Heaviside–Lorentz units explained

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 and Hendrik Antoon Lorentz. They share with the CGS-Gaussian system that the electric constant and magnetic constant 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 to use the speed of light instead.^[1]

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 appearing explicitly in Maxwell's equations.^[2] That this system is rationalized partly explains its appeal in quantum field theory: the Lagrangian underlying the theory does not have any factors of when this system is used.^[3] 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. It is often used in relativistic calculations,^[4] 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 (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 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.^{[5] [6]} In his 1893 book Electromagnetic Theory,^[7] 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 and the vacuum permeability . 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 and displacement field, respectively, and are the magnetic fields, is the polarization density, is the magnetization, is charge density, is current density, is the speed of light in vacuum, is the electric potential, is the magnetic vector potential, is the Lorentz force acting on a body of charge and velocity, is the permittivity, is the electric susceptibility, is the magnetic permeability, and is the magnetic susceptibility.

Maxwell's equations

See main article: Maxwell's equations.

Maxwell's equations in SI, Heaviside–Lorentz, and Gaussian quantities

Name	SI quantities	Heaviside–Lorentz quantities	Gaussian quantities
Gauss's law (macroscopic)	\nabla ⋅ D ^{sf{SI}=} \rho ^{sf{SI}} _{f}	\nabla ⋅ D ^{sf{HL}=} \rho ^{sf{HL}} _{f}	\nabla ⋅ D ^{sf{G}=} 4 \pi\rho ^{sf{G}} _{f}
Gauss's law (microscopic)	\nabla ⋅ E ^{sf{SI}=} \rho ^{sf{SI}/\varepsilon} _{0}	\nabla ⋅ E ^{sf{HL}=} \rho ^{sf{HL}}	\nabla ⋅ E ^{sf{G}=} 4 \pi\rho ^{sf{G}}
Gauss's law for magnetism	\nabla ⋅ B ^{sf{SI}=} 0	\nabla ⋅ B ^{sf{HL}=} 0	\nabla ⋅ B ^{sf{G}=} 0
Maxwell–Faraday equation (Faraday's law of induction)	\nabla x E ^{sf{SI}=} - (\partial B ^{sf{SI})/() }	\nabla x E ^{sf{HL}=} - (1)/(c) (\partial B ^{sf{HL})/() }	\nabla x E ^{sf{G}=} - (1)/(c) (\partial B ^{sf{G})/() }
Ampère–Maxwell equation (macroscopic)	\nabla x H ^{sf{SI}=} J ^{sf{SI} +} _{f} (\partial D ^{sf{SI})/() }	\nabla x H ^{sf{HL}=} (1)/(c) J ^{sf{HL} +} _{f} (1)/(c) (\partial D ^{sf{HL})/() }	\nabla x H ^{sf{G}=} (4 \pi)/(c) J ^{sf{G} +} _{f} (1)/(c) (\partial D ^{sf{G})/() }
Ampère–Maxwell equation (microscopic)	\nabla x B ^{sf{SI}=} \mu _{0} \left( J ^{sf{SI}+} \varepsilon _{0 (\partial E ^{sf{SI})/()} } \right)	\nabla x B ^{sf{HL}=} (1)/(c) J ^{sf{HL}+} (1)/(c) (\partial E ^{sf{HL})/() }	\nabla x B ^{sf{G}=} (4 \pi)/(c) J ^{sf{G}+} (1)/(c) (\partial E ^{sf{G})/() }

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 $\backslash mathbf\; =\; -\backslash nabla\backslash phi-\backslash frac$ in the SI system, but $\backslash mathbf\; =\; -\backslash nabla\backslash phi-\backslash frac\; \backslash frac$ in the HL or Gaussian systems.

Other basic laws

Other electrostatic laws in SI, Heaviside–Lorentz, and Gaussian quantities

Name	SI quantities	Heaviside–Lorentz quantities	Gaussian quantities
Lorentz force	F = q ^{sf{SI}\left(E} ^{sf{SI}+v x B} ^{sf{SI}\right)}	F = q ^{sf{HL}\left(E} ^{sf{HL}+ (1)/(c) v x B} ^{sf{HL}\right)}	F = q ^{sf{G}\left(E} ^{sf{G}+ (1)/(c) v x B} ^{sf{G}\right)}
Coulomb's law	F = (1)/(4 \pi\varepsilon _{0}) (q ^{sf{SI} _{1})/(q) ^{sf{SI}}{r} ^{2} ^{2}} \hatr	F = (1)/(4 \pi) (q ^{sf{HL} _{1})/(q) ^{sf{HL}}{r} ^{2} ^{2}} \hatr	F = (q ^{sf{G} _{1})/(q) ^{sf{G}}{r} ^{2} ^{2}} \hatr
Electric field of stationary point charge	E ^{sf{SI}=} (1)/(4 \pi\varepsilon _{0}) (q ^{sf{SI})/() } \mathbf	E ^{sf{HL}=} (1)/(4 \pi) (q ^{sf{HL})/() } \mathbf	E ^{sf{G}=} (q ^{sf{G})/() } \mathbf
Biot–Savart law	B ^{sf{SI}=} (\mu _{0})/(\oint 4 \pi) (I ^{sf{SI})/(d l) x \hatr }	B ^{sf{HL}=} (1)/(\oint 4 \pi c) (I ^{sf{HL})/(d l) x \hatr }	B ^{sf{G}=} (1)/(\oint c) (I ^{sf{G})/(d l) x \hatr }

Dielectric and magnetic materials

Below are the expressions for the macroscopic fields

D , P , H and M 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.

SI quantities	Heaviside–Lorentz quantities	Gaussian quantities
Dielectric	Magnetic	Dielectric	Magnetic	Dielectric	Magnetic
\begin{align} D ^{sf{SI}&=} \varepsilon _{0} E ^{sf{SI}+P} ^{sf{SI}\ &=} \varepsilon E ^{sf{SI}\end{align}}	\begin{align} B ^{sf{SI}&=} \mu _{0} ( H ^{sf{SI}+M} ^{sf{SI})\ &=} \mu ^{sf{SI}H} ^{sf{SI}\end{align}}	\begin{align} D ^{sf{HL}&=} E ^{sf{HL}+P} ^{sf{HL}\ &=} \varepsilon E ^{sf{HL}\end{align}}	\begin{align} B ^{sf{HL}&=} H ^{sf{HL}+M} ^{sf{HL}\ &=} \mu ^{sf{HL}H} ^{sf{HL}\end{align}}	\begin{align} D ^{sf{G}&=} E ^{sf{G}+4\piP} ^{sf{G}\ &=} \varepsilon E ^{sf{G}\end{align}}	\begin{align} B ^{sf{G}&=} H ^{sf{G}+4\piM} ^{sf{G}\ &=} \mu ^{sf{G}H} ^{sf{G}\end{align}}
P ^{sf{SI}=} \chi ^{sf{SI}} _{0 E}	M ^{sf{SI}=} \chi ^{sf{SI} H} _{m} ^{sf{SI}}	P ^{sf{HL}=} \chi ^{sf{HL} E} _{e} ^{sf{HL}}	M ^{sf{HL}=} \chi ^{sf{HL} H} _{m} ^{sf{HL}}	P ^{sf{G}=} \chi ^{sf{G} E} _{e} ^{sf{G}}	M ^{sf{G}=} \chi ^{sf{G} H} _{m} ^{sf{G}}
\varepsilon ^{sf{SI}/\varepsilon} _{0} = 1+ \chi ^{sf{SI}} _{e}	\mu ^{sf{SI}/\mu} _{0} = 1+ \chi ^{sf{SI}} _{m}	\varepsilon ^{sf{HL}=} 1+ \chi ^{sf{HL}} _{e}	\mu ^{sf{HL}=} 1+ \chi ^{sf{HL}} _{m}	\varepsilon ^{sf{G}=} 1+4 \pi\chi ^{sf{G}} _{e}	\mu ^{sf{G}=} 1+4 \pi\chi ^{sf{G}} _{m}

Note that The quantities \varepsilon ^{sf{SI}/\varepsilon} _{0} , \varepsilon ^{sf{HL}} and \varepsilon ^{sf{G}} are dimensionless, and they have the same numeric value. By contrast, the electric susceptibility \chi _{e} is dimensionless in all the systems, but has for the same material:$$\backslash chi\_\backslash text^\backslash textsf\; =\; \backslash chi\_\backslash text^\backslash textsf\; =\; 4\backslash pi\; \backslash chi\_\backslash text^\backslash textsf$$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 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, "[o]ld 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 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 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.

Equivalence expressions between SI, Heaviside–Lorentz, and Gaussian unit systems

Name	SI units	Heaviside–Lorentz units	Gaussian units
electric field, electric potential	\sqrt{\varepsilon _{0}} \left( E ^{sf{SI},\varphi} ^{sf{SI}\right)}	\left( E ^{sf{HL},\varphi} ^{sf{HL}\right)}	(1)/(\sqrt{ 4 \pi) } \left(\mathbf^\textsf, \varphi^\textsf\right)
displacement field	(1)/(\sqrt{\varepsilon _{0}) }\mathbf^\textsf	D ^{sf{HL}}	(1)/(\sqrt{ 4 \pi) }\mathbf^\textsf
charge, charge density, current, current density, polarization density, electric dipole moment	(1)/(\sqrt{\varepsilon _{0}) }\left(q^\textsf, \rho^\textsf, I^\textsf, \mathbf^\textsf,\mathbf^\textsf,\mathbf^\textsf\right)	\left(q ^{sf{HL},\rho} ^{sf{HL},I} ^{sf{HL},J} ^{sf{HL},P} ^{sf{HL},p} ^{sf{HL}\right)}	\sqrt{ 4 \pi} \left(q ^{sf{G},\rho} ^{sf{G},I} ^{sf{G},J} ^{sf{G},P} ^{sf{G},p} ^{sf{G}\right)}
magnetic field, magnetic flux,magnetic vector potential	(1)/(\sqrt{\mu _{0}) }\left(\mathbf^\textsf, \Phi_\text^\textsf,\mathbf^\textsf\right)	\left( B ^{sf{HL}, A} _{m} ^{sf{HL}\right)}	(1)/(\sqrt{ 4 \pi) } \left(\mathbf^\textsf, \Phi_\text^\textsf,\mathbf^\textsf\right)
magnetic field	\sqrt{\mu ^{sf{SI}} _{0 } H}	H ^{sf{HL}}	(1)/(\sqrt{ 4 \pi) } \mathbf^\textsf
magnetic moment, magnetization	\sqrt{\mu ^{sf{SI}, M} _{0 }\left( m} ^{sf{SI}\right)}	\left( m ^{sf{HL},M} ^{sf{HL}\right)}	\sqrt{ 4 \pi} \left( m ^{sf{G},M} ^{sf{G}\right)}
relative permittivity, relative permeability	\left( (\varepsilon ^{sf{SI})/() }, \frac\right)	\left(\varepsilon ^{sf{HL},\mu} ^{sf{HL}\right)}	\left(\varepsilon ^{sf{G},\mu} ^{sf{G}\right)}
electric susceptibility, magnetic susceptibility	\left(\chi ^{sf{SI}\right)} _{m}	\left(\chi ^{sf{HL}\right)} _{m}	4 \pi \left(\chi ^{sf{G}\right)} _{m}
conductivity, conductance, capacitance	(1)/(\varepsilon _{0}) \left(\sigma ^{sf{SI},S} ^{sf{SI},C} ^{sf{SI}\right)}	\left(\sigma ^{sf{HL},S} ^{sf{HL},C} ^{sf{HL}\right)}	4 \pi \left(\sigma ^{sf{G},S} ^{sf{G},C} ^{sf{G}\right)}
resistivity, resistance, inductance	\varepsilon ^{sf{SI},R} _{0 \left(\rho} ^{sf{SI},L} ^{sf{SI}\right)}	\left(\rho ^{sf{HL},R} ^{sf{HL},L} ^{sf{HL}\right)}	(1)/(4 \pi) \left(\rho ^{sf{G},R} ^{sf{G},L} ^{sf{G}\right)}

As an example, starting with the equation$$\backslash nabla\; \backslash cdot\; \backslash mathbf^\backslash textsf\; =\; \backslash rho^\backslash textsf/\backslash varepsilon\_0,$$and the equations from the table

Moving the factor across in the latter identities and substituting, the result is$$\backslash nabla\; \backslash cdot\; \backslash left(\backslash frac\; \backslash mathbf^\backslash textsf\backslash right)\; =\; \backslash left(\backslash sqrt\; \backslash rho^\backslash textsf\backslash right)/\backslash varepsilon\_0,$$which then simplifies to$$\backslash nabla\; \backslash cdot\; \backslash mathbf^\backslash textsf\; =\; \backslash rho^\backslash textsf\; .$$

Notes and References

1. Silsbee . Francis . Systems of Electrical Units . Journal of Research of the National Bureau of Standards Section C . 66C . 2 . 137–183 . April–June 1962 . 10.6028/jres.066C.014 . free .
2. Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity, " The Physics Teacher 24(2): 97–99. Alternate web link (subscription required)
3. Web site: Robert Grayson Littlejohn. Gaussian, SI and Other Systems of Units in Electromagnetic Theory . Physics 221A, University of California, Berkeley lecture notes . Littlejohn, Robert . Fall 2011 . 2008-05-06 .
4. As used by Einstein, such as in his book: Book: Einstein, Albert . The Meaning of Relativity. Princeton University Press (2005). 2005. 21 ff. 1956, 5th.
5. Heaviside . O. . Oliver Heaviside . 1882-11-18 . dmy-all . The relations between magnetic force and electric current . . London, UK.
6. Web site: System of measurement units . 24 April 2012 . wiki . Engineering and Technology History (ETHW.org) . 2021-12-23 . dmy-all.
7. Book: Heaviside, Oliver . Oliver Heaviside . 1893 . Electromagnetic Theory . The D. van Nostrand Company . London, UK . 1 . . Google Books.

   Alternate source for the same text:Web site: Heaviside 1893: Electromagnetic Theory . OCR text . wiki . Open Source Ecology Germany (wiki.opensourceecology.de) .