$$\require{cancel}$$

# 6.9: The Magnetic Field H

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 $$\textbf{D}=\epsilon \textbf{E}$$.