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Engineering LibreTexts

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 [m0003_Magnetic_Flux_Density]) 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:

    \[{\bf B} \triangleq \mu {\bf H} \label{m0012_eHDef}\]


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

    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,1 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. 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 [m0023_Boundary_Conditions_on_H], 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 [m0058_Magnetic_Materials].

    1. The concept of current density is not essential to understand this section; however, a primer can be found in Section [m0101_Current_Distributions].