# 2.7: Magnetic Field Intensity

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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.

## Contributors and Attributions

Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 Licensed with CC BY-SA 4.0 https://creativecommons.org/licenses/by-sa/4.0. Report adoption of this book here. If you are a professor reviewing, adopting, or adapting this textbook please help us understand a little more about your use by filling out this form.