| Value ofε0 | Unit |
|---|---|
| 8.8541878188(14)×10−12 | F⋅m−1 |
| C2⋅kg−1⋅m−3⋅s2 | |
| 55.26349406 | e2⋅eV−1⋅μm−1 |
Vacuum permittivity, commonly denotedε0 (pronounced "epsilon nought" or "epsilon zero"), is the value of theabsolute dielectric permittivity ofclassical vacuum. It may also be referred to as thepermittivity of free space, theelectric constant, or the distributed capacitance of the vacuum. It is an ideal (baseline)physical constant. ItsCODATA value is:
It is a measure of how dense of anelectric field is "permitted" to form in response to electric charges and relates the units forelectric charge to mechanical quantities such as length and force.[2] For example, the force between two separated electric charges with spherical symmetry (in thevacuum of classical electromagnetism) is given byCoulomb's law:Here,q1 andq2 are the charges,r is the distance between their centres, and the value of the constant fraction 1/(4πε0) is approximately9×109 N⋅m2⋅C−2. Likewise,ε0 appears inMaxwell's equations, which describe the properties ofelectric andmagnetic fields andelectromagnetic radiation, and relate them to their sources. Inelectrical engineering,ε0 itself is used as a unit to quantify the permittivity of variousdielectric materials.
The value ofε0 obeys the formula[3]wherec is the defined value for thespeed of light inclassical vacuum inSI units,[4]: 127 andμ0 is the parameter that international standards organizations refer to as themagnetic constant (also calledvacuum permeability or the permeability of free space). Sinceμ0 has an approximate value of 4π × 10−7 H/m (by the former definition of theampere),[4][5] andc has thedefined value299792458 m/s, it follows thatε0 can be expressed numerically as[6]The relative deviation of the recommended measured value (1.3×10−10 or 0.13 parts per billion) from the former defined value is within its uncertainty (1.6×10−10, in relative terms, or 0.16 parts per billion).
The historical origins of the electric constantε0, and its value, are explained in more detail below.
Theelementary charge was redefined exactly in terms of the coulomb as from 20 May 2019,[4] with the effect that the vacuum electric permittivity and themagnetic vacuum permeability no longer have exactly determined values in SI units. The value of the electron charge became a numerically defined quantity, makingε0 andμ0 measured quantities, neither of them exact, but related by the equationε0μ0c2 = 1. These values are determined by the experimentally determinedfine-structure constantα:withe being theelementary charge,h being thePlanck constant, andc being thespeed of light invacuum, each with exactly defined values. The relative uncertainty in the values of each ofε0 andμ0 are therefore the same as that for thefine-structure constant, namely1.6×10−10.[7]
Historically, the parameterε0 has been known by many different names. The terms "vacuum permittivity" or its variants, such as "permittivity in/of vacuum",[8][9] "permittivity of empty space",[10] or "permittivity offree space"[11] are widespread. Standards organizations also use "electric constant" as a term for this quantity.[12][13]
Another historical synonym was "dielectric constant of vacuum", as "dielectric constant" was sometimes used in the past for the absolute permittivity.[14][15] However, in modern usage "dielectric constant" typically refers exclusively to arelative permittivityε/ε0 and even this usage is considered "obsolete" by some standards bodies in favor ofrelative static permittivity.[13][16] Hence, the term "dielectric constant of vacuum" for the electric constantε0 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 eitherε0 orϵ0, using either of the commonglyphs for the letterepsilon.
As indicated above, the parameterε0 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 whyε0 has the value it does requires a brief understanding of the history.
The experiments ofCoulomb and others showed that the forceF between two, equal, point-like "amounts" of electricity that are situated a distancer apart in free space, should be given by a formula that has the formwhereQ is a quantity that represents the amount of electricity present at each of the two points, andke depends on the units. If one is starting with no constraints, then the value ofke may be chosen arbitrarily.[17] For each different choice ofke there is a different "interpretation" ofQ: 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 constantke was taken equal to 1, and a quantity now called "Gaussian electric charge"qs was defined by the resulting equation
The unit of Gaussian charge, thestatcoulomb, 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, thedyne. Thus, the unit of Gaussian charge can also be written 1 dyne1/2⋅cm. "Gaussian electric charge" is not the same mathematical quantity as modern (MKS and subsequently theSI) 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 quantitiesqs′ andke′ are not the same as those in the older convention. Puttingke′ = 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 symbolq, 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. The new quantityq is given the name "RMKS electric charge", or (nowadays) just "electric charge".[citation needed] The quantityqs used in the old cgs esu system is related to the new quantityq by:
In the2019 revision of the SI, the elementary charge is fixed at1.602176634×10−19 C and the value of the vacuum permittivity must be determined experimentally.[18]: 132
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 parameterε0 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 ofε0, one makes use of the fact that if one uses the rationalized forms of Coulomb's law andAmpère's force law (and other ideas) to developMaxwell's equations, then the relationship stated above is found to exist betweenε0,μ0 andc0. 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 ofε0 is determined by the values ofc0 andμ0, as stated above. For a brief explanation of how the value ofμ0 is decided, seeVacuum permeability.
By convention, the electric constantε0 appears in the relationship that defines theelectric displacement fieldD in terms of theelectric fieldE and classical electricalpolarization densityP of the medium. In general, this relationship has the form:
For a linear dielectric,P is assumed to be proportional toE, but a delayed response is permitted and a spatially non-local response, so one has:[19]
In the event that nonlocality and delay of response are not important, the result is:whereε is thepermittivity andεr therelative static permittivity. In thevacuum of classical electromagnetism, the polarizationP =0, soεr = 1 andε =ε0.