Vacuum permittivity, commonly denoted (pronounced "epsilon nought" or "epsilon zero"), is the value of the
absolute dielectric permittivity of
classical vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
. 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. Its
CODATA value is:
: (
farads per
meter
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its pref ...
), with a relative uncertainty of
It is a measure of how dense of an
electric field is "permitted" to form in response to electric charges, and relates the units for
electric charge
Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...
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 law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
:
:
Here, ''q''
1 and ''q''
2 are the charges, ''r'' is the distance between their centres, and the value of the constant fraction
(known as the
Coulomb constant, ''k''
e) is approximately 9 × 10
9 N⋅m
2⋅C
−2. 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, and electric circuits.
...
, which describe the properties of
electric
Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as described by ...
and
magnetic fields and
electromagnetic radiation
In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
, and relate them to their sources. In electrical engineering, ''ε''
0 itself is used as a unit to quantify the permittivity of various dielectric 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 that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in
classical vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
in
SI units,
and ''μ''
0 is the parameter that international Standards Organizations call the "
magnetic constant" (commonly called vacuum permeability or the permeability of free space). Since ''μ''
0 has an approximate value 4π × 10
−7 H/
m, and ''c'' has the ''defined'' value m⋅s
−1, it follows that ''ε''
0 can be expressed numerically as
:
:(or
A2⋅
s4⋅
kg−1⋅
m−3 in
SI base units, or
C2⋅
N−1⋅
m−2 or
C⋅
V−1⋅
m−1 using other SI coherent units).
[.]
The historical origins of the electric constant ''ε''
0, and its value, are explained in more detail below.
Redefinition of the SI units
The
ampere was redefined by defining the
elementary charge as an exact number of coulombs as from 20 May 2019,
with the effect that the vacuum electric permittivity no longer has an exactly determined value in SI units. The value of the electron charge became a numerically defined quantity, not measured, making ''μ''
0 a measured quantity. Consequently, ''ε''
0 is not exact. As before, it is defined by the equation , and is thus determined by the value of ''μ''
0, the
magnetic vacuum permeability which in turn is determined by the experimentally determined dimensionless
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between el ...
''α'':
:
with ''e'' being the
elementary charge, ''h'' being the
Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, and ''c'' being the
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
in
vacuum
A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
, each with exactly defined values. The relative uncertainty in the value of ''ε''
0 is therefore the same as that for the dimensionless
fine-structure constant
In physics, the fine-structure constant, also known as the Sommerfeld constant, commonly denoted by (the Greek letter ''alpha''), is a fundamental physical constant which quantifies the strength of the electromagnetic interaction between 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 is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Physicists often dis ...
"
[B. E. A. Saleh and M. C. Teich, ''Fundamentals of Photonics'' (Wiley, 1991)] are widespread. Standards Organizations worldwide now use "electric constant" as a uniform term for this quantity,
and official standards documents have adopted the term (although they continue to list the older terms as synonyms).
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 electric permittivity of a vacuum. A dielectric is an insulating material, and the dielectric constant of an insul ...
''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 insulat ...
.
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.
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 the
Coulomb constant. 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 "centimeter–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 franklin (Fr) or statcoulomb (statC) electrostatic unit of charge (esu) is the physical unit for electrical charge used in the cgs-esu and Gaussian units. It is a derived unit given by
: 1 statC = 1 dyn1/2⋅cm = 1 cm3/2⋅g1/2⋅s−1.
Tha ...
, is such that two units, a distance of 1 centimeter apart, repel each other with a force equal to the cgs unit of force, the
dyne
The dyne (symbol: dyn; ) is a derived unit of force specified in the centimetre–gram–second (CGS) system of units, a predecessor of the modern SI.
History
The name dyne was first proposed as a CGS unit of force in 1873 by a Committee of ...
. 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 meter–kilogram–second (rmks) equation system, or "meter–kilogram–second–ampere (mksa)" equation system. This is the system used to define the SI units.
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:
:
.
Determination of a value for ''ε''0
One now adds the requirement that one wants force to be measured in newtons, distance in meters, 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 meter").
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, the force of attraction or repulsion between two current-carrying wires (see first figure below) is often called Ampère's force law. The physical origin of this force is that each wire generates a magnetic field, followin ...
(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, and electric circuits.
...
, 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''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
''.
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) or electric induction is a vector field that appears in Maxwell's equations. It accounts for the effects of free and bound charge within materials. "D" stands for "displacement", as in ...
D in terms of the
electric field E and classical electrical
polarization density
In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is ...
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 ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in ...
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 insulat ...
. In the
vacuum of classical electromagnetism, the polarization , so and .
See also
*
Casimir effect
In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who pr ...
*
Coulomb's law
Coulomb's inverse-square law, or simply Coulomb's law, is an experimental law of physics that quantifies the amount of force between two stationary, electrically charged particles. The electric force between charged bodies at rest is convention ...
*
Electromagnetic wave equation
*
ISO 31-5
*
Mathematical descriptions of the electromagnetic field
There are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equation ...
*
Relative 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 insul ...
*
Sinusoidal plane-wave solutions of the electromagnetic wave equation
*
Wave impedance The wave impedance of an electromagnetic wave is the ratio of the transverse components of the electric and magnetic fields (the transverse components being those at right angles to the direction of propagation). For a transverse-electric-magnetic ...
*
Vacuum permeability
The vacuum magnetic permeability (variously ''vacuum permeability'', ''permeability of free space'', ''permeability of vacuum''), also known as the magnetic constant, is the magnetic permeability in a classical vacuum. It is a physical constant, ...
Notes
{{DEFAULTSORT:Vacuum Permittivity
Electromagnetism
Fundamental constants