# Definition:Vacuum Permittivity

## Physical Constant

The vacuum permittivity is the physical constant denoted $\varepsilon_0$ defined as:

$\varepsilon_0 := \dfrac {e^2} {2 \alpha h c}$

where:

$e$ is the elementary charge
$\alpha$ is the fine-structure constant
$h$ is Planck's constant
$c$ is the speed of light defined in $\mathrm m \, \mathrm s^{-1}$

Of the above, only the fine-structure constant $\alpha$ is a measured value; the others are defined.

It can be defined as the capability of an electric field to permeate a vacuum.

From Value of Vacuum Permittivity, it has the value:

$\varepsilon_0 = 8 \cdotp 85418 \, 78128 (13) \times 10^{-12} \, \mathrm F \, \mathrm m^{-1}$ (farads per metre)

It can equivalently be defined as:

$\varepsilon_0 := \dfrac 1 {\mu_0 c^2}$

where:

$\mu_0$ is the vacuum permeability defined in $\mathrm H \, \mathrm m^{-1}$ (henries per metre)
$c$ is the speed of light defined in $\mathrm m \, \mathrm s^{-1}$

### Dimension

The vacuum permittivity has the dimension $\mathsf {M^{-1} L^{-3} T^4 I^2}$.

This arises from its definition as capacitance per unit length:

$\dfrac {\mathsf {M^{-1} L^{-2} T^4 I^2} } {\mathsf L}$

## Also known as

The vacuum permittivity is also known by the terms:

permittivity of free space
permittivity of empty space
permittivity in (or of) vacuum
distributed capacitance of the vacuum

The term electric constant has now apparently been accepted by many standards organisations worldwide.

However, as from $20$ May $2019$, the definition of $\varepsilon_0$ is no longer as a defined constant, but derived ultimately from the fine-structure constant, which is a measured value.

Hence many authorities (including $\mathsf{Pr} \infty \mathsf{fWiki}$) prefer not to use electric constant.

The following terms are more or less obsolete:

dielectric constant
dielectric constant of vacuum

Note that the term dielectric constant is also still used sometimes to mean the absolute permittivity (of a material), and so suffers the additional problem of being ambiguous.

Some sources use $\epsilon_0$ instead of $\varepsilon_0$.

Either symbol is acceptable, but $\mathsf{Pr} \infty \mathsf{fWiki}$ (having to choose one or the other) has settled on $\varepsilon_0$ as standard.

Some sources denote the vacuum permittivity with the symbol $\Gamma_e$.

## Also see

• Results about vacuum permittivity can be found here.

## Historical Note

Before the redefinition of the SI base units on $20$ May $2019$, the vacuum permittivity was:

$\varepsilon_0 = 8 \cdotp 85418 \, 78176 \, 2039 \times 10^{-12} \, \mathrm F \, \mathrm m^{-1}$ (farads per metre)

which was derived from the equation:

$\varepsilon_0 := \dfrac 1 {\mu_0 c^2}$

where:

$\mu_0$ is the vacuum permeability defined as exactly $4 \pi \times 10^{-7} \, \mathrm H \, \mathrm m^{-1}$ (henries per metre)
$c$ is the speed of light defined as exactly $299 \, 792 \, 458 \, \mathrm m \, \mathrm s^{-1}$

However, since $20$ May $2019$, the vacuum permeability has been redefined to be dependent upon the newly redefined electric charge on the electron, as follows:

$\mu_0 = \dfrac {2 \alpha} {e^2} \dfrac h c$

where:

$\alpha$ is the fine-structure constant
$e$ is the elementary charge
$h$ is Planck's constant
$c$ is the speed of light.

As a consequence, $\mu_0$ is now dependent upon the measured quantity $\alpha$, and its value is approximately:

$\mu_0 \approx 4 \pi \times 1 \cdotp 00000 \, 00005 \, 5 (15) \times 10^{-7} \, \mathrm H \, \mathrm m^{-1}$

Some older sources interject the following:

$\varepsilon_0 = \dfrac 1 {36 \pi} \times 10^{-9}$

based on the well-known estimate of the speed of light $3 \times 10^8 \mathrm {m \, s^{-1} }$.

This works out at:

$\varepsilon_0 \approx 8 \cdotp 84194 \, 1283 \ldots$