6.9: The Magnetic Field H

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If you look at the various formulas for the magnetic field \(B\) near various geometries of conductor, such as equations 6.5.3, 6.6.2, 6.7.1, 6.8.4, you will see that there is always a \(\mu\) on the right hand side. It is often convenient to define a quantity \(H = B/\mu \). Then these equations become just

\[H=\frac{I}{2\pi a},\]

\[H=\frac{I}{2a},\]

\[H=\frac{NIa^2}{2}\left ( \frac{1}{[a^2 + (c-x)^2]^{3/2}}+\frac{1}{[a^2+(c+x)^2]^{3/2}}\right ) ,\]

\[H=nI . \]

It is easily seen from any of these equations that the SI units of \(H\)are \(\text{A m}^{-1}\), or amps per metre, and the dimensions are \(\text{QT}^{-1}\text{M}^{-1}\).

Of course the magnetic field, whether represented by the quantity \(B\) or by \(H\), is a vector quantity, and the relation between the two representations can be written

\[\textbf{B}=\mu \textbf{H}.\]

In an isotropic medium \(\textbf{B}\) and \(\textbf{H}\) are parallel, but in an anisotropic medium they are not parallel (except in the directions of the eigenvectors of the permeability tensor), and permeability is a tensor. This was discussed in section 1.7.1 with respect to the equation \(\textbf{D}=\epsilon \textbf{E}\).