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Heaviside–Lorentz units (or Lorentz–Heaviside units) constitute a system of units and quantities that extends the CGS with a particular set of equations that defines electromagnetic quantities, named for
Oliver Heaviside Oliver Heaviside ( ; 18 May 1850 – 3 February 1925) was an English mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vector calculus, an ...
and
Hendrik Antoon Lorentz Hendrik Antoon Lorentz ( ; ; 18 July 1853 – 4 February 1928) was a Dutch theoretical physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for their discovery and theoretical explanation of the Zeeman effect. He derived ...
. They share with the CGS-Gaussian system that the electric constant and
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 ...
do not appear in the defining equations for electromagnetism, having been incorporated implicitly into the electromagnetic quantities. Heaviside–Lorentz units may be thought of as normalizing and , while at the same time revising
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 ...
to use 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 ...
instead. The Heaviside–Lorentz unit system, like the
International System of Quantities The International System of Quantities (ISQ) is a standard system of Quantity, 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 ...
upon which the SI system is based, but unlike the CGS-Gaussian system, is ''rationalized'', with the result that there are no factors of appearing explicitly 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 ...
.Kowalski, Ludwik, 1986,
A Short History of the SI Units in Electricity
" ''The Physics Teacher'' 24(2): 97–99
Alternate web link (subscription required)
/ref> That this system is rationalized partly explains its appeal in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
: the Lagrangian underlying the theory does not have any factors of when this system is used. Consequently, electromagnetic quantities in the Heaviside–Lorentz system differ by factors of in the definitions of the electric and magnetic fields and of
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 ...
. It is often used in relativistic calculations,As used by Einstein, such as in his book: and are used in
particle physics Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
. They are particularly convenient when performing calculations in spatial dimensions greater than three such as in
string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...
.


Motivation

In the mid-late 19th century, electromagnetic measurements were frequently made in either the ''so-named''
electrostatic Electrostatics is a branch of physics that studies slow-moving or stationary electric charges. Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word (), mean ...
(ESU) or
electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...
(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 appearing in formulas for electromagnetic results, including those without any circular or spherical symmetry. For example, in the CGS-Gaussian system, the capacitance of sphere of radius is while that of a parallel plate capacitor is , where is the area of the smaller plate and is their separation. Heaviside, who was an important, though somewhat isolated, early theorist of electromagnetism, suggested in 1882 that the ''irrational'' appearance of in these sorts of relations could be removed by redefining the units for charges and fields. In his 1893 book ''Electromagnetic Theory'', : Alternate source for the same text: Heaviside wrote in the introduction:


Length–mass–time framework

As in the Gaussian system (), the Heaviside–Lorentz system () 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 in the Gaussian system, and in the HL system. The unit of charge then connects to , where 'HLC' is the HL unit of charge. The HL quantity describing a charge is then 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, the
vacuum permittivity 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 const ...
() and the
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 ...
(). These can be used to convert SI units to their corresponding Heaviside–Lorentz values, as detailed below. For example, SI charge is . When one puts , , , and , this evaluates to , the SI-equivalent of the Heaviside–Lorentz unit of charge.


Comparison of Heaviside–Lorentz with other systems of units

This section has a list of the basic formulas of electromagnetism, given in the SI, Heaviside–Lorentz, and Gaussian systems. Here and are 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 ...
and displacement field, respectively, and are the
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
s, is the polarization density, is the
magnetization In classical electromagnetism, magnetization is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. Accordingly, physicists and engineers usually define magnetization as the quanti ...
, is
charge density In electromagnetism, charge density is the amount of electric charge per unit length, surface area, or volume. Volume charge density (symbolized by the Greek letter ρ) is the quantity of charge per unit volume, measured in the SI system in co ...
, is
current density In electromagnetism, current density is the amount of charge per unit time that flows through a unit area of a chosen cross section. The current density vector is defined as a vector whose magnitude is the electric current per cross-sectional ...
, is 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, is the
electric potential Electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as electric potential energy per unit of electric charge. More precisely, electric potential is the amount of work (physic ...
, is the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often denoted A) is the vector quantity defined so that its curl is equal to the magnetic field, B: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the ma ...
, is the
Lorentz force In electromagnetism, the Lorentz force is the force exerted on a charged particle by electric and magnetic fields. It determines how charged particles move in electromagnetic environments and underlies many physical phenomena, from the operation ...
acting on a body of charge and velocity , 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 ...
, is the
electric susceptibility In electricity (electromagnetism), the electric susceptibility (\chi_; Latin: ''susceptibilis'' "receptive") is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applie ...
, is the
magnetic permeability In electromagnetism, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. It is the ratio of the magnetic ...
, and is the
magnetic susceptibility In electromagnetism, the magnetic susceptibility (; denoted , chi) is a measure of how much a material will become magnetized in an applied magnetic field. It is the ratio of magnetization (magnetic moment per unit volume) to the applied magnet ...
.


Maxwell's equations

The electric and magnetic fields can be written in terms of the potentials and . The definition of the magnetic field in terms of , , is the same in all systems of units, but the electric field is \mathbf = -\nabla\phi-\frac in the SI system, but \mathbf = -\nabla\phi-\frac \frac in the HL or Gaussian systems.


Other basic laws


Dielectric and magnetic materials

Below are the expressions for the macroscopic fields \mathbf, \mathbf, \mathbf and \mathbf in a material medium. It is assumed here for simplicity that the medium is homogeneous, linear, isotropic, and nondispersive, so that the susceptibilities are constants. Note that the quantities \varepsilon^\textsf/\varepsilon_0, \varepsilon^\textsf and \varepsilon^\textsf are dimensionless, and they have the same numeric value. By contrast, the
electric susceptibility In electricity (electromagnetism), the electric susceptibility (\chi_; Latin: ''susceptibilis'' "receptive") is a dimensionless proportionality constant that indicates the degree of polarization of a dielectric material in response to an applie ...
\chi_\text is dimensionless in all the systems, but has for the same material: \chi_\text^\textsf = \chi_\text^\textsf = 4\pi \chi_\text^\textsf The same statements apply for the corresponding magnetic quantities.


Advantages and disadvantages of Heaviside–Lorentz units


Advantages

* The formulas above are clearly simpler in units compared to either or Gaussian units. As Heaviside proposed, removing the from the Gauss law and putting it in the Force law considerably reduces the number of places the appears compared to Gaussian CGS units. * Removing the explicit from the Gauss law makes it clear that the inverse-square force law arises by the field spreading out over the surface of a sphere. This allows a straightforward extension to other dimensions. For example, the case of long, parallel wires extending straight in the direction can be considered a two-dimensional system. Another example is in string theory, where more than three spatial dimensions often need to be considered. * The equations are free of the constants and that are present in the SI system. (In addition and are overdetermined, because .) The below points are true in both Heaviside–Lorentz and Gaussian systems, but not SI. * The electric and magnetic fields and have the same dimensions in the Heaviside–Lorentz system, meaning it is easy to recall where factors of go in the Maxwell equation. Every time derivative comes with a , which makes it dimensionally the same as a space derivative. In contrast, in SI units is . * Giving the and fields the same dimension makes the assembly into the
electromagnetic tensor In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. Th ...
more transparent. There are no factors of that need to be inserted when assembling the tensor out of the three-dimensional fields. Similarly, and have the same dimensions and are the four components of the 4-potential. * The fields , , , and also have the same dimensions as and . For vacuum, any expression involving can simply be recast as the same expression with . In SI units, and have the same units, as do and , but they have different units from each other and from and .


Disadvantages

* Despite Heaviside's urgings, it proved difficult to persuade people to switch from the established units. He believed that if the units were changed, " d style instruments would very soon be in a minority, and then disappear ...". Persuading people to switch was already difficult in 1893, and in the meanwhile there have been more than a century's worth of additional textbooks printed and voltmeters built. * Heaviside–Lorentz units, like the Gaussian CGS units by which they generally differ by a factor of about 3.5, are frequently of rather inconvenient sizes. The ampere (coulomb/second) is reasonable unit for measuring currents commonly encountered, but the ESU/s, as demonstrated above, is far too small. The Gaussian CGS unit of electric potential is named a statvolt. It is about , a value which is larger than most commonly encountered potentials. The henry, the SI unit for
inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The electric current produces a magnetic field around the conductor. The magnetic field strength depends on the magnitude of the ...
is already on the large side compared to most inductors; the Gaussian unit is 12 orders of magnitude larger. * A few of the Gaussian CGS units have names; none of the Heaviside–Lorentz units do. Textbooks in theoretical physics use Heaviside–Lorentz units nearly exclusively, frequently in their natural form (see below), 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 the fact by inserting appropriate factors of and . 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.


Translating formulas between systems

To convert any formula between 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 by \varepsilon_0 \mu_0 or vice versa. This will reproduce any of the specific formulas given in the list above. As an example, starting with the equation \nabla \cdot \mathbf^\textsf = \rho^\textsf/\varepsilon_0 , and the equations from the table \begin \sqrt \ \mathbf^\textsf &= \mathbf^\textsf \\ \frac \rho^\textsf &= \rho^\textsf \,. \end Moving the factor across in the latter identities and substituting, the result is \nabla \cdot \left(\frac \mathbf^\textsf\right) = \left(\sqrt \rho^\textsf\right)/\varepsilon_0 , which then simplifies to \nabla \cdot \mathbf^\textsf = \rho^\textsf .


Notes


References

{{DEFAULTSORT:Heaviside-Lorentz units Special relativity Electromagnetism Hendrik Lorentz