# 2.6: Permeability

[m0009_Permeability]

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: $\mu_0 = 4\pi \times 10^{-7} ~\mbox{H/m}$ It is common practice to describe the permeability of materials in terms of their relative permeability: $\mu_r \triangleq \frac{\mu}{\mu_0}$ 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 [m0136_Table_of_Permeabilities].

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.