnetic field in terms of an equivalent (but not actual) current.

The magnetic field intensity $${\bf H}$$ (A/m), defined using Equation \ref{m0012_eHDef}, is a description of the magnetic field independent from material properties.

It may appear that $${\bf H}$$ is redundant information given $${\bf B}$$ and $$\mu$$, but this is true only in homogeneous media. The concept of magnetic field intensity becomes important – and decidedly not redundant – when we encounter boundaries between media having different permeabilities. As we shall see in Section 7.11, boundary conditions on $${\bf H}$$ constrain the component of the magnetic field which is tangent to the boundary separating two otherwise-homogeneous regions. If one ignores the characteristics of the magnetic field represented by $${\bf H}$$ and instead considers only $${\bf B}$$, then only the perpendicular component of the magnetic field is constrained.

The concept of magnetic field intensity also turns out to be useful in a certain problems in which $$\mu$$ is not a constant, but rather is a function of magnetic field strength. In this case, the magnetic behavior of the material is said to be nonlinear. For more on this, see Section 7.16.