### What is Electrical Permittivity?

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

### Electric energy

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

Second, electric energy can be stored in matter in the form of displaced charge. When the electric field \(\vec{E}\) is applied to a dielectric medium, it applies a force on atomic scale charges. Electrons and protons have opposite charge so they are pushed in opposite directions. However, the electrons are still bound to the positive charge nuclei due to the electrostatic attraction and the system acts like a mass on stretched spring. A stretched spring stores mechanical energy. When charges at the atomic scale are displaced from their equilibrium position due to an applied electric field, the material stores electric energy. In this state the material is said to be polarized and is quantified by the polarization vector \(\vec{P}\). The polarization is related to the applied electric field through

\(\vec{P} = \varepsilon_0 \chi_{\textrm{e}} \vec{E}\)

Eq. (1)

where \(\chi_{\textrm{e}}\) is the electric susceptibility and is a measure of how easily a material is polarized in response to an applied electric field. The electric flux density \(\vec{D}\) is the all-inclusive quantity that accounts for both types of stored electric energy. It is most closely related to charge.

\(\vec{D} = \varepsilon_0 \vec{E} + \vec{P}\)

Eq. (2)

The constant \(\varepsilon_0\) is the vacuum permittivity and simply converts \(\vec{E}\) into the units and scale of \(\vec{D}\). Putting the above two equations together gives

\(\vec{D} = \varepsilon_0\vec{E} + \varepsilon_0\chi_{\textrm{e}}\vec{E} = \varepsilon_0 \left( 1 + \chi_{\textrm{e}} \right) \vec{E}\)

Eq. (3)

The electric flux density \(\vec{D}\) and electric field intensity \(\vec{E}\) are also related through the permittivity \(\varepsilon\). This equation is called the constitutive relation.

\(\vec{D} = \varepsilon \vec{E}\)

Eq. (4)

The permittivity is a very small number and inconvenient to communicate. It is more convenient to communicate in terms of the relative permittivity \(\varepsilon_{\textrm{r}}\), commonly called the dielectric constant. The dielectric constant is always greater than 1.0.

\(\varepsilon = \varepsilon_0 \varepsilon_{\textrm{r}}\)

Eq. (5)

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

\(\vec{D} = \varepsilon_0 \varepsilon_{\textrm{r}} \vec{E}\)

Eq. (6)

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

\(\vec{D} = \varepsilon_0 \varepsilon_{\textrm{r}} \vec{E} = \varepsilon_0 \left( 1 + \chi_{\textrm{e}} \right) \vec{E}\)

Eq. (7)

This means

\(\varepsilon_{\textrm{r}} = 1 + \chi_{\textrm{e}} \)

Eq. (8)

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

Since the permittivity arises due to something analogous to a mass on a spring, it is a resonant phenomenon. This means the permittivity 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 permittivity can have crazy values and interesting properties, but the material will also be very lossy.