Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the
absolute dielectric permittivity of
classical vacuum. It may also be referred to as the permittivity of free space, the electric constant, or the distributed capacitance of the vacuum. It is an ideal (baseline)
physical constant
A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that cannot be explained by a theory and therefore must be measured experimentally. It is distinct from a mathematical constant, which has a ...
. Its
CODATA
The Committee on Data of the International Science Council (CODATA) was established in 1966 as the Committee on Data for Science and Technology, originally part of the International Council of Scientific Unions, now part of the International ...
value is:
It is a measure of how dense of an
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
is "permitted" to form in response to electric charges and relates the units for
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
to mechanical quantities such as length and force. For example, the force between two separated electric charges with spherical symmetry (in the
vacuum of classical electromagnetism) is given by
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
:
Here, ''q''
1 and ''q''
2 are the charges, ''r'' is the distance between their centres, and the value of the constant fraction 1/(4π''ε''
0) is approximately . Likewise, ''ε''
0 appears in
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
, which describe the properties of
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter possessing an electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by Maxwel ...
and
magnetic
Magnetism is the class of physical attributes that occur through a magnetic field, which allows objects to attract or repel each other. Because both electric currents and magnetic moments of elementary particles give rise to a magnetic field, m ...
fields and
electromagnetic radiation
In physics, electromagnetic radiation (EMR) is a self-propagating wave of the electromagnetic field that carries momentum and radiant energy through space. It encompasses a broad spectrum, classified by frequency or its inverse, wavelength ...
, and relate them to their sources. In
electrical engineering
Electrical engineering is an engineering discipline concerned with the study, design, and application of equipment, devices, and systems that use electricity, electronics, and electromagnetism. It emerged as an identifiable occupation in the l ...
, ''ε''
0 itself is used as a unit to quantify the permittivity of various
dielectric
In electromagnetism, a dielectric (or dielectric medium) is an Insulator (electricity), electrical insulator that can be Polarisability, polarised by an applied electric field. When a dielectric material is placed in an electric field, electric ...
materials.
Value
The value of ''ε''
0 is ''defined'' by the formula
[
The approximate numerical value is found at:
This formula determining the exact value of ''ε''0 is found in Table 1, p. 637 of ]
where ''c'' is the defined value for the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
in
classical vacuum in
SI units
The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official st ...
,
and ''μ''
0 is the parameter that international standards organizations refer to as the
magnetic constant
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum'', ''magnetic constant'') is the magnetic permeability in a classical vacuum. It is a physical constant, conventionall ...
(also called
vacuum permeability
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum'', ''magnetic constant'') is the magnetic permeability in a classical vacuum. It is a physical constant, conventionally ...
or the permeability of free space). Since ''μ''
0 has an approximate value of 4π × 10
−7 H/
m (by the former definition of the
ampere
The ampere ( , ; symbol: A), often shortened to amp,SI supports only the use of symbols and deprecates the use of abbreviations for units. is the unit of electric current in the International System of Units (SI). One ampere is equal to 1 c ...
),
and ''c'' has the ''defined'' value , it follows that ''ε''
0 can be expressed numerically as
The relative deviation of the recommended measured value ( or 0.13 parts per billion) from the former defined value is within its uncertainty (, in relative terms, or 0.16 parts per billion).
The historical origins of the electric constant ''ε''
0, and its value, are explained in more detail below.
Revision of the SI
The
elementary charge
The elementary charge, usually denoted by , is a fundamental physical constant, defined as the electric charge carried by a single proton (+1 ''e'') or, equivalently, the magnitude of the negative electric charge carried by a single electron, ...
was redefined exactly in terms of the coulomb as from 20 May 2019,
with the effect that the vacuum electric permittivity and the
magnetic vacuum permeability no longer have exactly determined values in SI units. The value of the electron charge became a numerically defined quantity, making ''ε''
0 and ''μ''
0 measured quantities, neither of them exact, but related by the equation . These values are determined by the experimentally determined
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...
''α'':
with ''e'' being the
elementary charge
The elementary charge, usually denoted by , is a fundamental physical constant, defined as the electric charge carried by a single proton (+1 ''e'') or, equivalently, the magnitude of the negative electric charge carried by a single electron, ...
, ''h'' being the
Planck constant
The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
, and ''c'' being the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
in
vacuum
A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
, each with exactly defined values. The relative uncertainty in the values of each of ''ε''
0 and ''μ''
0 are therefore the same as that for the
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Alpha, Greek letter ''alpha''), is a Dimensionless physical constant, fundamental physical constant that quantifies the strength of the el ...
, namely
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",
[
][
] "permittivity of empty space",
[
] or "permittivity of
free space
A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
"
[B. E. A. Saleh and M. C. Teich, ''Fundamentals of Photonics'' (Wiley, 1991)] are widespread. Standards organizations also use "electric constant" as a term for this quantity.
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
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the vacuum permittivity, electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric co ...
''ε''/''ε''
0 and even this usage is considered "obsolete" by some standards bodies in favor of
relative static permittivity
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insula ...
.
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 ''ε''
0 or '
0, using either of the common
glyph
A glyph ( ) is any kind of purposeful mark. In typography, a glyph is "the specific shape, design, or representation of a character". It is a particular graphical representation, in a particular typeface, of an element of written language. A ...
s for the letter
epsilon
Epsilon (, ; uppercase , lowercase or ; ) is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel or . In the system of Greek numerals it also has the value five. It was derived from the Phoenic ...
.
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.
Rationalization of units
The experiments of
Coulomb
The coulomb (symbol: C) is the unit of electric charge in the International System of Units (SI).
It is defined to be equal to the electric charge delivered by a 1 ampere current in 1 second, with the elementary charge ''e'' as a defining c ...
and others showed that the force ''F'' between two, equal, point-like "amounts" of electricity that are 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 depends on the units. If one is starting with no constraints, then the value of ''k''
e may be chosen arbitrarily.
[For an introduction to the subject of choices for independent units, see
] 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
The statcoulomb (statC), franklin (Fr), or electrostatic unit of charge (esu) is the unit of measurement for electrical charge used in the centimetre–gram–second electrostatic units variant (CGS-ESU) and Gaussian systems of units. In term ...
, 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, the
dyne
The dyne (symbol: dyn; ) is a derived units of measurement, unit of force (physics), force specified in the centimetre–gram–second system of units, centimetre–gram–second (CGS) system of units, a predecessor of the modern International S ...
. 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 (
MKS and subsequently the
SI) 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 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. The new quantity ''q'' is given the name "RMKS electric charge", or (nowadays) just "electric charge". The quantity ''q''
s used in the old cgs esu system is related to the new quantity ''q'' by:
In the
2019 revision of the SI
In 2019, four of the seven SI base units specified in the International System of Quantities were redefined in terms of natural physical constants, rather than human artefacts such as the standard kilogram.
Effective 20 May 2019, the 144th ...
, the elementary charge is fixed at and the value of the vacuum permittivity must be determined experimentally.
[
]
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 an equivalent unit – in practice, farad 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
In magnetostatics, Ampère's force law describes the force of attraction or repulsion between two current-carrying wires. The physical origin of this force is that each wire generates a magnetic field, following the Biot–Savart law, and th ...
(and other ideas) to develop
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, Electrical network, electr ...
, 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 ''
Vacuum permeability
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum'', ''magnetic constant'') is the magnetic permeability in a classical vacuum. It is a physical constant, conventionally ...
''.
Permittivity of real media
By convention, the electric constant ''ε''
0 appears in the relationship that defines the
electric displacement field
In physics, the electric displacement field (denoted by D), also called electric flux density, is a vector field that appears in Maxwell's equations. It accounts for the electromagnetic effects of polarization and that of an electric field, com ...
D in terms of the
electric field
An electric field (sometimes called E-field) is a field (physics), physical field that surrounds electrically charged particles such as electrons. In classical electromagnetism, the electric field of a single charge (or group of charges) descri ...
E and classical electrical
polarization density
In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the volumetric density of permanent or induced electric dipole moments in a dielectric material. When a die ...
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
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more ...
and ''ε''
r the
relative static permittivity
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insula ...
. In the
vacuum of classical electromagnetism, the polarization , so and .
See also
*
Casimir effect
In quantum field theory, the Casimir effect (or Casimir force) is a physical force (physics), force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of a field (physics), field. The term Casim ...
*
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental scientific law, law of physics that calculates the amount of force (physics), force between two electric charge, electrically charged particles at rest. This electric for ...
*
Electromagnetic wave equation
*
ISO 31-5
*
Mathematical descriptions of the electromagnetic field
*
Relative permittivity
The relative permittivity (in older texts, dielectric constant) is the permittivity of a material expressed as a ratio with the vacuum permittivity, electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric co ...
*
Sinusoidal plane-wave solutions of the electromagnetic wave equation
*
Wave impedance
*
Vacuum permeability
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum'', ''magnetic constant'') is the magnetic permeability in a classical vacuum. It is a physical constant, conventionally ...
Notes
{{DEFAULTSORT:Vacuum Permittivity
Electromagnetism
Fundamental constants