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51-741: The Heaviside–Lorentz unit system, like the International System of Quantities upon which the SI system is based, but unlike the CGS-Gaussian system, is rationalized , with the result that there are no factors of 4 π appearing explicitly in Maxwell's equations . That this system is rationalized partly explains its appeal in quantum field theory : the Lagrangian underlying the theory does not have any factors of 4 π when this system

102-512: A dimensionless quantity (a term that is still commonly used); all its dimensional exponents are zero and its dimension symbol is 1 {\displaystyle 1} . Such a quantity can be regarded as a derived quantity in the form of the ratio of two quantities of the same dimension. The named dimensionless units " radian " (rad) and " steradian " (sr) are acceptable for distinguishing dimensionless quantities of different kind, respectively plane angle and solid angle . The level of

153-665: A derived result contingent upon the validity of Maxwell's equations. Conversely, as the permittivity is related to the fine-structure constant ( α ), the permeability can be derived from the latter (using the Planck constant , h , and the elementary charge , e ): μ 0 = 2 α e 2 h c = 4 π × α ℏ e 2 c . {\displaystyle \mu _{0}={\frac {2\alpha }{e^{2}}}{\frac {h}{c}}=4\pi \times {\frac {\alpha \hbar }{e^{2}c}}.} In

204-433: A given system of physical quantities is a subset of those quantities, where no base quantity can be expressed in terms of the others, but where every quantity in the system can be expressed in terms of the base quantities. Within this constraint, the set of base quantities is chosen by convention. There are seven ISQ base quantities . The symbols for them, as for other quantities, are written in italics. The dimension of

255-531: A known weight and known separation of the wires, defined in terms of the international standards of mass, length and time in order to produce a standard for the ampere (and this is what the Kibble balance was designed for). In the 2019 revision of the SI , the ampere is defined exactly in terms of the elementary charge and the second , and the value of μ 0 {\displaystyle \mu _{0}}

306-457: A physical quantity does not include magnitude or units. The conventional symbolic representation of the dimension of a base quantity is a single upper-case letter in roman (upright) sans-serif type. A derived quantity is a quantity in a system of quantities that is defined in terms of only the base quantities of that system. The ISQ defines many derived quantities and corresponding derived units . The conventional symbolic representation of

357-493: A quantity is defined as the logarithm of the ratio of the quantity with a stated reference value of that quantity. Within the ISQ it is differently defined for a root-power quantity (also known by the deprecated term field quantity ) and for a power quantity. It is not defined for ratios of quantities of other kinds. Within the ISQ, all levels are treated as derived quantities of dimension 1. Several units for levels are defined by

408-404: Is " magnetic permittivity of vacuum ". See, for example, Servant et al. Variations thereof, such as "permeability of free space", remain widespread. The name "magnetic constant" was briefly used by standards organizations in order to avoid use of the terms "permeability" and "vacuum", which have physical meanings. The change of name had been made because μ 0 was a defined value, and was not

459-587: Is a standard system of quantities used in physics and in modern science in general. It includes basic quantities such as length and mass and the relationships between those quantities. This system underlies the International System of Units (SI) but does not itself determine the units of measurement used for the quantities. The system is formally described in a multi-part ISO standard ISO/IEC 80000 (which also defines many other quantities used in science and technology), first completed in 2009 and subsequently revised and expanded. The base quantities of

510-480: Is an accepted version of this page 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 written as μ 0 (pronounced "mu nought" or "mu zero"). It quantifies the strength of the magnetic field induced by an electric current . Expressed in terms of SI base units , it has

561-463: Is determined experimentally; 4 π  × 0.999 999 999 87 (16) × 10  H⋅m is the 2022 CODATA value in the new system (and the Kibble balance has become an instrument for measuring weight from a known current, rather than measuring current from a known weight). From 1948 to 2019, μ 0 had a defined value (per the former definition of the SI ampere ), equal to: The deviation of

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612-514: Is equal in size to the ampere in the emu system: μ 0 was defined to be 4 π × 10 H / m . Historically, several different systems (including the two described above) were in use simultaneously. In particular, physicists and engineers used different systems, and physicists used three different systems for different parts of physics theory and a fourth different system (the engineers' system) for laboratory experiments. In 1948, international decisions were made by standards organizations to adopt

663-506: Is homogeneous, linear, isotropic, and nondispersive, so that the susceptibilities are constants. Note that The quantities ε SI / ε 0 {\displaystyle \varepsilon ^{\textsf {SI}}/\varepsilon _{0}} , ε HL {\displaystyle \varepsilon ^{\textsf {HL}}} and ε G {\displaystyle \varepsilon ^{\textsf {G}}} are dimensionless, and they have

714-497: Is the speed of light in vacuum, ϕ is the electric potential , A is the magnetic vector potential , F is the Lorentz force acting on a body of charge q and velocity v , ε is the permittivity , χ e is the electric susceptibility , μ is the magnetic permeability , and χ m is the magnetic susceptibility . The electric and magnetic fields can be written in terms of the potentials A and ϕ . The definition of

765-483: Is used. Consequently, electromagnetic quantities in the Heaviside–Lorentz system differ by factors of √ 4 π in the definitions of the electric and magnetic fields and of electric charge . It is often used in relativistic calculations, and are used in particle physics . They are particularly convenient when performing calculations in spatial dimensions greater than three such as in string theory . In

816-589: Is very complicated. Briefly, the basic reason why μ 0 has the value it does is as follows. Ampère's force law describes the experimentally-derived fact that, for two thin, straight, stationary, parallel wires, a distance r apart, in each of which a current I flows, the force per unit length, F m / L , that one wire exerts upon the other in the vacuum of free space would be given by F m L ∝ I 2 r . {\displaystyle {\frac {F_{\mathrm {m} }}{L}}\propto {\frac {I^{2}}{r}}.} Writing

867-492: Is well recognised that the 4 π was an unfortunate and mischievous mistake, the source of many evils. In plain English, the common system of electrical units involves an irrationality of the same kind as would be brought into the metric system of weights and measures, were we to define the unit area to be the area, not of a square with unit side, but of a circle of unit diameter. The constant π would then obtrude itself into

918-1335: The ampere was defined as "that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 × 10 newton per metre of length". This is equivalent to a definition of μ 0 {\displaystyle \mu _{0}} of exactly 4 π × 10   H / m , since F m L = μ 0 2 π ( 1 A ) 2 1 m {\displaystyle {\frac {\mathbf {F} _{\text{m}}}{L}}={\mu _{0} \over 2\pi }\mathrm {(1\,A)^{2} \over {1\,m}} } 2 × 10 − 7   N / m = μ 0 2 π ( 1 A ) 2 1 m {\displaystyle {2\times 10^{-7}\ \mathrm {N/m} }={\mu _{0} \over 2\pi }\mathrm {(1\,A)^{2} \over {1\,m}} } μ 0 = 4 π × 10 − 7  H/m {\displaystyle \mu _{0}=4\pi \times 10^{-7}{\text{ H/m}}} The current in this definition needed to be measured with

969-420: The speed of light in vacuum, c , is related to the magnetic constant and the electric constant (vacuum permittivity) , ε 0 , by the equation: c 2 = 1 μ 0 ε 0 . {\displaystyle c^{2}={1 \over {\mu _{0}\varepsilon _{0}}}.} This relation can be derived using Maxwell's equations of classical electromagnetism in

1020-463: The vacuum permittivity ( ε 0 ) and the vacuum permeability ( μ 0 ). These can be used to convert SI units to their corresponding Heaviside–Lorentz values, as detailed below. For example, SI charge is √ ε 0 L M / T . When one puts ε 0 = 8.854 pF/m , L = 1 cm , M = 1 g , and T = 1 s , this evaluates to 9.409 669 × 10  C , the SI-equivalent of

1071-444: The CGS-Gaussian system, the capacitance of sphere of radius r is r while that of a parallel plate capacitor is ⁠ A / 4 πd ⁠ , where A is the area of the smaller plate and d is their separation. Heaviside , who was an important, though somewhat isolated, early theorist of electromagnetism, suggested in 1882 that the irrational appearance of 4 π in these sorts of relations could be removed by redefining

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1122-493: The Gaussian system ( G ), the Heaviside–Lorentz system ( HL ) uses the length–mass–time dimensions. This means that all of the units of electric and magnetic quantities are expressible in terms of the units of the base quantities length, time and mass. Coulomb's equation, used to define charge in these systems, is F = q 1 q 2 / r in the Gaussian system, and F = q 1 q 2 / (4 πr ) in

1173-452: The HL or Gaussian systems. Below are the expressions for the macroscopic fields D {\displaystyle \mathbf {D} } , P {\displaystyle \mathbf {P} } , H {\displaystyle \mathbf {H} } and M {\displaystyle \mathbf {M} } in a material medium. It is assumed here for simplicity that the medium

1224-419: The HL system. The unit of charge then connects to 1 dyn⋅cm = 1 statC = 4 π HLC , where 'HLC' is the HL unit of charge. The HL quantity q describing a charge is then √ 4 π times larger than the corresponding Gaussian quantity. There are comparable relationships for the other electromagnetic quantities (see below). The commonly used set of units is the called the SI , which defines two constants,

1275-410: The Heaviside–Lorentz unit of charge. This section has a list of the basic formulas of electromagnetism, given in the SI, Heaviside–Lorentz, and Gaussian systems. Here E and D are the electric field and displacement field , respectively, B and H are the magnetic fields , P is the polarization density , M is the magnetization , ρ is charge density , J is current density , c

1326-478: The ISQ. ISO/IEC 80000 defines physical quantities that are measured with the SI units and also includes many other quantities in modern science and technology. The name "International System of Quantities" is used by the General Conference on Weights and Measures (CGPM) to describe the system of quantities that underlie the International System of Units . Vacuum permeability This

1377-402: The SI and classified as "non-SI units accepted for use with the SI units". An example of level is sound pressure level , with the unit of decibel . Units of logarithmic frequency ratio include the octave , corresponding to a factor of 2 in frequency (precisely) and the decade , corresponding to a factor 10. The ISQ recognizes another logarithmic quantity, information entropy , for which

1428-408: The SI, Heaviside–Lorentz system or Gaussian system, the corresponding expressions shown in the table below can be equated and hence substituted for each other. Replace 1 / c 2 {\displaystyle 1/c^{2}} by ε 0 μ 0 {\displaystyle \varepsilon _{0}\mu _{0}} or vice versa. This will reproduce any of

1479-422: The ampere, was then defined as equal to one tenth of the electromagnetic unit of current. In another system, the "rationalized metre–kilogram–second (rmks) system" (or alternatively the "metre–kilogram–second–ampere (mksa) system"), k m is written as μ 0 /2 π , where μ 0 is a measurement-system constant called the "magnetic constant". The value of μ 0 was chosen such that the rmks unit of current

1530-455: The area of a rectangle, and everywhere it should not be, and be a source of great confusion and inconvenience. So it is in the common electrical units, which are truly irrational. Now, to make a mistake is easy and natural to man. But that is not enough. The next thing is to correct it: When a mistake has once been started, it is not necessary to go on repeating it for ever and ever with cumulative inconvenience. — Oliver Heaviside (1893) As in

1581-536: The coherent unit is the natural unit of information (symbol nat). The system is formally described in a multi-part ISO standard ISO/IEC 80000 , first completed in 2009 but subsequently revised and expanded, which replaced standards published in 1992, ISO 31 and ISO 1000 . Working jointly, ISO and IEC have formalized parts of the ISQ by giving information and definitions concerning quantities, systems of quantities, units, quantity and unit symbols, and coherent unit systems, with particular reference to

Heaviside–Lorentz units - Misplaced Pages Continue

1632-417: The constant of proportionality as k m gives F m L = k m I 2 r . {\displaystyle {\frac {F_{\mathrm {m} }}{L}}=k_{\mathrm {m} }{\frac {I^{2}}{r}}.} The form of k m needs to be chosen in order to set up a system of equations, and a value then needs to be allocated in order to define the unit of current. In

1683-440: The corresponding magnetic quantities. The below points are true in both Heaviside–Lorentz and Gaussian systems, but not SI. Textbooks in theoretical physics use Heaviside–Lorentz units nearly exclusively, frequently in their natural form (see below), HL system's conceptual simplicity and compactness significantly clarify the discussions, and it is possible if necessary to convert the resulting answers to appropriate units after

1734-558: The dimension of a derived quantity is the product of powers of the dimensions of the base quantities according to the definition of the derived quantity. The dimension of a quantity is denoted by L a M b T c I d Θ e N f J g {\displaystyle {\mathsf {L}}^{a}{\mathsf {M}}^{b}{\mathsf {T}}^{c}{\mathsf {I}}^{d}{\mathsf {\Theta }}^{e}{\mathsf {N}}^{f}{\mathsf {J}}^{g}} , where

1785-417: The dimensional exponents are positive, negative, or zero. The dimension symbol may be omitted if its exponent is zero. For example, in the ISQ, the quantity dimension of velocity is denoted L T − 1 {\displaystyle {\mathsf {LT}}^{-1}} . The following table lists some quantities defined by the ISQ. A quantity of dimension one is historically known as

1836-412: The fact by inserting appropriate factors of c and ε 0 . Some textbooks on classical electricity and magnetism have been written using Gaussian CGS units, but recently some of them have been rewritten to use SI units. Outside of these contexts, including for example magazine articles on electric circuits, Heaviside–Lorentz and Gaussian CGS units are rarely encountered. To convert any formula between

1887-839: The factor across in the latter identities and substituting, the result is ∇ ⋅ ( 1 ε 0 E HL ) = ( ε 0 ρ HL ) / ε 0 , {\displaystyle \nabla \cdot \left({\frac {1}{\sqrt {\varepsilon _{0}}}}\mathbf {E} ^{\textsf {HL}}\right)=\left({\sqrt {\varepsilon _{0}}}\rho ^{\textsf {HL}}\right)/\varepsilon _{0},} which then simplifies to ∇ ⋅ E HL = ρ HL . {\displaystyle \nabla \cdot \mathbf {E} ^{\textsf {HL}}=\rho ^{\textsf {HL}}.} International System of Quantities The International System of Quantities ( ISQ )

1938-454: The fundamental definitions of current units have been related to the definitions of mass, length, and time units, using Ampère's force law . However, the precise way in which this has "officially" been done has changed many times, as measurement techniques and thinking on the topic developed. The overall history of the unit of electric current, and of the related question of how to define a set of equations for describing electromagnetic phenomena,

1989-632: The magnetic field in terms of A , B = ∇ × A , is the same in all systems of units, but the electric field is E = − ∇ ϕ − ∂ A ∂ t {\textstyle \mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}} in the SI system, but E = − ∇ ϕ − 1 c ∂ A ∂ t {\textstyle \mathbf {E} =-\nabla \phi -{\frac {1}{c}}{\frac {\partial \mathbf {A} }{\partial t}}} in

2040-403: The medium of classical vacuum . Between 1948 and 2018, this relation was used by BIPM (International Bureau of Weights and Measures) and NIST (National Institute of Standards and Technology) as a definition of ε 0 in terms of the defined numerical value for c and, prior to 2018, the defined numerical value for μ 0 . During this period of standards definitions, it was not presented as

2091-474: The mid-late 19th century, electromagnetic measurements were frequently made in either the so-named electrostatic (ESU) or electromagnetic (EMU) systems of units. These were based respectively on Coulomb's and Ampere's Law. Use of these systems, as with to the subsequently developed Gaussian CGS units, resulted in many factors of 4 π appearing in formulas for electromagnetic results, including those without any circular or spherical symmetry. For example, in

Heaviside–Lorentz units - Misplaced Pages Continue

2142-399: The old "electromagnetic (emu)" system of units , defined in the late 19th century, k m was chosen to be a pure number equal to 2, distance was measured in centimetres, force was measured in the cgs unit dyne , and the currents defined by this equation were measured in the "electromagnetic unit (emu) of current", the " abampere ". A practical unit to be used by electricians and engineers,

2193-440: The recommended measured value from the former defined value is within its uncertainty. NIST/CODATA refers to μ 0 as the vacuum magnetic permeability . Prior to the 2019 revision, it was referred to as the magnetic constant . Historically, the constant μ 0 has had different names. In the 1987 IUPAP Red book, for example, this constant was called the permeability of vacuum . Another, now rather rare and obsolete, term

2244-434: The relationship that defines the magnetic H -field in terms of the magnetic B -field. In real media, this relationship has the form: H = B μ 0 − M , {\displaystyle \mathbf {H} ={\mathbf {B} \over \mu _{0}}-\mathbf {M} ,} where M is the magnetization density. In vacuum , M = 0 . In the International System of Quantities (ISQ),

2295-407: The result of experimental measurement (see below). In the new SI system, the permeability of vacuum no longer has a defined value, but is a measured quantity, with an uncertainty related to that of the (measured) dimensionless fine structure constant. In principle, there are several equation systems that could be used to set up a system of electrical quantities and units. Since the late 19th century,

2346-510: The rmks system, and its related set of electrical quantities and units, as the single main international system for describing electromagnetic phenomena in the International System of Units . The magnetic constant μ 0 appears in Maxwell's equations , which describe the properties of electric and magnetic fields and electromagnetic radiation , and relate them to their sources. In particular, it appears in relationship to quantities such as permeability and magnetization density , such as

2397-560: The same numeric value. By contrast, the electric susceptibility χ e {\displaystyle \chi _{\text{e}}} is dimensionless in all the systems, but has different numeric values for the same material: χ e SI = χ e HL = 4 π χ e G {\displaystyle \chi _{\text{e}}^{\textsf {SI}}=\chi _{\text{e}}^{\textsf {HL}}=4\pi \chi _{\text{e}}^{\textsf {G}}} The same statements apply for

2448-884: The specific formulas given in the list above. As an example, starting with the equation ∇ ⋅ E SI = ρ SI / ε 0 , {\displaystyle \nabla \cdot \mathbf {E} ^{\textsf {SI}}=\rho ^{\textsf {SI}}/\varepsilon _{0},} and the equations from the table ε 0   E SI = E HL 1 ε 0 ρ SI = ρ HL . {\displaystyle {\begin{aligned}{\sqrt {\varepsilon _{0}}}\ \mathbf {E} ^{\textsf {SI}}&=\mathbf {E} ^{\textsf {HL}}\\{\frac {1}{\sqrt {\varepsilon _{0}}}}\rho ^{\textsf {SI}}&=\rho ^{\textsf {HL}}\,.\end{aligned}}} Moving

2499-550: The unit kg⋅m⋅s ⋅A . It can be also expressed in terms of SI derived units , N ⋅A . Since the revision of the SI in 2019 (when the values of e and h were fixed as defined quantities), μ 0 is an experimentally determined constant, its value being proportional to the dimensionless fine-structure constant , which is known to a relative uncertainty of 1.6 × 10 , with no other dependencies with experimental uncertainty. Its value in SI units as recommended by CODATA is: The terminology of permeability and susceptibility

2550-430: The units for charges and fields. In his 1893 book Electromagnetic Theory , Heaviside wrote in the introduction: It is not long since it was taken for granted that the common electrical units were correct. That curious and obtrusive constant 4 π was considered by some to be a sort of blessed dispensation, without which all electrical theory would fall to pieces. I believe that this view is now nearly extinct, and that it

2601-747: Was introduced by William Thomson, 1st Baron Kelvin in 1872. The modern notation of permeability as μ and permittivity as ε has been in use since the 1950s. Two thin, straight, stationary, parallel wires, a distance r apart in free space , each carrying a current I , will exert a force on each other. Ampère's force law states that the magnetic force F m per length L is given by | F m | L = μ 0 2 π I 2 | r | . {\displaystyle {\frac {|\mathbf {F} _{\text{m}}|}{L}}={\mu _{0} \over 2\pi }{I^{2} \over |{\boldsymbol {r}}|}.} From 1948 until 2019

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