Vacuum permittivity

This article is about the electric constant. For the analogous magnetic constant, see vacuum permeability. For the ordinal number ε₀, see epsilon numbers (mathematics).

The physical constant ε₀ (pronounced as “epsilon nought” or “epsilon zero”), commonly called the vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum, is an ideal, (baseline) physical constant, which is the value of the absolute dielectric permittivity of classical vacuum. It has an exactly defined value that can be approximated as

ε₀ = 6988885418781699999♠8.854187817...×10⁻¹² F⋅m⁻¹ (farads per metre).

It is the capability of the vacuum to permit electric field lines. This constant relates the units for electric charge to mechanical quantities such as length and force. For example, the force between two separated electric charges (in the vacuum of classical electromagnetism) is given by Coulomb's law:

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

The value of the constant fraction is approximately 9 × 10⁹ N⋅m²⋅C⁻², q₁ and q₂ are the charges, and r is the distance between them. 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.

Value

The value of ε₀ is currently defined by the formula

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

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

$\varepsilon _{0}={\frac {1}{(4\pi \times 10^{-7}\,{\textrm {N/A}}^{2})(299792458\,{\textrm {m/s}})^{2}}}={\frac {625000}{22468879468420441\pi }}\,{\textrm {F/m}}\approx 8.85418781762039\times 10^{-12}\,{\textrm {Fm}}^{-1}$ (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).

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

Redefinition of the SI units

Main article: New SI definitions

Under the accepted proposals to redefine the ampere as a fixed number of elementary charges per second in 2019, the electric constant will no longer have an exact fixed value. The value of the electron charge becomes a defined number, not measured, making μ₀ a measured quantity. Consequently, ε₀ will not be exact. As before, it will be 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:

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

with e the exact elementary charge, h the exact Planck constant, and c the exact speed of light in vacuum.

The relative uncertainty in the value of ε₀ therefore would be the same as that for the dimensionless fine-structure constant, 1/137.035999046(27) according to a new measurement.

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", "permittivity of empty space", or "permittivity of free space" are widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity, and official standards documents have adopted the term (although they continue to list the older terms as synonyms). In the new SI system, the permittivity of vacuum will not be a constant anymore, but a measured quantity, related to the (measured) dimensionless fine structure constant.

Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity. 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. 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 $\varepsilon _{0}\,$ or $\epsilon _{0}\,$, 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.

Rationalization of units

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

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

where Q is a quantity that represents the amount of electricity present at each of the two points, and k_e is Coulomb's constant. If one is starting with no constraints, then the value of k_e may be chosen arbitrarily. 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

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

The unit of gaussian charge, the statcoulomb, is such that two units, 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:

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

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:

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

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. This is the system used to define the SI units. The new quantity q is given the name "rmks electric charge", or (nowadays) just "electric charge". Clearly, the quantity q_s used in the old cgs esu system is related to the new quantity q by

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

Determination of a value for ε₀

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 equivalent units – in practice "farads 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 the article about μ₀.

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:

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

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:

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

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

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

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