Heaviside–Lorentz units
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Heaviside–Lorentz units (or Lorentz–Heaviside units) constitute a system of units (particularly electromagnetic units) within CGS, named for
Hendrik Antoon Lorentz Hendrik Antoon Lorentz (; 18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorentz t ...
and
Oliver Heaviside Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
. They share with CGS-Gaussian units the property that the
electric constant 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 consta ...
and
magnetic constant 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, ...
do not appear, having been incorporated implicitly into the electromagnetic quantities by the way they are defined. Heaviside–Lorentz units may be regarded 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, and electric circuits. ...
to use 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 ...
instead. Heaviside–Lorentz units, like SI units but unlike
Gaussian units Gaussian units constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the Gaussian unit system, Gaussian-cgs unit ...
, are ''rationalized'', meaning 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, and electric circuits. ...
.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 these units are rationalized partly explains their appeal in
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
: the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
underlying the theory does not have any factors of in these units. Consequently, Heaviside–Lorentz units differ by factors of in the definitions of the electric and magnetic fields and of
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 ...
. They are 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 fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
. 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 interac ...
.


Motivation

In the mid-late 19th Century, electromagnetic measurements were frequently made in either the so-called
electrostatic Electrostatics is a branch of physics that studies electric charges at rest (static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for amber ...
(ESU) or
electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
(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, even in examples without circular or spherical symmetry. For example, in Gaussian CGS units, the capacitance of sphere of radius is while that of a
parallel plate capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of a c ...
is , where is the area of the plates 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 unit of the charges and fields. In 1893 Heaviside wrote (See also https://wiki.opensourceecology.de/Heaviside_1893)


Length–mass–time framework

As in the Gaussian units (), the Heaviside–Lorentz () units use the ''length–mass–time'' dimensions. This means that all of the electric and magnetic units are expressible in terms of the base units of length, time and mass. Coulomb's equation, used to define charge in these systems, is / in the Gaussian system, and in the system. The unit of charge then connects to , where is the unit of charge. The quantity describing a charge is then 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 system, and it defines two constants, called 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 consta ...
() and the
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, ...
(). These can be used to convert units to their corresponding Heaviside–Lorentz values, as detailed below. For example, charge is When one puts , , , and , this evaluates to , which size 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 SI, Heaviside–Lorentz, and Gaussian units. Here \mathbf and \mathbf are the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
and displacement field, respectively, \mathbf and \mathbf are the
magnetic field A magnetic field is a vector 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 to its own velocity and to ...
s, \mathbf is the
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 ...
, \mathbf 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. Movement within this field is described by direction and is either Axial or Di ...
, \mathbf 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 ...
, \mathbf 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 ar ...
, c is the
speed of light in vacuum 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 f ...
, \phi is the
electric potential The electric potential (also called the ''electric field potential'', potential drop, the electrostatic potential) is defined as the amount of work energy needed to move a unit of electric charge from a reference point to the specific point in ...
, \mathbf is the
magnetic vector potential In classical electromagnetism, magnetic vector potential (often called A) is the vector quantity defined so that its curl is equal to the magnetic field: \nabla \times \mathbf = \mathbf. Together with the electric potential ''φ'', the magnetic v ...
, \mathbf is the
Lorentz force In physics (specifically in electromagnetism) the Lorentz force (or electromagnetic force) is the combination of electric and magnetic force on a point charge due to electromagnetic fields. A particle of charge moving with a velocity in an elect ...
acting on a body of charge q and velocity v, \epsilon 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 ...
, \chi_\text 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 ...
. \mu is the
magnetic permeability In electromagnetism, permeability is the measure of magnetization that a material obtains in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter ''μ''. The term was coined by William ...
, and \chi_\text is the
magnetic susceptibility In electromagnetism, the magnetic susceptibility (Latin: , "receptive"; denoted ) 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 ap ...
.


Maxwell's equations

The electric and magnetic fields can be written in terms of the potentials \mathbf and \phi. The definition of the magnetic field in terms of \mathbf, \mathbf = \nabla \times \mathbf, is the same in all systems of units, but the electric field is \mathbf = -\nabla\phi-\frac in the system, but \mathbf = -\nabla\phi-\frac \frac in the or 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 \epsilon^\textsf/\epsilon_0, \epsilon^\textsf and \epsilon^\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 ''different numeric values'' 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 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 \mathbf 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 system. (In addition and are overdetermined, because = .) The below points are true in both and systems, but not . * The electric and magnetic fields \mathbf and \mathbf have the same dimensions in the 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 units mathbf/ mathbf/math> is . * Giving the \mathbf and \mathbf 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, \phi and \mathbf have the same dimensions and are the four components of the 4-potential. * The fields \mathbf, \mathbf, \mathbf and \mathbf also have the same dimensions as \mathbf and \mathbf. In a vacuum, any expression involving \mathbf can simply be recast as the same expression with \mathbf. In units, \mathbf and \mathbf have the same units, as do \mathbf and \mathbf, but they have different units from each other and from \mathbf and \mathbf.


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 , are frequently of rather inconvenient sizes. The ampere (coulomb/sec) 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 300 V, a value which is larger than most commonly encountered potentials. The henry, the unit for
inductance Inductance is the tendency of an electrical conductor to oppose a change in the electric current flowing through it. The flow of electric current creates a magnetic field around the conductor. The field strength depends on the magnitude of the ...
is already on the large side compared to most inductors, the 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), because the 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 c and \epsilon_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 units. Outside of these contexts, including for example magazine articles on electric circuits, and units are rarely encountered.


Translating expressions and formulas between systems

To convert any expression or formula between SI, Heaviside–Lorentz or Gaussian systems, the corresponding quantities shown in the table below can be directly equated and hence substituted. 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/\epsilon_0 , and the equations from the table : \sqrt \ \mathbf^\textsf = \mathbf^\textsf : \frac \rho^\textsf = \rho^\textsf , 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)/\epsilon_0 , which then simplifies to :\nabla \cdot \mathbf^\textsf = \rho^\textsf .


Notes


References

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