What is the Vacuum Permittivity?

by Alexander FufaeV

The electric constant (or vacuum permittivity) is a physical constant and is notated as $ \varepsilon_0 $ (pronounced: "epsilon zero"). It has the following value:

Value of the vacuum permittivity

$$ \begin{align} \varepsilon_0 ~=~ 8.854 \, 187 \, 8128 ~\cdot~ 10^{-12} \, \frac{\mathrm{As}}{\mathrm{Vm}} \end{align} $$

The unit of $\varepsilon_0$ is for example ampere-second per voltmeter or farad per meter:

$$ \begin{align} \frac{\mathrm{As}}{\mathrm{Vm}} ~=~ \frac{\mathrm{F}}{\mathrm{m}} ~=~ \frac{\mathrm{A}^2\,\mathrm{s}^4}{\mathrm{kg}\,\mathrm{m}^3} \end{align} $$

The vacuum permittivity appears in equations that have to do with electric fields. For example, in Coulomb's law or in the wave equation for electromagnetic waves.

In our universe, the vacuum permittivity determines how strongly electric charges are allowed to attract or repel each other.

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The attractive and repulsive force between electric charges.

The vacuum permittivity, together with the vacuum permeability $\mu_0$ (magnetic constant), determines how large the speed of light $c$ should be in vacuum:

$$ \begin{align} c ~=~ \sqrt{\frac{1}{\mu_0 \, \varepsilon_0}} \end{align} $$

Vacuum permittivity $\varepsilon_0$ specifies how easy / hard it is for electric fields to penetrate the vacuum. Permittivity $\varepsilon$ of water, for example, is about eighty times the permittivity of vacuum: $ \varepsilon = 80 \cdot \varepsilon_0 $. Here the factor $ \varepsilon_{\text r} = 80 $ is called relative permittivity.

How can the vacuum permittivity be determined experimentally?:
The vacuum permittivity $ \varepsilon_0 $ can be determined experimentally, for example, with a so-called Coulomb's torsion balance. Similar to the Eötvös gravitational balance, it exploits the torque exerted on each other by two known charges $ q_1 $ and $ q_2 $.

From the experimentally determined electric force $ F_{\text e} $ between the two charges, the value of the vacuum permittivity can be found using Coulomb's law:

$$ \begin{align} \varepsilon_0 ~=~ \frac{1}{4\pi F_{\text e}} \, \frac{q_1 \, q_2}{r^2} \end{align} $$