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6.9: The Magnetic Field H

Last updated

Mar 5, 2022
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- 6.8: Field on the Axis of a Long Solenoid
- 6.10: Flux

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

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