Vacuum permittivity, commonly denoted ε₀ (pronounced "epsilon nought" or "epsilon zero"), is the value of the absolute dielectric permittivity of classical vacuum. It may also be referred to as the permittivity of free space, the electric constant, or the distributed capacitance of the vacuum. It is an ideal (baseline) physical constant. Its CODATA value is:

ε₀ = 8.8541878128(13)×10⁻¹² F⋅m⁻¹ (farads per meter), with a relative uncertainty of 1.5×10⁻¹⁰.^[1]

It is a measure of how dense of an electric field is "permitted" to form in response to electric charges and relates the units for electric charge to mechanical quantities such as length and force.^[2] For example, the force between two separated electric charges with spherical symmetry (in the vacuum of classical electromagnetism) is given by Coulomb's law:

${\displaystyle F_{\text{C}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}}$

Here, q₁ and q₂ are the charges, r is the distance between their centres, and the value of the constant fraction ${\displaystyle 1/(4\pi \varepsilon _{0})}$ (known as the Coulomb constant, k_e) is approximately 9 × 10⁹ N⋅m²⋅C⁻². Likewise, ε₀ appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources. In electrical engineering, ε₀ itself is used as a unit to quantify the permittivity of various dielectric materials.

Value

The value of ε₀ is defined by the formula^[3]

${\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}c^{2}}}}$

where c is the defined value for the speed of light in classical vacuum in SI units,^[4]: 127 and μ₀ is the parameter that international Standards Organizations call the "magnetic constant" (commonly called vacuum permeability or the permeability of free space). Since μ₀ has an approximate value 4π × 10⁻⁷ H/m,^[5] and c has the defined value 299792458 m⋅s⁻¹, it follows that ε₀ can be expressed numerically as

${\displaystyle {\begin{aligned}\varepsilon _{0}&\approx {\frac {1}{\left(4\pi \times 10^{-7}\,{\textrm {N/A}}^{2}\right)\left(299792458\,{\textrm {m/s}}\right)^{2}}}\\[2pt]&={\frac {625000}{22468879468420441\pi }}\,{\textrm {F/m}}\\[2pt]&\approx 8.85418781762039\times 10^{-12}\,{\textrm {F}}{\cdot }{\textrm {m}}^{-1}\end{aligned}}}$

(or A²⋅s⁴⋅kg⁻¹⋅m⁻³ in SI base units, or C²⋅N⁻¹⋅m⁻² or C⋅V⁻¹⋅m⁻¹ using other SI coherent units).^[6][7]

The historical origins of the electric constant ε₀, and its value, are explained in more detail below.

Terminology

Historically, the parameter ε₀ has been known by many different names. The terms "vacuum permittivity" or its variants, such as "permittivity in/of vacuum",^[9][10] "permittivity of empty space",^[11] or "permittivity of free space"^[12] are widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity,^[6] and official standards documents have adopted the term (although they continue to list the older terms as synonyms).^[13][14]

Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.^[15][16] However, in modern usage "dielectric constant" typically refers exclusively to a relative permittivity ε/ε₀ and even this usage is considered "obsolete" by some standards bodies in favor of relative static permittivity.^[14][17] Hence, the term "dielectric constant of vacuum" for the electric constant ε₀ is considered obsolete by most modern authors, although occasional examples of continuing usage can be found.

As for notation, the constant can be denoted by either ε₀ or ϵ₀, using either of the common glyphs for the letter epsilon.

Historical origin of the parameter ε₀

As indicated above, the parameter ε₀ is a measurement-system constant. Its presence in the equations now used to define electromagnetic quantities is the result of the so-called "rationalization" process described below. But the method of allocating a value to it is a consequence of the result that Maxwell's equations predict that, in free space, electromagnetic waves move with the speed of light. Understanding why ε₀ has the value it does requires a brief understanding of the history.

Permittivity of real media

By convention, the electric constant ε₀ appears in the relationship that defines the electric displacement field D in terms of the electric field E and classical electrical polarization density P of the medium. In general, this relationship has the form:

${\displaystyle \mathbf {D} =\varepsilon _{0}\mathbf {E} +\mathbf {P} .}$

For a linear dielectric, P is assumed to be proportional to E, but a delayed response is permitted and a spatially non-local response, so one has:^[20]

${\displaystyle \mathbf {D} (\mathbf {r} ,\ t)=\int _{-\infty }^{t}dt'\int d^{3}\mathbf {r} '\ \varepsilon \left(\mathbf {r} ,\ t;\mathbf {r} ',\ t'\right)\mathbf {E} \left(\mathbf {r} ',\ t'\right).}$

In the event that nonlocality and delay of response are not important, the result is:

${\displaystyle \mathbf {D} =\varepsilon \mathbf {E} =\varepsilon _{\text{r}}\varepsilon _{0}\mathbf {E} }$

where ε is the permittivity and ε_r the relative static permittivity. In the vacuum of classical electromagnetism, the polarization P = 0, so ε_r = 1 and ε = ε₀.

See also

• Casimir effect
• Coulomb's law
• Electromagnetic wave equation
• ISO 31-5
• Mathematical descriptions of the electromagnetic field
• Relative permittivity
• Sinusoidal plane-wave solutions of the electromagnetic wave equation
• Wave impedance
• Vacuum permeability

Notes