# Vacuum permittivity

The physical constant *ε*_{0}, commonly called the **vacuum permittivity**, **permittivity of free space** or **electric constant** is an ideal, (baseline) physical constant, which is the value of the absolute (*not* relative) dielectric permittivity of classical vacuum. Its value is:^{[1]}

The ellipsis ‘...’ does not represent an experimental inaccuracy (the value is exact) but the error introduced by truncation of a non-terminating decimal value.

This constant relates the units for electric charge to mechanical quantities such as length and force.^{[2]} For example, the force between two separated electric charges (in the vacuum of classical electromagnetism) is given by Coulomb's law:

where *q*_{1} and *q*_{2} are the charges, and *r* is the distance between them. Likewise, *ε*_{0} appears in Maxwell's equations, which describe the properties of electric and magnetic fields and electromagnetic radiation, and relate them to their sources.

## Contents

## Value

The value of *ε*_{0} is *defined* by the formula^{[3]}

where *c*_{0} is the defined value for the speed of light in classical vacuum in SI units,^{[4]} and *μ*_{0} is the parameter that international Standards Organizations call the "magnetic constant" (commonly called vacuum permeability). Since *μ*_{0} has the *defined* value 4π × 10^{−7} H m^{−1},^{[5]} and *c*_{0} has the *defined* value 299792458 m·s^{−1},^{[6]} it follows that *ε*_{0} has a *defined* value given approximately by

*ε*_{0}≈ 8.854187817620... × 10^{−12}F·m^{−1}(or A^{2}·s^{4}·kg^{−1}·m^{−3}in SI base units, or C^{2}·N^{−1}·m^{−2}or C·V^{−1}·m^{−1}using other SI coherent units).^{[7]}^{[8]}

The ellipsis (...) does not indicate experimental uncertainty, but the arbitrary termination of a nonrecurring decimal. The historical origins of the electric constant *ε*_{0}, and its value, are explained in more detail below.

### Redefinition of the SI units

Under the proposals to redefine the ampere as a fixed number of elementary charges per second,^{[9]} the electric constant would no longer have an exact fixed value. The value of the electron charge would become a defined number, not measured, making μ_{0} a measured quantity. Consequently, ε_{0} also would not be exact. As before, it would be defined by the equation ε_{0}= 1/(μ_{0}c_{0}^{2}), but now with a measurement error related to the error related to that in μ_{0}, the magnetic constant. This measurement error can be related to that in the fine structure constant α:

with *e* the exact elementary charge, *h* the exact Planck constant, and *c*_{0} the exact speed of light in vacuum. Here use is made of the relation for the fine structure constant:

The relative uncertainty in the value of ε_{0} therefore would be the same as that for the fine structure constant, currently 6.8×10^{−10}.^{[7]}

## Terminology

Historically, the parameter *ε*_{0} has been known by many different names. The terms "vacuum permittivity" or its variants, such as "permittivity in/of vacuum",^{[10]}^{[11]} "permittivity of empty space",^{[12]} or "permittivity of free space"^{[13]} are widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity,^{[7]} and official standards documents have adopted the term (although they continue to list the older terms as synonyms).^{[14]}^{[15]}

Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.^{[16]}^{[17]} However, in modern usage "dielectric constant" typically refers exclusively to a relative permittivity *ε*/*ε*_{0} and even this usage is considered "obsolete" by some standards bodies in favor of relative static permittivity.^{[15]}^{[18]} Hence, the term "dielectric constant of vacuum" for the electric constant *ε*_{0} 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 or , using either of the common glyphs for the letter epsilon.

## Historical origin of the parameter *ε*_{0}

As indicated above, the parameter *ε*_{0} 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 *ε*_{0} has the value it does requires a brief understanding of the history of how electromagnetic measurement systems developed.

### 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

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.^{[19]} 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

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:

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:

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.^{[20]}
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

### Determination of a value for *ε*_{0}

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 *ε*_{0} should be allocated the unit
C^{2}·N^{−1}·m^{−2} (or equivalent units - in practice "farads per metre").

In order to establish the numerical value of *ε*_{0}, 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 *ε*_{0}, *μ*_{0} and *c*_{0}. 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 *ε*_{0} is determined by the values of *c*_{0} and *μ*_{0}, as stated above. For a brief explanation of how the value of *μ*_{0} is decided, see the article about *μ*_{0}.

## Permittivity of real media

By convention, the electric constant *ε*_{0} 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:

- .

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:^{[21]}

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

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* = *ε*_{0}.

## See also

- Casimir effect
- Coulomb's law
- Electromagnetic wave equation
- ISO 31-5
- Mathematical descriptions of the electromagnetic field
- Sinusoidal plane-wave solutions of the electromagnetic wave equation

## Notes

- ↑
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*c*_{0}(or sometimes simply*c*) is the conventional symbol for the speed of light in vacuum. " See NIST*Special Publication 330*, p. 18 - ↑ See the last sentence of the NIST definition of ampere.
- ↑ See the last sentence of the NIST definition of meter.
- ↑
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