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.