Electric constant
Encyclopedia
The physical constant
ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, (baseline) physical constant, which is the value of the absolute (not relative) dielectric permittivity of classical vacuum. Its value is:
The ellipsis ‘...’ does not represent an experimental inaccuracy (the value is exact) but the error introduced by truncation of a non-terminating decimal value.
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 (in vacuum) 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.
where c0 is the 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
" (commonly called vacuum permeability). Since μ0 has the defined value 4π × 10−7 H m−1, and c0 has the defined value m·s−1, it follows that ε0 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 ε0, and its value, are explained in more detail below.
Under the proposals to redefine the ampere
as a fixed number of elementary charge
s per second, the electric constant would no longer have an exact fixed value. Instead, it would be defined by the equation
where e is the elementary charge, α is the fine structure constant and h is the Planck constant
. The relative uncertainty in the value would be the same as that of the fine structure constant, currently 6.8.
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 glyph
s for the letter epsilon
.
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 cgs esu system), 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 (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 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 (rmks) equation system, or "metre-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". Clearly, the quantity qs used in the old cgs esu system is related to the new quantity q by
C2·N−1·m−2 (or equivalent units - in practice "farads per metre").
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
(and other ideas) 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
.
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, so one has
where ε is the permittivity
and εr the relative static permittivity. In vacuum, the polarization P = 0, so εr = 1 and ε = ε0.
Physical constant
A physical constant is a physical quantity that is generally believed to be both universal in nature and constant in time. It can be contrasted with a mathematical constant, which is a fixed numerical value but does not directly involve any physical measurement.There are many physical constants in...
ε0, commonly called the vacuum permittivity, permittivity of free space or electric constant is an ideal, (baseline) physical constant, which is the value of the absolute (not relative) dielectric permittivity of classical vacuum. Its value is:
- ε0 ≈ 8.854187817620... × 10−12 F·m−1.
The ellipsis ‘...’ does not represent an experimental inaccuracy (the value is exact) but the error introduced by truncation of a non-terminating decimal value.
This constant relates the units for electric charge
Electric charge
Electric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...
to mechanical quantities such as length and force. For example, the force between two separated electric charges (in vacuum) is given by Coulomb's law
Coulomb's law
Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
:
where q1 and q2 are the charges, and r is the distance between them. Likewise, ε0 appears in Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
, which describe the properties of electric
Electric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
and magnetic
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
fields and electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...
, and relate them to their sources.
Value
The value of ε0 is defined by the formulawhere c0 is the is the defined value for the speed of light
Speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
in classical vacuum in SI units, and μ0 is the parameter that international Standards Organizations call the "magnetic constant
Vacuum permeability
The physical constant μ0, commonly called the vacuum permeability, permeability of free space, or magnetic constant is an ideal, physical constant, which is the value of magnetic permeability in a classical vacuum...
" (commonly called vacuum permeability). Since μ0 has the defined value 4π × 10−7 H m−1, and c0 has the defined value m·s−1, it follows that ε0 has a defined value given approximately by
- ε0 ≈ ... × 10−12 FFaradThe farad is the SI unit of capacitance. The unit is named after the English physicist Michael Faraday.- Definition :A farad is the charge in coulombs which a capacitor will accept for the potential across it to change 1 volt. A coulomb is 1 ampere second...
·m−1 (or AAmpereThe ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...
2·sSecondThe second is a unit of measurement of time, and is the International System of Units base unit of time. It may be measured using a clock....
4·kgKilogramThe kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...
−1·m−3 in SI base unitSI base unitThe International System of Units defines seven units of measure as a basic set from which all other SI units are derived. These SI base units and their physical quantities are:* metre for length...
s, or C2·N−1·m−2 or C·VVoltThe volt is the SI derived unit for electric potential, electric potential difference, and electromotive force. The volt is named in honor of the Italian physicist Alessandro Volta , who invented the voltaic pile, possibly the first chemical battery.- Definition :A single volt is defined as the...
−1·m−1 using other SI coherent units).
The ellipsis (...) does not indicate experimental uncertainty, but the arbitrary termination of a nonrecurring decimal. The historical origins of the electric constant ε0, and its value, are explained in more detail below.
Under the proposals to redefine the ampere
Ampere
The ampere , often shortened to amp, is the SI unit of electric current and is one of the seven SI base units. It is named after André-Marie Ampère , French mathematician and physicist, considered the father of electrodynamics...
as a fixed number of elementary charge
Elementary charge
The elementary charge, usually denoted as e, is the electric charge carried by a single proton, or equivalently, the absolute value of the electric charge carried by a single electron. This elementary charge is a fundamental physical constant. To avoid confusion over its sign, e is sometimes called...
s per second, the electric constant would no longer have an exact fixed value. Instead, it would be defined by the equation
where e is the elementary charge, α is the fine structure constant and h is the Planck constant
Planck constant
The Planck constant , also called Planck's constant, is a physical constant reflecting the sizes of energy quanta in quantum mechanics. It is named after Max Planck, one of the founders of quantum theory, who discovered it in 1899...
. The relative uncertainty in the value would be the same as that of the fine structure constant, currently 6.8.
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 (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 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 glyphGlyph
A glyph is an element of writing: an individual mark on a written medium that contributes to the meaning of what is written. A glyph is made up of one or more graphemes....
s for the letter epsilon
Epsilon
Epsilon is the fifth letter of the Greek alphabet, corresponding phonetically to a close-mid front unrounded vowel . In the system of Greek numerals it has a value of 5. It was derived from the Phoenician letter He...
.
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 of how electromagnetic measurement systems developed.Rationalization of units
The experiments of CoulombCharles-Augustin de Coulomb
Charles-Augustin de Coulomb was a French physicist. He is best known for developing Coulomb's law, the definition of the electrostatic force of attraction and repulsion. The [SI unit] of charge, the coulomb, was named after him....
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
Coulomb's law
Coulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
. 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 cgs esu system), 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
Statcoulomb
The statcoulomb or franklin or electrostatic unit of charge is the physical unit for electrical charge used in the centimetre-gram-second system of units and Gaussian units. It is a derived unit given by...
, 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
Dyne
In physics, the dyne is a unit of force specified in the centimetre-gram-second system of units, a predecessor of the modern SI. One dyne is equal to exactly 10 µN...
. 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 (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 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 (rmks) equation system, or "metre-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". 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 unitC2·N−1·m−2 (or equivalent units - in practice "farads per metre").
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
Ampère's force law
In magnetostatics, the force of attraction or repulsion between two current-carrying wires is often called Ampère's force law...
(and other ideas) to develop Maxwell's equations
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.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
Vacuum permeability
The physical constant μ0, commonly called the vacuum permeability, permeability of free space, or magnetic constant is an ideal, physical constant, which is the value of magnetic permeability in a classical vacuum...
.
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 fieldElectric field
In physics, an electric field surrounds electrically charged particles and time-varying magnetic fields. The electric field depicts the force exerted on other electrically charged objects by the electrically charged particle the field is surrounding...
E and classical electrical polarization density
Polarization density
In classical electromagnetism, polarization density is the vector field that expresses the density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its molecules gain electric dipole moment and the 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, so one has
where ε is the permittivity
Permittivity
In electromagnetism, absolute permittivity is the measure of the resistance that is encountered when forming an electric field in a medium. In other words, permittivity is a measure of how an electric field affects, and is affected by, a dielectric medium. The permittivity of a medium describes how...
and εr the relative static permittivity. In vacuum, the polarization P = 0, so εr = 1 and ε = ε0.
See also
- Casimir effectCasimir effectIn quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field. The typical example is of two uncharged metallic plates in a vacuum, like capacitors placed a few micrometers apart, without any external electromagnetic field...
- Coulomb's lawCoulomb's lawCoulomb's law or Coulomb's inverse-square law, is a law of physics describing the electrostatic interaction between electrically charged particles. It was first published in 1785 by French physicist Charles Augustin de Coulomb and was essential to the development of the theory of electromagnetism...
- Electromagnetic wave equationElectromagnetic wave equationThe electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum...
- ISO 31-5ISO 31-5ISO 31-5 is the part of international standard ISO 31 that defines names and symbols for quantities and units related to electricity and magnetism.Some of its definitions are below, with values taken from :...
- Mathematical descriptions of the electromagnetic fieldMathematical descriptions of the electromagnetic fieldThere are various mathematical descriptions of the electromagnetic field that are used in the study of electromagnetism, one of the four fundamental forces of nature. In this article four approaches are discussed.-Vector field approach:...
- Sinusoidal plane-wave solutions of the electromagnetic wave equationSinusoidal plane-wave solutions of the electromagnetic wave equationSinusoidal plane-wave solutions are particular solutions to the electromagnetic wave equation.The general solution of the electromagnetic wave equation in homogeneous, linear, time-independent media can be written as a linear superposition of plane-waves of different frequencies and...
- Dimensional formula