# 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

moving at velocity

gives rise to a magnetic flux density:

where

is the unit vector pointing from the charged particle to the field point

,

is this distance, and “

” is the cross product. Note that

increases with charge and speed, which makes sense since moving charge is the source of the magnetic field. Also note that

is inversely proportional to

, indicating that

decreases in proportion to the area of a sphere surrounding the charge, also known as the *inverse square law*. The remaining factor,

, is the constant of proportionality that captures the effect of material. We refer to

as the *permeability* of the material. Since

can be expressed in units of

/m^{2} and the units of

are m/s, we see that

must have units of henries per meter (H/m). (To see this, note that

.)

Permeability (

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

In free space, we find that the permeability

where:

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

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 10.2.

Note that

is approximately 1 for all but a small class of materials. These are known as *magnetic materials*, and may exhibit values of

as large as

10^{6}. A commonly-encountered category of magnetic materials is *ferromagnetic* material, of which the best-known example is iron.

## Additional Reading

- “Permeability (electromagnetism)” on Wikipedia.
- Section 7.16 (“Magnetic Materials”).
- Appendix 10.2 (“Permeability of Some Common Materials”).

Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 CC BY-SA 4.0

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