# 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

- Interconnection between Vacuum Permittivity and Vacuum Permeability
- Value of Vacuum Permittivity
- Definition:Vacuum Permeability

- 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:

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

## Sources

- 1990: I.S. Grant and W.R. Phillips:
*Electromagnetism*(2nd ed.) ... (previous) ... (next): Chapter $1$: Force and energy in electrostatics: $1.1$ Electric Charge