# 2.6: Permeability

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Permeability describes the effect of material in determining the magnetic flux density. All else being equal, magnetic flux density increases in proportion to permeability.

To illustrate the concept, consider that a particle bearing charge \(q\) moving at velocity \({\bf v}\) gives rise to a magnetic flux density:

\[{\bf B}({\bf r}) = \mu ~ \frac{q {\bf v}}{4\pi R^2} \times \hat{\bf R} %\label{eMatB} \]

where \(\hat{\bf R}\) is the unit vector pointing from the charged particle to the field point \({\bf r}\), \(R\) is this distance, and “\(\times\)” is the cross product. Note that \({\bf B}\) increases with charge and speed, which makes sense since moving charge is the source of the magnetic field. Also note that \({\bf B}\) is inversely proportional to \(4\pi R^2\), indicating that \(\left|{\bf B}\right|\) decreases in proportion to the area of a sphere surrounding the charge, also known as the *inverse square law*. The remaining factor, \(\mu\), is the constant of proportionality that captures the effect of material. We refer to \(\mu\) as the *permeability* of the material. Since \({\bf B}\) can be expressed in units of Wb/m\(^2\) and the units of \({\bf v}\) are m/s, we see that \(\mu\) must have units of henries per meter (H/m). (To see this, note that 1 H \(\triangleq\) 1 Wb/A.)

Permeability (\(\mu\), H/m) describes the effect of material in determining the magnetic flux density.

In free space, we find that the permeability \(\mu=\mu_0\) where:

\[\mu_0 = 4\pi \times 10^{-7} ~\mbox{H/m} \nonumber \]

It is common practice to describe the permeability of materials in terms of their *relative permeability*:

\[\mu_r \triangleq \frac{\mu}{\mu_0} \nonumber \]

which gives the permeability relative to the minimum possible value; i.e., that of free space. Relative permeability for a few representative materials is given in Appendix A2.

Note that \(\mu_r\) is approximately 1 for all but a small class of materials. These are known as *magnetic materials*, and may exhibit values of \(\mu_r\) as large as \(\sim 10^6\). A commonly-encountered category of magnetic materials is *ferromagnetic* material, of which the best-known example is iron.