View source for Heaviside–Lorentz units

{{more citations needed|date=November 2012}}

'''Heaviside–Lorentz units''' (or '''Lorentz–Heaviside units''') constitute a system of units and quantities that extends the [[Centimetre–gram–second system of units|CGS]] with a particular set of equations that defines electromagnetic quantities, named for [[Oliver Heaviside]] and [[Hendrik Antoon Lorentz]]. They share with the [[Gaussian units|CGS-Gaussian system]] that the [[electric constant]] {{math|1=''ε''<sub>0</sub>}} and [[magnetic constant]] {{math|1=''µ''<sub>0</sub>}} 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 {{math|1=''ε''<sub>0</sub> = 1}} and {{math|1=''µ''<sub>0</sub> = 1}}, while at the same time revising [[Maxwell's equations]] to use the [[speed of light]] {{math|1=''c''}} instead.<ref>{{cite journal |last=Silsbee |first=Francis |title=Systems of Electrical Units |journal=Journal of Research of the National Bureau of Standards Section C |volume=66C |issue=2 |pages=137–183 |date=April–June 1962 |doi =10.6028/jres.066C.014 |doi-access=free }}</ref>

The Heaviside–Lorentz unit system, like the [[International System of Quantities]] upon which the [[SI]] system is based, but unlike the [[Gaussian units|CGS-Gaussian system]], is ''rationalized'', with the result that there are no factors of {{math|1=4''π''}} appearing explicitly in [[Maxwell's equations]].<ref name=Kowalski>Kowalski, Ludwik, 1986, "[http://alpha.montclair.edu/~kowalskiL/SI/SI_PAGE.HTML A Short History of the SI Units in Electricity], {{webarchive|url=https://web.archive.org/web/20090429035624/http://alpha.montclair.edu/~kowalskiL/SI/SI_PAGE.HTML |date=2009-04-29 }}"  ''The Physics Teacher'' 24(2): 97–99. [https://dx.doi.org/10.1119/1.2341955 Alternate web link (subscription required)]</ref> That this system is rationalized partly explains its appeal in [[quantum field theory]]: the [[Lagrangian (field theory)|Lagrangian]] underlying the theory does not have any factors of {{math|1=4''π''}} when this system is used.<ref name=Littlejohn>{{cite web|author-link1=Robert Grayson Littlejohn| url=http://bohr.physics.berkeley.edu/classes/221/1112/notes/emunits.pdf | title=Gaussian, SI and Other Systems of Units in Electromagnetic Theory | work=Physics 221A, University of California, Berkeley lecture notes  | author=Littlejohn, Robert | date=Fall 2011 | access-date=2008-05-06 }}</ref> Consequently, electromagnetic quantities in the Heaviside–Lorentz system differ by factors of {{math|1={{sqrt|4''π''}}}} in the definitions of the electric and magnetic fields and of [[electric charge]].  It is often used in [[Special relativity|relativistic]] calculations,<ref group="note">As used by Einstein, such as in his book: {{cite book |url=https://archive.org/details/B-001-001-738/page/n0 | title =The Meaning of Relativity| last =Einstein | first =Albert | publisher =Princeton University Press (2005)| year =2005|pages=21 ff| edition =1956, 5th}}</ref> and are used in [[particle physics]].  They are particularly convenient when performing calculations in spatial dimensions greater than three such as in [[string theory]].

== Motivation ==

In the mid-late 19th century, electromagnetic measurements were frequently made in either the ''so-named'' [[Electrostatic units|electrostatic]] (ESU) or [[Electromagnetic units|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 {{math| 4''π'' }} 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 {{mvar|r}} is {{mvar|r}} while that of a [[Capacitor#Parallel-plate capacitor|parallel plate capacitor]] is {{big|{{math| {{sfrac| ''A'' | 4 ''π d'' }} }} }}, where {{mvar|A}} is the area of the smaller plate and {{mvar|d}} is their separation.

[[Oliver Heaviside|Heaviside]], who was an important, though somewhat isolated,{{citation needed|reason=Heaviside apparently actively campaigned and was involved in the development of the unit systems of the time; how true is this claim?|date=February 2023}} early theorist of electromagnetism, suggested in 1882 that the ''irrational'' appearance of {{math|4''π''}} in these sorts of relations could be removed by redefining the units for charges and fields.<ref name=Heaviside1>
{{cite magazine
 |last=Heaviside |first=O. |author-link=Oliver Heaviside
 |date=1882-11-18  |df=dmy-all
 |title=The relations between magnetic force and electric current
 |magazine=[[The Electrician]]
 |location=London, UK
}}
</ref><ref name=ETHW>
{{cite web
 |title=System of measurement units
 |date=24 April 2012
 |type=wiki 
 |website=Engineering and Technology History (ETHW.org)
 |url=https://ethw.org/System_of_Measurement_Units
 |access-date=2021-12-23  |df=dmy-all
}}
</ref>
In his 1893 book ''Electromagnetic Theory'',<ref name=Heaviside2>
{{cite book
 |last=Heaviside |first=Oliver |author-link=Oliver Heaviside
 |year=1893
 |title=Electromagnetic Theory
 |publisher=The D. van Nostrand Company
 |place=London, UK
 |volume=1 |page={{mvar|xi}}
 |url=https://books.google.com/books?id=_Zk3AAAAMAAJ |via=Google Books
}}
: Alternate source for the same text:
{{cite web
 |title=Heaviside 1893: ''Electromagnetic Theory'' {{nobr|volume 1 ...}}
 |type=[[optical character recognition|OCR]] text
 |department=wiki
 |website=Open Source Ecology Germany (wiki.opensourceecology.de)
 |url=https://wiki.opensourceecology.de/Heaviside_1893
}}
</ref>
Heaviside wrote in the introduction:
{{blockquote|
It is not long since it was taken for granted that the common electrical units were correct. That curious and obtrusive constant {{math|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 is well recognised that the {{math|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 {{math|''π''}} would then obtrude itself into 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 {{nobr|inconvenience. — Oliver Heaviside (1893)<ref name=Heaviside2/>}}
}}

== Length–mass–time framework ==

As in the Gaussian system ({{abbr|G|Gaussian unit system}}), the Heaviside–Lorentz system ({{abbr|HL|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 {{math|1=''F'' = ''q''{{su|p=G|b=1|lh=1em}}''q''{{su|p=G|b=2|lh=1em}} / ''r''{{i sup|2}}}} in the Gaussian system, and {{math|1=''F'' = ''q''{{su|p=HL|b=1|lh=1em}}''q''{{su|p=HL|b=2|lh=1em}} / (4''πr''{{i sup|2}})}} in the HL system. The unit of charge then connects to {{math|1=1 dyn⋅cm{{sup|2}} = 1 statC{{sup|2}} = 4''π'' HLC{{sup|2}} }}, where 'HLC' is the HL unit of charge. The HL quantity {{math|''q''{{sup|HL}}}} describing a charge is then {{math|{{sqrt|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 [[International System of Units|SI]], which defines two constants, the [[vacuum permittivity]] ({{math|''ε''{{sub|0}}}}) and the [[vacuum permeability]] ({{math|''μ''{{sub|0}}}}). These can be used to convert SI units to their corresponding Heaviside–Lorentz values, as detailed below.  For example, SI charge is {{math|{{sqrt|''ε''{{sub|0}}''L''{{sup|3}}''M'' / ''T''{{sup|2}}}}}}.  When one puts {{math|1=''ε''{{sub|0}} = 8.854 pF/m}}, {{math|1=''L'' = 1 cm}}, {{math|1=''M'' = 1 g}}, and {{math|''T'' {{=}} 1 s}}, this evaluates to {{val|9.409669|e=-11|ul=C}}, 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 {{math|'''E'''}} and {{math|'''D'''}} are the [[electric field]] and [[Electric displacement field|displacement field]], respectively,
{{math|'''B'''}} and {{math|'''H'''}} are the [[magnetic field]]s,
{{math|'''P'''}} is the [[polarization density]],
{{math|'''M'''}} is the [[magnetization]],
{{mvar|ρ}} is [[charge density]],
{{math|'''J'''}} is [[current density]],
{{mvar|c}} is the [[speed of light]] in vacuum,
{{mvar|ϕ}} is the [[electric potential]],
{{math|'''A'''}} is the [[magnetic vector potential]],
{{math|'''F'''}} is the [[Lorentz force]] acting on a body of charge {{mvar|q}} and velocity {{mvar|v}},
{{mvar|ε}} is the [[permittivity]],
{{math|χ{{sub|e}}}} is the [[electric susceptibility]],
{{mvar|μ}} is the [[magnetic permeability]], and
{{math|χ{{sub|m}}}} is the [[magnetic susceptibility]].

=== Maxwell's equations ===
{{main|Maxwell's equations}}

{| class="wikitable plainrowheaders"
|+ Maxwell's equations in SI, Heaviside–Lorentz, and Gaussian quantities
|-
! scope="col" | Name
! scope="col" | SI quantities
! scope="col" | Heaviside–Lorentz quantities
! scope="col" | [[Gaussian units|Gaussian]] quantities
|-
! scope="row" | [[Gauss's law]]<br />(macroscopic)
| <math>\nabla \cdot \mathbf{D}^\textsf{SI} = \rho_\text{f}^\textsf{SI}</math>
| <math>\nabla \cdot \mathbf{D}^\textsf{HL} = \rho_\text{f}^\textsf{HL}</math>
| <math>\nabla \cdot \mathbf{D}^\textsf{G} = 4\pi\rho_\text{f}^\textsf{G}</math>
|-
! scope="row" | [[Gauss's law]]<br />(microscopic)
| <math>\nabla \cdot \mathbf{E}^\textsf{SI} = \rho^\textsf{SI}/\varepsilon_0</math>
| <math>\nabla \cdot \mathbf{E}^\textsf{HL} = \rho^\textsf{HL}</math>
| <math>\nabla \cdot \mathbf{E}^\textsf{G} = 4\pi\rho^\textsf{G}</math>
|-
! scope="row" | [[Gauss's law for magnetism]]
| <math>\nabla \cdot \mathbf{B}^\textsf{SI} = 0</math>    
| <math>\nabla \cdot \mathbf{B}^\textsf{HL} = 0</math>    
| <math>\nabla \cdot \mathbf{B}^\textsf{G} = 0</math>    
|-
! scope="row" | Maxwell–Faraday equation<br />([[Faraday's law of induction]])
| <math>\nabla \times \mathbf{E}^\textsf{SI} = -\frac{\partial \mathbf{B}^\textsf{SI}} {\partial t}</math>
| <math>\nabla \times \mathbf{E}^\textsf{HL} = -\frac{1}{c}\frac{\partial \mathbf{B}^\textsf{HL}} {\partial t}</math>          
| <math>\nabla \times \mathbf{E}^\textsf{G} = -\frac{1}{c}\frac{\partial \mathbf{B}^\textsf{G}} {\partial t}</math>     
|-
! scope="row" | [[Ampère–Maxwell equation]]<br /> (macroscopic)
| <math>\nabla \times \mathbf{H}^\textsf{SI} = \mathbf{J}_{\text{f}}^\textsf{SI} + \frac{\partial \mathbf{D}^\textsf{SI}} {\partial t}</math>
| <math>\nabla \times \mathbf{H}^\textsf{HL} = \frac{1}{c}\mathbf{J}_\text{f}^\textsf{HL} + \frac{1}{c}\frac{\partial \mathbf{D}^\textsf{HL}} {\partial t}</math>
| <math>\nabla \times \mathbf{H}^\textsf{G} = \frac{4\pi}{c}\mathbf{J}_\text{f}^\textsf{G} + \frac{1}{c}\frac{\partial \mathbf{D}^\textsf{G}} {\partial t}</math>
|-
! scope="row" | [[Ampère–Maxwell equation]]<br /> (microscopic)
| <math>\nabla \times \mathbf{B}^\textsf{SI} = \mu_0 \left( \mathbf{J}^\textsf{SI} + \varepsilon_0\frac{\partial \mathbf{E}^\textsf{SI}} {\partial t} \right)</math>
| <math>\nabla \times \mathbf{B}^\textsf{HL} = \frac{1}{c}\mathbf{J}^\textsf{HL} + \frac{1}{c}\frac{\partial \mathbf{E}^\textsf{HL}} {\partial t}</math>
| <math>\nabla \times \mathbf{B}^\textsf{G} = \frac{4\pi}{c}\mathbf{J}^\textsf{G} + \frac{1}{c}\frac{\partial \mathbf{E}^\textsf{G}} {\partial t}</math>
|}

The electric and magnetic fields can be written in terms of the potentials {{math|'''A'''}} and {{mvar|ϕ}}.
The definition of the magnetic field in terms of {{math|'''A'''}}, {{math|1='''B''' = ∇ × '''A'''}}, is the same in all systems of units, but the electric field is <math display="inline">\mathbf{E} = -\nabla\phi-\frac{\partial \mathbf{A}}{\partial t}</math> in the SI system, but <math display="inline">\mathbf{E} = -\nabla\phi-\frac{1}{c} \frac{\partial \mathbf{A}}{\partial t}</math> in the HL or Gaussian systems.

=== Other basic laws ===
{| class="wikitable plainrowheaders"
|+ Other electrostatic laws in SI, Heaviside–Lorentz, and Gaussian quantities
|-
! scope="col" | Name
! scope="col" | SI quantities
! scope="col" | Heaviside–Lorentz quantities
! scope="col" | Gaussian quantities
|-
! scope="row" | [[Lorentz force]]
| <math>\mathbf{F} = q^\textsf{SI}\left(\mathbf{E}^\textsf{SI}+\mathbf{v}\times\mathbf{B}^\textsf{SI}\right)</math>
| <math>\mathbf{F} = q^\textsf{HL}\left(\mathbf{E}^\textsf{HL}+\frac{1}{c}\mathbf{v}\times\mathbf{B}^\textsf{HL}\right)</math>
| <math>\mathbf{F} = q^\textsf{G}\left(\mathbf{E}^\textsf{G}+\frac{1}{c}\mathbf{v}\times\mathbf{B}^\textsf{G}\right)</math>
|-
! scope="row" | [[Coulomb's law]]
| <math>\mathbf{F} = \frac{1}{4\pi\varepsilon_0}\frac{q_1^\textsf{SI} q_2^\textsf{SI}}{r^2} \mathbf{\hat r}</math> <br />
| <math>\mathbf{F} = \frac{1}{4\pi}\frac{q_1^\textsf{HL} q_2^\textsf{HL}}{r^2} \mathbf{\hat r}</math>
| <math>\mathbf{F} = \frac{q_1^\textsf{G} q_2^\textsf{G}}{r^2} \mathbf{\hat r}</math>
|-
! scope="row" | Electric field of<br />[[Coulomb's law|stationary point charge]]
| <math>\mathbf{E}^\textsf{SI} = \frac{1}{4\pi\varepsilon_0}\frac{q^\textsf{SI}}{r^2} \mathbf{\hat r}</math>
| <math>\mathbf{E}^\textsf{HL} = \frac{1}{4\pi}\frac{q^\textsf{HL}}{r^2} \mathbf{\hat r}</math>
| <math>\mathbf{E}^\textsf{G} = \frac{q^\textsf{G}}{r^2} \mathbf{\hat r}</math>
|-
! scope="row" | [[Biot–Savart law]]
| <math> \mathbf{B}^\textsf{SI} = \frac{\mu_0}{4\pi} \oint\frac{I^\textsf{SI} d\mathbf{l} \times \mathbf{\hat r}}{r^2}</math>
| <math> \mathbf{B}^\textsf{HL} = \frac{1}{4\pi c} \oint\frac{I^\textsf{HL} d\mathbf{l} \times \mathbf{\hat r}}{r^2}</math>
| <math> \mathbf{B}^\textsf{G} = \frac{1}{c} \oint\frac{I^\textsf{G} d\mathbf{l} \times \mathbf{\hat r}}{r^2}</math>
|}

=== Dielectric and magnetic materials ===

Below are the expressions for the macroscopic fields <math>\mathbf{D}</math>, <math>\mathbf{P}</math>, <math>\mathbf{H}</math> and <math>\mathbf{M}</math> 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.
{| class="wikitable"
|-
! scope="col" colspan=2 | SI quantities
! scope="col" colspan=2 | Heaviside–Lorentz quantities
! scope="col" colspan=2 | Gaussian quantities
|-
! scope="col" | Dielectric
! scope="col" | Magnetic
! scope="col" | Dielectric
! scope="col" | Magnetic
! scope="col" | Dielectric
! scope="col" | Magnetic
|-
| <math>\begin{align}\mathbf{D}^\textsf{SI} &= \varepsilon_0 \mathbf{E}^\textsf{SI}+\mathbf{P}^\textsf{SI} \\ &= \varepsilon\mathbf{E}^\textsf{SI}\end{align}</math>
| <math>\begin{align}\mathbf{B}^\textsf{SI} &= \mu_0 (\mathbf{H}^\textsf{SI}+\mathbf{M}^\textsf{SI}) \\ &= \mu^\textsf{SI}\mathbf{H}^\textsf{SI}\end{align}</math>
| <math>\begin{align}\mathbf{D}^\textsf{HL} &= \mathbf{E}^\textsf{HL}+\mathbf{P}^\textsf{HL} \\ &= \varepsilon\mathbf{E}^\textsf{HL}\end{align}</math>
| <math>\begin{align}\mathbf{B}^\textsf{HL} &= \mathbf{H}^\textsf{HL}+\mathbf{M}^\textsf{HL} \\ &= \mu^\textsf{HL}\mathbf{H}^\textsf{HL}\end{align}</math>
| <math>\begin{align}\mathbf{D}^\textsf{G} &= \mathbf{E}^\textsf{G}+4\pi\mathbf{P}^\textsf{G} \\ &= \varepsilon\mathbf{E}^\textsf{G}\end{align}</math>
| <math>\begin{align}\mathbf{B}^\textsf{G} &= \mathbf{H}^\textsf{G}+4\pi\mathbf{M}^\textsf{G} \\ &= \mu^\textsf{G}\mathbf{H}^\textsf{G}\end{align}</math>
|-
| <math>\mathbf{P}^\textsf{SI} = \chi_\text{e}^\textsf{SI}\varepsilon_0\mathbf{E}^\textsf{SI}</math>
| <math>\mathbf{M}^\textsf{SI} = \chi_\text{m}^\textsf{SI}\mathbf{H}^\textsf{SI}</math>
| <math>\mathbf{P}^\textsf{HL} = \chi_\text{e}^\textsf{HL}\mathbf{E}^\textsf{HL}</math>
| <math>\mathbf{M}^\textsf{HL} = \chi_\text{m}^\textsf{HL}\mathbf{H}^\textsf{HL}</math>
| <math>\mathbf{P}^\textsf{G} = \chi_\text{e}^\textsf{G}\mathbf{E}^\textsf{G}</math>
| <math>\mathbf{M}^\textsf{G} = \chi_\text{m}^\textsf{G}\mathbf{H}^\textsf{G}</math>
|-
| <math>\varepsilon^\textsf{SI}/\varepsilon_0 = 1+\chi_\text{e}^\textsf{SI}</math>
| <math>\mu^\textsf{SI}/\mu_0 = 1+\chi_\text{m}^\textsf{SI}</math>
| <math>\varepsilon^\textsf{HL} = 1+\chi_\text{e}^\textsf{HL}</math>
| <math>\mu^\textsf{HL} = 1+\chi_\text{m}^\textsf{HL}</math>
| <math>\varepsilon^\textsf{G} = 1+4\pi\chi_\text{e}^\textsf{G}</math>
| <math>\mu^\textsf{G} = 1+4\pi\chi_\text{m}^\textsf{G}</math>
|}
Note that The quantities <math>\varepsilon^\textsf{SI}/\varepsilon_0</math>, <math>\varepsilon^\textsf{HL}</math> and <math>\varepsilon^\textsf{G}</math> are dimensionless, and they have the same numeric value. By contrast, the [[electric susceptibility]] <math>\chi_\text{e}</math> is dimensionless in all the systems, but has {{em|different numeric values}} for the same material:
<math display="block">\chi_\text{e}^\textsf{SI} = \chi_\text{e}^\textsf{HL} = 4\pi \chi_\text{e}^\textsf{G}</math>
The same statements apply for the corresponding magnetic quantities.

== Advantages and disadvantages of Heaviside–Lorentz units ==

=== Advantages ===

* The formulas above are clearly simpler in {{abbr|HL|Heaviside-Lorentz}} units compared to either {{abbr|SI|International System of Units}} or Gaussian units. As Heaviside proposed, removing the {{math|4''π''}} from the Gauss law and putting it in the Force law considerably reduces the number of places the {{math|''π''}} appears compared to Gaussian CGS units.
* Removing the explicit {{math|4''π''}} from the Gauss law makes it clear that the inverse-square force law arises by the {{math|'''E'''}} 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 {{mvar|z}} 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 {{math|''ε''{{sub|0}}}} and {{math|''μ''{{sub|0}}}} that are present in the SI system. (In addition {{math|''ε''{{sub|0}}}} and {{math|''μ''{{sub|0}}}} are overdetermined, because {{math|1= ''ε''{{sub|0}}{{thinsp}}''μ''{{sub|0}} = 1{{thinsp}}/{{thinsp}}''c''{{i sup|2}}}}.)

The below points are true in both Heaviside–Lorentz and Gaussian systems, but not SI.

* The electric and magnetic fields {{math|'''E'''}} and {{math|'''B'''}} have the same dimensions in the Heaviside–Lorentz system, meaning it is easy to recall where factors of {{mvar|c}} go in the Maxwell equation. Every time derivative comes with a {{math|1{{thinsp}}/{{thinsp}}''c''}}, which makes it dimensionally the same as a space derivative. In contrast, in SI units {{math|1=['''E'''] / ['''B''']}} is {{math|[''c'']}}.
* Giving the {{math|'''E'''}} and {{math|'''B'''}} fields the same dimension makes the assembly into the [[electromagnetic tensor]] more transparent. There are no factors of {{mvar|c}} that need to be inserted when assembling the tensor out of the three-dimensional fields. Similarly, {{mvar|ϕ}} and {{math|'''A'''}} have the same dimensions and are the four components of the 4-potential.
* The fields {{math|'''D'''}}, {{math|'''H'''}}, {{math|'''P'''}}, and {{math|'''M'''}} also have the same dimensions as {{math|'''E'''}} and {{math|'''B'''}}. For vacuum, any expression involving {{math|'''D'''}} can simply be recast as the same expression with {{math|'''E'''}}. In SI units, {{math|'''D'''}} and {{math|'''P'''}} have the same units, as do {{math|'''H'''}} and {{math|'''M'''}}, but they have different units from each other and from {{math|'''E'''}} and {{math|'''B'''}}.

=== 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, "[o]ld style instruments would very soon be in a minority, and then disappear&nbsp;...".<ref name=Heaviside2 /> 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 {{val|300|u=V}}, a value which is larger than most commonly encountered potentials. The henry, the SI unit for [[inductance]] 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), {{abbr|HL|Heaviside–Lorentz}} 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 {{mvar|c}} and {{math|''ε''{{sub|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.{{refn|group="note"|For example, the first and second editions of J. D. Jacksons's ''Classical Electrodynamics''<ref name=JDJ>{{cite book |last=Jackson|first=J.D.|date=1973|title=Classical Electrodynamics, Second Edition|publisher=John Wiley & Sons, New York|pages=811–821}}</ref> used Gaussian units exclusively, but in the third edition Jackson rewrote many of the chapters in SI units. Likewise, E. M. Purcell's ''Electricity and Magnetism'',<ref name=Purcell>{{cite book |last=Purcell|first=E.M.|date=1965|title=Electricity and Magnetism, Berkeley physics course |volume= 2|publisher=McGraw Hill, New York|pages=449–452|edition=first}}</ref> a commonly used textbook for introductory studies, was originally written in Gaussian units; the third edition was rewritten in SI units.}} Outside of these contexts, including for example magazine articles on electric circuits, Heaviside–Lorentz and Gaussian CGS units are rarely encountered.

==Translating expressions and formulas between systems==

To convert any expression or formula between the 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.

{| class="wikitable plainrowheaders"
|+ Equivalence expressions between SI, Heaviside–Lorentz, and Gaussian unit systems
|-
! scope="col" | Name
! scope="col" | SI units
! scope="col" | Heaviside–Lorentz units
! scope="col" | Gaussian units
|-
! scope="row" | electric field, electric potential
| <math> \sqrt{\varepsilon_0} \left(\mathbf{E}^\textsf{SI}, \varphi^\textsf{SI}\right) </math>
| <math> \left(\mathbf{E}^\textsf{HL}, \varphi^\textsf{HL}\right) </math>
| <math> \frac{1}{\sqrt{4\pi}} \left(\mathbf{E}^\textsf{G}, \varphi^\textsf{G}\right) </math>
|-
! scope="row" | displacement field
| <math> \frac{1}{\sqrt{\varepsilon_0}}\mathbf{D}^\textsf{SI} </math>
| <math> \mathbf{D}^\textsf{HL} </math>
| <math> \frac{1}{\sqrt{4\pi}}\mathbf{D}^\textsf{G} </math>
|-
! scope="row" | charge, charge density, <br>[[electric current|current]], current density,<br> polarization density, [[electric dipole moment]]
| <math> \frac{1}{\sqrt{\varepsilon_0}}\left(q^\textsf{SI}, \rho^\textsf{SI}, I^\textsf{SI}, \mathbf{J}^\textsf{SI},\mathbf{P}^\textsf{SI},\mathbf{p}^\textsf{SI}\right) </math>
| <math> \left(q^\textsf{HL}, \rho^\textsf{HL}, I^\textsf{HL}, \mathbf{J}^\textsf{HL},\mathbf{P}^\textsf{HL}, \mathbf{p}^\textsf{HL}\right) </math>
| <math> \sqrt{4\pi} \left(q^\textsf{G}, \rho^\textsf{G}, I^\textsf{G}, \mathbf{J}^\textsf{G},\mathbf{P}^\textsf{G}, \mathbf{p}^\textsf{G}\right) </math>
|-
! scope="row" | magnetic {{math|'''B'''}} field, [[magnetic flux]],<br />magnetic vector potential
| <math> \frac{1}{\sqrt{\mu_0}}\left(\mathbf{B}^\textsf{SI}, \Phi_\text{m}^\textsf{SI},\mathbf{A}^\textsf{SI}\right) </math>
| <math> \left(\mathbf{B}^\textsf{HL}, \Phi_\text{m}^\textsf{HL},\mathbf{A}^\textsf{HL}\right) </math>
| <math> \frac{1}{\sqrt{4\pi}} \left(\mathbf{B}^\textsf{G}, \Phi_\text{m}^\textsf{G},\mathbf{A}^\textsf{G}\right) </math>
|-
! scope="row" | magnetic {{math|'''H'''}} field
| <math> \sqrt{\mu_0}\ \mathbf{H}^\textsf{SI} </math>
| <math> \mathbf{H}^\textsf{HL} </math>
| <math> \frac{1}{\sqrt{4\pi}} \mathbf{H}^\textsf{G} </math>
|-
! scope="row" | [[magnetic moment]], magnetization
| <math> \sqrt{\mu_0}\left(\mathbf{m}^\textsf{SI}, \mathbf{M}^\textsf{SI}\right) </math>
| <math> \left(\mathbf{m}^\textsf{HL}, \mathbf{M}^\textsf{HL}\right) </math>
| <math> \sqrt{4\pi} \left(\mathbf{m}^\textsf{G}, \mathbf{M}^\textsf{G}\right) </math>
|-
! scope="row" | [[relative permittivity]],<br/>relative permeability
| <math> \left(\frac{\varepsilon^\textsf{SI}}{\varepsilon_0}, \frac{\mu^\textsf{SI}}{\mu_0}\right) </math>
| <math> \left(\varepsilon^\textsf{HL}, \mu^\textsf{HL}\right) </math>
| <math> \left(\varepsilon^\textsf{G}, \mu^\textsf{G}\right) </math>
|-
! scope="row" | electric susceptibility,<br />magnetic susceptibility
| <math> \left(\chi_\text{e}^\textsf{SI}, \chi_\text{m}^\textsf{SI}\right) </math>
| <math> \left(\chi_\text{e}^\textsf{HL}, \chi_\text{m}^\textsf{HL}\right) </math>
| <math> 4\pi \left(\chi_\text{e}^\textsf{G}, \chi_\text{m}^\textsf{G}\right) </math>
|-
! scope="row" | [[Electrical conductivity|conductivity]], [[Electrical conductance|conductance]], [[capacitance]]
| <math> \frac{1}{\varepsilon_0}\left(\sigma^\textsf{SI},S^\textsf{SI},C^\textsf{SI}\right) </math>
| <math> \left(\sigma^\textsf{HL}, S^\textsf{HL}, C^\textsf{HL}\right) </math>
| <math> 4\pi \left(\sigma^\textsf{G}, S^\textsf{G}, C^\textsf{G}\right) </math>
|-
! scope="row" | [[resistivity]], [[electrical resistance|resistance]], inductance
| <math> \varepsilon_0\left(\rho^\textsf{SI},R^\textsf{SI},L^\textsf{SI}\right) </math>
| <math> \left(\rho^\textsf{HL},R^\textsf{HL},L^\textsf{HL}\right) </math>
| <math> \frac{1}{4\pi} \left(\rho^\textsf{G},R^\textsf{G},L^\textsf{G}\right) </math>
|}

As an example, starting with the equation

<math display="block">\nabla \cdot \mathbf{E}^\textsf{SI} = \rho^\textsf{SI}/\varepsilon_0 ,</math>

and the equations from the table

<math display="block">\begin{align}
\sqrt{\varepsilon_0} \ \mathbf{E}^\textsf{SI}   &= \mathbf{E}^\textsf{HL} \\
\frac{1}{\sqrt{\varepsilon_0}} \rho^\textsf{SI} &= \rho^\textsf{HL} \,.
\end{align}</math>

Moving the factor across in the latter identities and substituting, the result is

<math display="block">\nabla \cdot \left(\frac{1}{\sqrt{\varepsilon_0}} \mathbf{E}^\textsf{HL}\right) = \left(\sqrt{\varepsilon_0} \rho^\textsf{HL}\right)/\varepsilon_0 ,</math>

which then simplifies to

<math display="block">\nabla \cdot \mathbf{E}^\textsf{HL} = \rho^\textsf{HL} .</math>

== Notes ==

{{Reflist|group="note"|1}}

== References ==
{{reflist|25em}}

{{Systems of measurement|state=collapsed}}

{{DEFAULTSORT:Heaviside-Lorentz units}}
[[Category:Special relativity]]
[[Category:Electromagnetism]]
[[Category:Hendrik Lorentz]]