Vacuum permittivity

This article is about a constant used in science. For the ordinal in mathematics ε₀, see epsilon nought.

The constant ε₀, commonly called the vacuum permittivityL, permittivity of free space or electric constant, relates the units for electric charge to mechanical quantities such as length and force in the International System of Units. For example, the force between two separated electric charges is given by Coulomb's law:

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

where 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.

The value of ε₀ is defined by the formula

$\varepsilon _{0}={\frac {1}{\mu _{0}c_{0}^{2}}}$ = 8.854 187 817... × 10⁻¹² F m⁻¹

where c₀ is the speed of light in vacuum and μ₀ is the magnetic constant or vacuum permeability.

Value

The value of ε₀ is defined by the formula

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

where c₀ is the speed of light in vacuum, 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⁻¹, and c₀ has the defined value 299792458 m s⁻¹, it follows that ε₀ has a defined value given approximately by

ε₀ ≈ 8.854 187 817... × 10⁻¹² F m⁻¹ (or A² s⁴ kg⁻¹ m⁻³ in SI base units, or C² N⁻¹m⁻² or C V⁻¹m⁻¹ using other SI coherent units).

The ellipsis (...) does not indicate experimental uncertainty, but the arbitrary termination of a nonrecurring decimal. The historical origins of the electric constant ε₀, and its value, are explained in more detail below.

Alternatively, ε₀ can be expressed as

ε₀ ≈ 0.055 263 5... eV V^{−2 nm−1.}

The view (sometimes encountered) that ε₀ is a physical constant that describes a physical property of a realizable "vacuum" is incorrect. Rather, ε₀ is a measurement-system constant introduced and defined as a result of international agreement. The value allocated to ε₀ relates to the velocity of light in a reference situation or benchmark, sometimes called free space, used as a baseline for comparison of measurements made in all types of real media. The physical properties of realizable vacuums such as outer space, ultra-high vacuum, QCD vacuum or quantum vacuum are experimental and theoretical matters, separate from ε₀. The meaning and value of ε₀ are metrology issues, not issues about properties of realizable vacuums. This potential for confusion is why many Standards Organizations now prefer to use the name "electric constant" for ε₀.

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).

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 already noted, the name now preferred by Standards Organizations is electric constant. This name avoids the use of the term permittivity in the name of ε₀, and also avoids the use of free space and of vacuum (which is not as simple a concept as once was thought, see free space). The name "electric constant" avoids the suggestion that ε₀, which is a derived quantity based upon the defined values of c₀ and μ₀ as indicated above, is a "property" of anything physical. It appears to be thought that the name "electric constant" is less likely to cause misunderstandings than the older names.

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 of how electromagnetic measurement systems developed.

In the following discussion, it may be noted that classically no distinction was made between "vacuum" and free space. Today, in the literature, the term "vacuum" may refer to a variety of experimental conditions and theoretical entities. In reading the literature, only context can decide what is meant. Below, the term "free space" is used to refer to the reference state called "vacuum" by Standards Organizations. For a description of how closely experimentally-realisable vacuum approximates free space, see the article on free space.

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

$F=\;k_{\mathrm {e} }Q^{2}/r^{2},$

where Q is a quantity that represents the amount of electricity present at each of the two points, and k_e is a 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 1800s, 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

$F={q_{s}}^{2}/r^{2}.$

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 (rmks) 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:

$F=\;k'_{\mathrm {e} }{q'_{s}}^{2}/4\pi r^{2}.$

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:

$\ F=q^{2}/4\pi \epsilon _{0}r^{2}.$

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 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

$\ q_{s}=q/(4\pi \epsilon _{0})^{1/2}.$

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. In real media this relationship has the form

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

where ε is the permittivity, ε_r the relative static permittivity, and P is the classical electrical polarization density of the medium. In the reference state of free space, called "vacuum" by Standards Organizations, the polarization P = 0.

See also

• Casimir effect
• Electromagnetic wave equation
• ISO 31-5
• Mathematical descriptions of the electromagnetic field
• Sinusoidal plane-wave solutions of the electromagnetic wave equation