Permittivity

Charged bodies have Coulomb forces acting upon them. We often refer to these charges as point-like and placed in a vacuum, however, in real life that's usually not the case. They are almost always surrounded by a medium that is a nonconductor of electricity - a dielectric, such as air. It's intuitive to conclude that in a dielectric medium, the forces involved decrease in comparison to those in a vacuum. To describe this relation, a coefficient that describes the extent of polarization of a material in an electric field is introduced. This coefficient is called electric permittivity, and it shows how many times the Coulomb force in a dielectric medium is smaller than in a vacuum. In this article, we'll discuss this property of matter in greater detail.

Permittivity Meaning

First, let's define what exactly is electric permittivity.

To better understand it, let's imagine the following situation. Two plates are placed in a medium with a distance \(d\) separating them.

Fig. 1 - Non-polarized dipoles outside an electric field are chaotically distributed. In the presence of an electric field \(\vec{E}\) these dipoles become polarized.

First, let's consider the case where this medium is air, made up of various molecules. These molecules are made up of sub-atomic positive nuclei and negative electrons and so in the presence of an electric field, these air molecules rotate in a way to create an electric dipole moment. The electric field created by the polarized dielectric is opposite to the direction of the external field.

The medium, which in this case is air, resists the electric field, and this resistance is known as relative permittivity. If we remove all the air molecules leaving the electric field in a vacuum, it would experience no resistance, also known as absolute permittivity. Let's look at each of these two types of permittivity separately in the next sections of this article.

Permittivity of Free Space

The first kind of permittivity is known as the absolute permittivity of free space. Permittivity of free space, or vacuum, (\(\varepsilon_0\)) is a constant value equal to \(8.85\times10^{-12}\,\frac{\mathrm{C}}{\mathrm{N}\,\mathrm{m}^2}\). Mathematically, it can be obtained using the following equation:

where \(\mu_0\) is vacuum permeability equal to \(4\pi\times10^{-7} \, \frac{\mathrm{T} \, \mathrm{m}}{\mathrm{A}}\), and \(c\) is the speed of light \(3.00\times10^8\,\frac{\mathrm{m}}{\mathrm{s}}\).

Relative Permittivity

As soon as we're dealing with a medium other than free space, the permittivity becomes different from \(\varepsilon_0\). The extent to which this permittivity differs depends on the specific matter's atomic composition and arrangement. This value is obtained by determining how easily the electrons can change their arrangement within the material.

Relative permittivity \(\varepsilon_\mathrm{r}\) is found using by combining the permittivity of a specific material \(\varepsilon\) and the absolute permittivity \(\varepsilon_0\), described earlier. Mathematically, they combine into the following expression:

\[\varepsilon_\mathrm{r}=\frac{\varepsilon}{\varepsilon_0}.\]

We can distinguish two types of materials, conductors and insulators. In conductors, the charge carriers can easily move through them. The best conductors of electricity are metals, such as copper and aluminum.

Fig. 2 - The permittivity of all metals, including those used in electric wires, is infinite.

The opposite occurs in insulators, where the charge carriers cannot easily move through. Another commonly used name for insulators is dielectrics, which we'll discuss in greater detail in the next section.

Dielectric Permittivity

First, let's define what exactly is a dielectric material.

Instead, the material itself becomes polarized in the presence of an external electric field.

A common example of a dielectric medium is air, which has an electric permittivity of \(1.00059\). Many other non-ionized gases and liquids are also dielectrics. In a dielectric, the Coulomb force acting between charges becomes \(\varepsilon\) times weaker than if the charges were in a vacuum. This relation was mathematically defined earlier by using relative permittivity.

So all the dielectric mediums, air, paper, and oil,—weaken the electric field created by the conductors. Some of the most commonly used dielectrics and their dielectric constants are combined in the table below.

Table 1 - Common dielectric materials and their dielectric constants.

Capacitance and Permittivity

Before we can look at the expression that combines capacitance and permittivity, we must explain the concept of parallel-plate capacitors.

Fig. 3 - Parallel plate capacitor.

Keeping that in mind, capacitance is then used to connect the amount of the charge stored on each plate and the electric potential difference created by separating those charges. Capacitance depends solely on the physical properties of the capacitor, such as its shape and material.

Mathematically, capacitance \(C\) can be expressed as

\[C=\kappa \varepsilon_0 \frac{A}{d}.\]

In other words, the capacitance of a parallel plate capacitor is proportional to the area \(A\) of one of its plates and inversely proportional to the separation \(d\) between its plates. Here, the constant of proportionality (\(\kappa \varepsilon_0\)) is what relates capacitance to permittivity.