Vacuum permittivity

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 also use "electric constant" as a term for this quantity.

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 ε₀ or ϵ₀, using either of the common glyphs for the letter epsilon.

History

Rationalization of units

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. 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: $${\displaystyle F=k_{\text{e}}^{\prime }\frac{{q_{\text{s}}^{\prime }}^2}{4\pi r^2}.}$$

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: $${\displaystyle F=\frac{1}{4\pi {\epsilon }_0}\frac{q^2}{r^2}.}$$

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". The quantity q_s used in the old cgs esu system is related to the new quantity q by: $${\displaystyle q_{\text{s}}=\frac{q}{\sqrt{4\pi {\epsilon }_0}}.}$$

Prior to the 2019 revision of the SI, μ₀ had an exact defined value of 4π × 10⁻⁷ H/m (by the former definition of the ampere), and c had (and retains) the defined value 299792458 m/s. Under that former definition, ε₀ could be expressed numerically as an exact derivative: $${\displaystyle {\epsilon }_0=\frac{1}{\left(4\pi \times {10}^{-7}\mathrm{N}/{\mathrm{A}}^2\right){\left(299792458\mathrm{m}/\mathrm{s}\right)}^2}\approx 8.8541878176\times {10}^{-12}\mathrm{F}/\mathrm{m}.}$$

Revision of the SI

The elementary charge was redefined exactly in terms of the coulomb as from 20 May 2019, with the effect that the vacuum electric permittivity and the magnetic vacuum permeability no longer have exact values in SI units. The value of the electron charge became a numerically defined quantity, making ε₀ and μ₀ measured quantities, neither of them exact, but related by the equation ε₀μ₀c² = 1. These values are determined by the experimentally determined fine-structure constant α: $${\displaystyle {\epsilon }_0=\frac{1}{{\mu }_0{c}^2}=\frac{e^2}{2\alpha hc},}$$ 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 values of each of ε₀ and μ₀ are therefore the same as that for the fine-structure constant, namely 1.6×10⁻¹⁰.

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.

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 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 c. 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 c and μ₀, as stated above. For a brief explanation of how the value of μ₀ is decided, see Vacuum permeability.

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: $${\displaystyle \mathbf{D}={\epsilon }_0\mathbf{E}+\mathbf{P}.}$$

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: $${\displaystyle \mathbf{D}(\mathbf{r},t)={\int }_{-\mathrm{\infty }}^{t}d{t}^{\prime }\int d^3{\mathbf{r}}^{\prime }\epsilon \left(\mathbf{r},t;{\mathbf{r}}^{\prime },t^{\prime }\right)\mathbf{E}\left({\mathbf{r}}^{\prime },t^{\prime }\right).}$$

In the event that nonlocality and delay of response are not important, the result is: $${\displaystyle \mathbf{D}=\epsilon \mathbf{E}={\epsilon }_{\text{r}}{\epsilon }_0\mathbf{E}}$$ 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.