The constant ε_{0}, commonly called the vacuum permittivity, permittivity of free space or electric constant^{[1]}, 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:
where q_{1} and q_{2} are the charges, and r is the distance between them.^{[2]} 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.
The value of ε_{0} is defined by the formula
where c_{0} is the speed of light in vacuum and μ_{0} is the magnetic constant or vacuum permeability.
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The value of ε_{0} is defined by the formula
where c_{0} is the speed of light in vacuum,^{[3]} 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},^{[4]} and c_{0} has the defined value 299792458 m·s^{−1},^{[5]} it follows that ε_{0} has a defined value given approximately by
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.
Alternatively, ε_{0} can be expressed as
The view (sometimes encountered) that ε_{0} is a physical constant that describes a physical property of a realizable "vacuum" is incorrect. Rather, ε_{0} is a measurementsystem constant introduced and defined as a result of international agreement. The value allocated to ε_{0} 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, ultrahigh vacuum, QCD vacuum or quantum vacuum are experimental and theoretical matters, separate from ε_{0}. The meaning and value of ε_{0} 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 ε_{0}.
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",^{[7]}^{[8]} "permittivity of empty space",^{[9]} or "permittivity of free space"^{[10]} are widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity,^{[1]} and official standards documents have adopted the term (although they continue to list the older terms as synonyms).^{[11]}^{[12]}
Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.^{[13 ]}^{[14]} 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.^{[12]}^{[15]} 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 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 ε_{0}, 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 ε_{0}, which is a derived quantity based upon the defined values of c_{0} and μ_{0} 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 or , using either of the common glyphs for the letter epsilon.
As indicated above, the parameter ε_{0} is a measurementsystem constant. Its presence in the equations now used to define electromagnetic quantities is the result of the socalled "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.
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 experimentallyrealisable vacuum approximates free space, see the article on free space.
The experiments of Coulomb and others showed that the force F between two equal pointlike "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 a constant. If one is starting with no constraints, then the value of k_{e} may be chosen arbitrarily.^{[16]} 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 "centimetregramsecond 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 metrekilogramsecond (rmks) equation system, or "metrekilogramsecondampere (mksa)" equation system. This is system used to define the SI units.^{[17]} 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
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}.
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. In real media this relationship has the form
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.
