Vacuum permittivity

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:

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 ${\displaystyle 1/(4\pi \varepsilon _{0})}$ 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.

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 refer to as the magnetic constant (also 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^[6]

$${\displaystyle \varepsilon _{0}\approx {\frac {1}{\left(4\pi \times 10^{-7}\,{\textrm {N/A}}^{2}\right)\left(299792458\,{\textrm {m/s}}\right)^{2}}}\approx 8.8541878176\times 10^{-12}\,{\textrm {F}}{\cdot }{\textrm {m}}^{-1}.}$$

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

The ampere was redefined by defining the elementary charge as an exact number of coulombs as from 20 May 2019,^[4] with the effect that the vacuum electric permittivity no longer has an exactly determined value in SI units. The value of the electron charge became a numerically defined quantity, not measured, making μ₀ a measured quantity. Consequently, ε₀ is not exact. As before, it is defined by the equation ε₀ = 1/(μ₀c²), and is thus determined by the value of μ₀, the magnetic vacuum permeability which in turn is determined by the experimentally determined dimensionless fine-structure constant α:

$${\displaystyle \varepsilon _{0}={\frac {1}{\mu _{0}c^{2}}}={\frac {e^{2}}{2\alpha hc}}\ ,}$$

with e being the elementary charge, h being the Planck constant, and c being the speed of light in vacuum, each with exactly defined values. The relative uncertainty in the value of ε₀ is therefore the same as that for the dimensionless fine-structure constant, namely 1.6×10⁻¹⁰.^[7]

Historically, the parameter ε₀ has been known by many different names. The terms "vacuum permittivity" or its variants, such as "permittivity in/of vacuum",^[8][9] "permittivity of empty space",^[10] or "permittivity of free space"^[11] are widespread. Standards organizations also use "electric constant" as a term for this quantity.^[12][13]

Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.^[14][15] 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.^[13][16] 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.

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.

The experiments of Coulomb and others showed that the force F between two, equal, point-like "amounts" of electricity that are situated a distance r apart in free space, should be given by a formula that has the form

$${\displaystyle F=k_{\text{e}}{\frac {Q^{2}}{r^{2}}},}$$

where Q is a quantity that represents the amount of electricity present at each of the two points, and k_e depends on the units. If one is starting with no constraints, then the value of k_e may be chosen arbitrarily.^[17] For each different choice of k_e there is a different "interpretation" of Q: to avoid confusion, each different "interpretation" has to be allocated a distinctive name and symbol.

In one of the systems of equations and units agreed in the late 19th century, called the "centimetre–gram–second electrostatic system of units" (the cgs esu system), the constant k_e was taken equal to 1, and a quantity now called "Gaussian electric charge" q_s was defined by the resulting equation

$${\displaystyle F={\frac {{q_{\text{s}}}^{2}}{r^{2}}}.}$$

The unit of Gaussian charge, the statcoulomb, is such that two units, at a distance of 1 centimetre apart, repel each other with a force equal to the cgs unit of force, the dyne. Thus, the unit of Gaussian charge can also be written 1 dyne^1/2⋅cm. "Gaussian electric charge" is not the same mathematical quantity as modern (MKS and subsequently the SI) electric charge and is not measured in coulombs.

The idea subsequently developed that it would be better, in situations of spherical geometry, to include a factor 4π in equations like Coulomb's law, and write it in the form:

$${\displaystyle F=k'_{\text{e}}{\frac {{q'_{\text{s}}}^{2}}{4\pi r^{2}}}.}$$

This idea is called "rationalization". The quantities q_s′ and k_e′ are not the same as those in the older convention. Putting k_e′ = 1 generates a unit of electricity of different size, but it still has the same dimensions as the cgs esu system.

The next step was to treat the quantity representing "amount of electricity" as a fundamental quantity in its own right, denoted by the symbol q, and to write Coulomb's law in its modern form:

$${\displaystyle \ F={\frac {1}{4\pi \varepsilon _{0}}}{\frac {q^{2}}{r^{2}}}.}$$

The system of equations thus generated is known as the rationalized metre–kilogram–second (RMKS) equation system, or "metre–kilogram–second–ampere (MKSA)" equation system. The new quantity q is given the name "RMKS electric charge", or (nowadays) just "electric charge".^{[citation needed]} The quantity q_s used in the old cgs esu system is related to the new quantity q by:

$${\displaystyle \ q_{\text{s}}={\frac {q}{\sqrt {4\pi \varepsilon _{0}}}}.}$$

In the 2019 revision of the SI, the elementary charge is fixed at 1.602176634×10⁻¹⁹ C and the value of the vacuum permittivity must be determined experimentally.^[18]: 132

One now adds the requirement that one wants force to be measured in newtons, distance in metres, and charge to be measured in the engineers' practical unit, the coulomb, which is defined as the charge accumulated when a current of 1 ampere flows for one second. This shows that the parameter ε₀ should be allocated the unit C²⋅N⁻¹⋅m⁻² (or an equivalent unit – in practice, farad per metre).

In order to establish the numerical value of ε₀, one makes use of the fact that if one uses the rationalized forms of Coulomb's law and Ampère's force law (and other ideas) to develop Maxwell's equations, then the relationship stated above is found to exist between ε₀, μ₀ and c₀. In principle, one has a choice of deciding whether to make the coulomb or the ampere the fundamental unit of electricity and magnetism. The decision was taken internationally to use the ampere. This means that the value of ε₀ is determined by the values of c₀ and μ₀, as stated above. For a brief explanation of how the value of μ₀ is decided, see Vacuum permeability.

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:^[19]

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

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.

• 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