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