### What is Magnetic Permeability?

Some of the most common questions I see online about electromagnetics is “what is permittivity” and “what is permeability.” I have a separate post answering the question about permittivity, so here I will do the same with permeability. If you read both articles, you may notice much of the language is very similar. That is intentional to illustrate the parallel concepts between the two.

The strict definition of permeability is that it is a measure of how well a medium stores magnetic energy. This definition does little to give a person any insight about what permeability is or why it is important. The general topic here is the magnetic response of materials.

### Magnetic Energy

There are two ways to store magnetic energy. First, magnetic energy is stored in the magnetic field itself. We know this because electromagnetic waves can transport energy through the vacuum of space. Think of this as the magnetic field intensity \(\vec{H}\), which is the field quantity most closely associated with electric current.

Second, magnetic energy can be stored in matter in the form of tilted magnetic dipoles (i.e. circulating charges). Some atoms and molecules are structured in a way that electrons flow in more circular paths than symmetric spheres. This forms a loop current that induces a magnetic field through the axis of the loop. This is a magnetic dipole.

When a magnetic field \(\vec{H}\) is applied to a medium with magnetic dipoles, it applies a force that tilts the magnetic dipoles away from their equilibrium positions. This stretches the molecules a bit so there is a force that wants to pull the magnetic dipole back to its equilibrium position. This makes the system act like a mass on stretched spring. A stretched spring stores mechanical energy. When magnetic dipoles at the atom scale are displaced from their equilibrium position due to an applied magnetic field, the material stores magnetic energy. In this state the material is said to be magnetized and is quantified by the magnetization polarization vector \(\vec{M}\). The magnetic polarization is related to the applied magnetic field through

\(\vec{M} = \mu_0 \chi_{\textrm{m}} \vec{H}\)

Eq. (1)

where \(\chi_{\textrm{m}}\) is the magnetic susceptibility and is a measure of how easily a material is magnetically polarized in response to an applied magnetic field. The magnetic flux density \(\vec{B}\) is the all-inclusive quantity that accounts for both types of stored magnetic energy. It is most closely related to force. The needle of a compass points in the direction of \(\vec{B}\), not \(\vec{H}\).

\(\vec{B} = \mu_0\vec{H} + \vec{M}\)

Eq. (2)

The constant \(\mu_0\) is the vacuum permeability and simply converts \(\vec{H}\) into the units and scale of \(\vec{B}\). Putting the above two equations together gives

\(\vec{B} = \mu_0\vec{H} + \mu_0 \chi_{\textrm{m}}\vec{H} = \mu_0 \left( 1 + \chi_{\textrm{m}} \right)\vec{H}\)

Eq. (3)

The magnetic flux density \(\vec{B}\) and magnetic field intensity \(\vec{H}\) are also related through the permeability \(\mu\). This equation is called the constitutive relation.

\(\vec{B} = \mu \vec{H}\)

Eq. (4)

The permeability is a very small number and inconvenient to communicate. It is more convenient to communicate in terms of the relative permeability \(\mu_{\textrm{r}}\). The relative permeability is always greater than 1.0.

\(\mu = \mu_0 \mu_{\textrm{r}}\)

Eq. (5)

The constitutive relation can now be written in terms of the relative permeability as

\(\vec{B} = \mu_0 \mu_{\textrm{r}} \vec{H}\)

Eq. (6)

Comparing Eq. (6) to Eq. (3) shows that

\(\vec{B} = \mu_0 \mu_{\textrm{r}} \vec{H} = \mu_0 \left( 1 + \chi_{\textrm{m}} \right)\vec{H}\)

Eq. (7)

This means

\(\mu_{\textrm{r}} = 1 + \chi_{\textrm{m}}\)

Eq. (8)

The relative permeability is proportional to how well a medium stores magnetic energy. The ‘1’ in Eq. (8) represents the ability to store magnetic energy in the field itself. The \(\chi_{\textrm{m}}\) term represents the ability to store magnetic energy in matter. So the relative permeability, and in turn the permeability, is an all-inclusive measurement of how well a medium can store magnetic energy.

Since the permeability arises due to something analogous to a mass on a spring, it is a resonant phenomenon. This means the permeability cannot be a constant. In fact, it is usually increasing with frequency because materials are typically used at frequencies below the resonant frequency. Near resonance, the permeability can have crazy values and interesting properties, but the material will also be very lossy.

Unlike dielectric materials, the resonant mechanisms that produce permeability are much slower and so they exhibit much lower resonant frequencies. In practice it is virtually impossible for a natural material to exhibit strong permeability above a gigahertz or so. If this is needed, either the operating frequency has to be lowered or the designer should look at things like metamaterials. You can read my blog post on metamaterials for more about this.