The physical constant ε0, commonly called the vacuum permittivity, permittivity of free
space or electric constant, is an ideal, physical constant, which is the value of the absolute
dielectric permittivity of classical vacuum. Its value is:
ε0 = 8.854 187 817... x 10−12 [F/m]. This constant relates the units for electric
charge to mechanical quantities such as length and force. For example, the force between
two separated electric charges is given by Coulomb's law:
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.
Value The value of ε0 is currently defined by the
formula
where c is the defined value for the speed of light in classical vacuum in SI units,
and μ0 is the parameter that international Standards Organizations call the "magnetic
constant". Since μ0 has the defined value 4π × 10−7 H·m−1, and c has the defined
value 299792458 m·s−1, it follows that ε0 has a defined value given approximately
by ε0 ≈ 8.854187817620... × 10−12 F·m−1.
The historical origins of the electric constant ε0, 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, 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 μ0 a measured quantity.
Consequently, ε0 also would not be exact. As before, it would be defined by the equation
ε0 = 1/(μ0c2), but now with a measurement error related to the error related to that
in μ0, 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 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 ε0 therefore would be the same as that for the
fine-structure constant, currently 6.8×10−10. 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. 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 or , 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. 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 ke is Coulomb's constant. If one is starting with no constraints, then
the value of ke may be chosen arbitrarily. For each different choice of ke 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 constant ke 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, 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 dyne1/2 cm. "Gaussian electric charge" is
not the same mathematical quantity as modern 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 qs' and ke' are not the same as
those in the older convention. Putting ke'=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
equation system, or "metre-kilogram-second-ampere" equation system. This is the system used to
define the SI units. The new quantity q is given the name "rmks electric charge", or
just "electric charge". Clearly, the quantity qs used in the old cgs esu system is related
to the new quantity q by
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. 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 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 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:
In the event that nonlocality and delay of response are not important, the result is:
where ε is the permittivity and εr the relative static permittivity. In the vacuum of classical
electromagnetism, the polarization P = 0, so εr = 1 and ε = ε0.
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 Wave impedance