Heaviside–Lorentz units

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 Template:Math and magnetic constant Template:Math 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 Template:Math and Template:Math, while at the same time revising Maxwell's equations to use the speed of light Template:Math instead.

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 Template:Math 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 Template:Math when this system is used. Consequently, electromagnetic quantities in the Heaviside–Lorentz system differ by factors of Template:Math 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.

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 Template:Math 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 Template:Mvar is Template:Mvar while that of a parallel plate capacitor is Template:Big, where Template:Mvar is the area of the smaller plate and Template:Mvar is their separation.

Heaviside, who was an important, though somewhat isolated,Template:Citation needed early theorist of electromagnetism, suggested in 1882 that the irrational appearance of Template:Math in these sorts of relations could be removed by redefining the units for charges and fields. In his 1893 book Electromagnetic Theory, Heaviside wrote in the introduction: Template:Blockquote

Length–mass–time framework

As in the Gaussian system (Template:Abbr), the Heaviside–Lorentz system (Template:Abbr) 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 Template:Math in the Gaussian system, and Template:Math in the HL system. The unit of charge then connects to Template:Math, where 'HLC' is the HL unit of charge. The HL quantity Template:Math describing a charge is then Template:Math 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 (Template:Math) and the vacuum permeability (Template:Math). These can be used to convert SI units to their corresponding Heaviside–Lorentz values, as detailed below. For example, SI charge is Template:Math. When one puts Template:Math, Template:Math, Template:Math, and Template:Math, this evaluates to Template:Val, 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 Template:Math and Template:Math are the electric field and displacement field, respectively, Template:Math and Template:Math are the magnetic fields, Template:Math is the polarization density, Template:Math is the magnetization, Template:Mvar is charge density, Template:Math is current density, Template:Mvar is the speed of light in vacuum, Template:Mvar is the electric potential, Template:Math is the magnetic vector potential, Template:Math is the Lorentz force acting on a body of charge Template:Mvar and velocity Template:Mvar, Template:Mvar is the permittivity, Template:Math is the electric susceptibility, Template:Mvar is the magnetic permeability, and Template:Math is the magnetic susceptibility.

Maxwell's equations

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

Name	SI	Heaviside–Lorentz system	Gaussian system
Gauss's law (macroscopic)	$\mathrm{\nabla }\cdot {𝐃}^{\mathsf{\text{SI}}}={\rho }_{\text{f}}^{\mathsf{\text{SI}}}$	$\mathrm{\nabla }\cdot {𝐃}^{\mathsf{\text{HL}}}={\rho }_{\text{f}}^{\mathsf{\text{HL}}}$	$\mathrm{\nabla }\cdot {𝐃}^{\mathsf{\text{G}}}=4\pi {\rho }_{\text{f}}^{\mathsf{\text{G}}}$
Gauss's law (microscopic)	$\mathrm{\nabla }\cdot {𝐄}^{\mathsf{\text{SI}}}={\rho }^{\mathsf{\text{SI}}}/{\epsilon }_0$	$\mathrm{\nabla }\cdot {𝐄}^{\mathsf{\text{HL}}}={\rho }^{\mathsf{\text{HL}}}$	$\mathrm{\nabla }\cdot {𝐄}^{\mathsf{\text{G}}}=4\pi {\rho }^{\mathsf{\text{G}}}$
Gauss's law for magnetism	$\mathrm{\nabla }\cdot {𝐁}^{\mathsf{\text{SI}}}=0$	$\mathrm{\nabla }\cdot {𝐁}^{\mathsf{\text{HL}}}=0$	$\mathrm{\nabla }\cdot {𝐁}^{\mathsf{\text{G}}}=0$
Maxwell–Faraday equation (Faraday's law of induction)	$\mathrm{\nabla }\times {𝐄}^{\mathsf{\text{SI}}}=-\frac{\partial {𝐁}^{\mathsf{\text{SI}}}}{\partial t}$	$\mathrm{\nabla }\times {𝐄}^{\mathsf{\text{HL}}}=-\frac{1}{c}\frac{\partial {𝐁}^{\mathsf{\text{HL}}}}{\partial t}$	$\mathrm{\nabla }\times {𝐄}^{\mathsf{\text{G}}}=-\frac{1}{c}\frac{\partial {𝐁}^{\mathsf{\text{G}}}}{\partial t}$
Ampère–Maxwell equation (macroscopic)	$\mathrm{\nabla }\times {𝐇}^{\mathsf{\text{SI}}}={𝐉}_{\text{f}}^{\mathsf{\text{SI}}}+\frac{\partial {𝐃}^{\mathsf{\text{SI}}}}{\partial t}$	$\mathrm{\nabla }\times {𝐇}^{\mathsf{\text{HL}}}=\frac{1}{c}{𝐉}_{\text{f}}^{\mathsf{\text{HL}}}+\frac{1}{c}\frac{\partial {𝐃}^{\mathsf{\text{HL}}}}{\partial t}$	$\mathrm{\nabla }\times {𝐇}^{\mathsf{\text{G}}}=\frac{4\pi }{c}{𝐉}_{\text{f}}^{\mathsf{\text{G}}}+\frac{1}{c}\frac{\partial {𝐃}^{\mathsf{\text{G}}}}{\partial t}$
Ampère–Maxwell equation (microscopic)	$\mathrm{\nabla }\times {𝐁}^{\mathsf{\text{SI}}}={\mu }_0({𝐉}^{\mathsf{\text{SI}}}+{\epsilon }_0\frac{\partial {𝐄}^{\mathsf{\text{SI}}}}{\partial t})$	$\mathrm{\nabla }\times {𝐁}^{\mathsf{\text{HL}}}=\frac{1}{c}{𝐉}^{\mathsf{\text{HL}}}+\frac{1}{c}\frac{\partial {𝐄}^{\mathsf{\text{HL}}}}{\partial t}$	$\mathrm{\nabla }\times {𝐁}^{\mathsf{\text{G}}}=\frac{4\pi }{c}{𝐉}^{\mathsf{\text{G}}}+\frac{1}{c}\frac{\partial {𝐄}^{\mathsf{\text{G}}}}{\partial t}$

The electric and magnetic fields can be written in terms of the potentials Template:Math and Template:Mvar. The definition of the magnetic field in terms of Template:Math, Template:Math, is the same in all systems of units, but the electric field is $𝐄=-\mathrm{\nabla }\varphi -\frac{\partial 𝐀}{\partial t}$ in the SI system, but $𝐄=-\mathrm{\nabla }\varphi -\frac{1}{c}\frac{\partial 𝐀}{\partial t}$ in the HL or Gaussian systems.

Other basic laws

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

Name	SI	Heaviside–Lorentz system	Gaussian system
Lorentz force	$𝐅=q^{\mathsf{\text{SI}}}({𝐄}^{\mathsf{\text{SI}}}+𝐯\times {𝐁}^{\mathsf{\text{SI}}})$	$𝐅=q^{\mathsf{\text{HL}}}({𝐄}^{\mathsf{\text{HL}}}+\frac{1}{c}𝐯\times {𝐁}^{\mathsf{\text{HL}}})$	$𝐅=q^{\mathsf{\text{G}}}({𝐄}^{\mathsf{\text{G}}}+\frac{1}{c}𝐯\times {𝐁}^{\mathsf{\text{G}}})$
Coulomb's law	$𝐅=\frac{1}{4\pi {\epsilon }_0}\frac{q_1^{\mathsf{\text{SI}}}{q}_2^{\mathsf{\text{SI}}}}{r^2}\widehat{𝐫}$	$𝐅=\frac{1}{4\pi }\frac{q_1^{\mathsf{\text{HL}}}{q}_2^{\mathsf{\text{HL}}}}{r^2}\widehat{𝐫}$	$𝐅=\frac{q_1^{\mathsf{\text{G}}}{q}_2^{\mathsf{\text{G}}}}{r^2}\widehat{𝐫}$
Electric field of stationary point charge	${𝐄}^{\mathsf{\text{SI}}}=\frac{1}{4\pi {\epsilon }_0}\frac{q^{\mathsf{\text{SI}}}}{r^2}\widehat{𝐫}$	${𝐄}^{\mathsf{\text{HL}}}=\frac{1}{4\pi }\frac{q^{\mathsf{\text{HL}}}}{r^2}\widehat{𝐫}$	${𝐄}^{\mathsf{\text{G}}}=\frac{q^{\mathsf{\text{G}}}}{r^2}\widehat{𝐫}$
Biot–Savart law	${𝐁}^{\mathsf{\text{SI}}}=\frac{{\mu }_0}{4\pi }\oint \frac{I^{\mathsf{\text{SI}}}d𝐥\times \widehat{𝐫}}{r^2}$	${𝐁}^{\mathsf{\text{HL}}}=\frac{1}{4\pi c}\oint \frac{I^{\mathsf{\text{HL}}}d𝐥\times \widehat{𝐫}}{r^2}$	${𝐁}^{\mathsf{\text{G}}}=\frac{1}{c}\oint \frac{I^{\mathsf{\text{G}}}d𝐥\times \widehat{𝐫}}{r^2}$

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		Heaviside–Lorentz system		Gaussian system
Dielectric	Magnetic	Dielectric	Magnetic	Dielectric	Magnetic
$\begin{array}{cc}{𝐃}^{\mathsf{\text{SI}}}& ={\epsilon }_0{𝐄}^{\mathsf{\text{SI}}}+{𝐏}^{\mathsf{\text{SI}}}\\ & =\epsilon {𝐄}^{\mathsf{\text{SI}}}\end{array}$	$\begin{array}{cc}{𝐁}^{\mathsf{\text{SI}}}& ={\mu }_0({𝐇}^{\mathsf{\text{SI}}}+{𝐌}^{\mathsf{\text{SI}}})\\ & ={\mu }^{\mathsf{\text{SI}}}{𝐇}^{\mathsf{\text{SI}}}\end{array}$	$\begin{array}{cc}{𝐃}^{\mathsf{\text{HL}}}& ={𝐄}^{\mathsf{\text{HL}}}+{𝐏}^{\mathsf{\text{HL}}}\\ & =\epsilon {𝐄}^{\mathsf{\text{HL}}}\end{array}$	$\begin{array}{cc}{𝐁}^{\mathsf{\text{HL}}}& ={𝐇}^{\mathsf{\text{HL}}}+{𝐌}^{\mathsf{\text{HL}}}\\ & ={\mu }^{\mathsf{\text{HL}}}{𝐇}^{\mathsf{\text{HL}}}\end{array}$	$\begin{array}{cc}{𝐃}^{\mathsf{\text{G}}}& ={𝐄}^{\mathsf{\text{G}}}+4\pi {𝐏}^{\mathsf{\text{G}}}\\ & =\epsilon {𝐄}^{\mathsf{\text{G}}}\end{array}$	$\begin{array}{cc}{𝐁}^{\mathsf{\text{G}}}& ={𝐇}^{\mathsf{\text{G}}}+4\pi {𝐌}^{\mathsf{\text{G}}}\\ & ={\mu }^{\mathsf{\text{G}}}{𝐇}^{\mathsf{\text{G}}}\end{array}$
${𝐏}^{\mathsf{\text{SI}}}={\chi }_{\text{e}}^{\mathsf{\text{SI}}}{\epsilon }_0{𝐄}^{\mathsf{\text{SI}}}$	${𝐌}^{\mathsf{\text{SI}}}={\chi }_{\text{m}}^{\mathsf{\text{SI}}}{𝐇}^{\mathsf{\text{SI}}}$	${𝐏}^{\mathsf{\text{HL}}}={\chi }_{\text{e}}^{\mathsf{\text{HL}}}{𝐄}^{\mathsf{\text{HL}}}$	${𝐌}^{\mathsf{\text{HL}}}={\chi }_{\text{m}}^{\mathsf{\text{HL}}}{𝐇}^{\mathsf{\text{HL}}}$	${𝐏}^{\mathsf{\text{G}}}={\chi }_{\text{e}}^{\mathsf{\text{G}}}{𝐄}^{\mathsf{\text{G}}}$	${𝐌}^{\mathsf{\text{G}}}={\chi }_{\text{m}}^{\mathsf{\text{G}}}{𝐇}^{\mathsf{\text{G}}}$
${\epsilon }^{\mathsf{\text{SI}}}/{\epsilon }_0=1+{\chi }_{\text{e}}^{\mathsf{\text{SI}}}$	${\mu }^{\mathsf{\text{SI}}}/{\mu }_0=1+{\chi }_{\text{m}}^{\mathsf{\text{SI}}}$	${\epsilon }^{\mathsf{\text{HL}}}=1+{\chi }_{\text{e}}^{\mathsf{\text{HL}}}$	${\mu }^{\mathsf{\text{HL}}}=1+{\chi }_{\text{m}}^{\mathsf{\text{HL}}}$	${\epsilon }^{\mathsf{\text{G}}}=1+4\pi {\chi }_{\text{e}}^{\mathsf{\text{G}}}$	${\mu }^{\mathsf{\text{G}}}=1+4\pi {\chi }_{\text{m}}^{\mathsf{\text{G}}}$

Note that The quantities ${\epsilon }^{\mathsf{\text{SI}}}/{\epsilon }_0$, ${\epsilon }^{\mathsf{\text{HL}}}$ and ${\epsilon }^{\mathsf{\text{G}}}$ are dimensionless, and they have the same numeric value. By contrast, the electric susceptibility ${\chi }_{\text{e}}$ is dimensionless in all the systems, but has Template:Em for the same material: $${\chi }_{\text{e}}^{\mathsf{\text{SI}}}={\chi }_{\text{e}}^{\mathsf{\text{HL}}}=4\pi {\chi }_{\text{e}}^{\mathsf{\text{G}}}$$ The same statements apply for the corresponding magnetic quantities.

Advantages and disadvantages of Heaviside–Lorentz units

Advantages

• The formulas above are clearly simpler in Template:Abbr units compared to either Template:Abbr or Gaussian units. As Heaviside proposed, removing the Template:Math from the Gauss law and putting it in the Force law considerably reduces the number of places the Template:Math appears compared to Gaussian CGS units.
• Removing the explicit Template:Math from the Gauss law makes it clear that the inverse-square force law arises by the Template:Math 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 Template:Mvar 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 Template:Math and Template:Math that are present in the SI system. (In addition Template:Math and Template:Math are overdetermined, because Template:Math.)

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

• The electric and magnetic fields Template:Math and Template:Math have the same dimensions in the Heaviside–Lorentz system, meaning it is easy to recall where factors of Template:Mvar go in the Maxwell equation. Every time derivative comes with a Template:Math, which makes it dimensionally the same as a space derivative. In contrast, in SI units Template:Math is Template:Math.
• Giving the Template:Math and Template:Math fields the same dimension makes the assembly into the electromagnetic tensor more transparent. There are no factors of Template:Mvar that need to be inserted when assembling the tensor out of the three-dimensional fields. Similarly, Template:Mvar and Template:Math have the same dimensions and are the four components of the 4-potential.
• The fields Template:Math, Template:Math, Template:Math, and Template:Math also have the same dimensions as Template:Math and Template:Math. For vacuum, any expression involving Template:Math can simply be recast as the same expression with Template:Math. In SI units, Template:Math and Template:Math have the same units, as do Template:Math and Template:Math, but they have different units from each other and from Template:Math and Template:Math.

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 Template:Val, 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), Template:Abbr 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 Template:Mvar and Template:Math. 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.Template:Refn 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 ${\epsilon }_0{\mu }_0$ or vice versa. This will reproduce any of the specific formulas given in the list above.

Equivalent expressions between SI, Heaviside–Lorentz system, and Gaussian system

Name	SI	Heaviside–Lorentz system	Gaussian system
electric field, electric potential	$\sqrt{{\epsilon }_0}({𝐄}^{\mathsf{\text{SI}}},{\phi }^{\mathsf{\text{SI}}})$	$({𝐄}^{\mathsf{\text{HL}}},{\phi }^{\mathsf{\text{HL}}})$	$\frac{1}{\sqrt{4\pi }}({𝐄}^{\mathsf{\text{G}}},{\phi }^{\mathsf{\text{G}}})$
displacement field	$\frac{1}{\sqrt{{\epsilon }_0}}{𝐃}^{\mathsf{\text{SI}}}$	${𝐃}^{\mathsf{\text{HL}}}$	$\frac{1}{\sqrt{4\pi }}{𝐃}^{\mathsf{\text{G}}}$
charge, charge density, Template:Brcurrent, current density,Template:Br polarization density, electric dipole moment	$\frac{1}{\sqrt{{\epsilon }_0}}(q^{\mathsf{\text{SI}}},{\rho }^{\mathsf{\text{SI}}},I^{\mathsf{\text{SI}}},{𝐉}^{\mathsf{\text{SI}}},{𝐏}^{\mathsf{\text{SI}}},{𝐩}^{\mathsf{\text{SI}}})$	$(q^{\mathsf{\text{HL}}},{\rho }^{\mathsf{\text{HL}}},I^{\mathsf{\text{HL}}},{𝐉}^{\mathsf{\text{HL}}},{𝐏}^{\mathsf{\text{HL}}},{𝐩}^{\mathsf{\text{HL}}})$	$\sqrt{4\pi }(q^{\mathsf{\text{G}}},{\rho }^{\mathsf{\text{G}}},I^{\mathsf{\text{G}}},{𝐉}^{\mathsf{\text{G}}},{𝐏}^{\mathsf{\text{G}}},{𝐩}^{\mathsf{\text{G}}})$
magnetic Template:Math field, magnetic flux,Template:Brmagnetic vector potential	$\frac{1}{\sqrt{{\mu }_0}}({𝐁}^{\mathsf{\text{SI}}},{\mathrm{\Phi }}_{\text{m}}^{\mathsf{\text{SI}}},{𝐀}^{\mathsf{\text{SI}}})$	$({𝐁}^{\mathsf{\text{HL}}},{\mathrm{\Phi }}_{\text{m}}^{\mathsf{\text{HL}}},{𝐀}^{\mathsf{\text{HL}}})$	$\frac{1}{\sqrt{4\pi }}({𝐁}^{\mathsf{\text{G}}},{\mathrm{\Phi }}_{\text{m}}^{\mathsf{\text{G}}},{𝐀}^{\mathsf{\text{G}}})$
magnetic Template:Math field	$\sqrt{{\mu }_0}{𝐇}^{\mathsf{\text{SI}}}$	${𝐇}^{\mathsf{\text{HL}}}$	$\frac{1}{\sqrt{4\pi }}{𝐇}^{\mathsf{\text{G}}}$
magnetic moment, magnetization	$\sqrt{{\mu }_0}({𝐦}^{\mathsf{\text{SI}}},{𝐌}^{\mathsf{\text{SI}}})$	$({𝐦}^{\mathsf{\text{HL}}},{𝐌}^{\mathsf{\text{HL}}})$	$\sqrt{4\pi }({𝐦}^{\mathsf{\text{G}}},{𝐌}^{\mathsf{\text{G}}})$
relative permittivity, relative permeability	$(\frac{{\epsilon }^{\mathsf{\text{SI}}}}{{\epsilon }_0},\frac{{\mu }^{\mathsf{\text{SI}}}}{{\mu }_0})$	$({\epsilon }^{\mathsf{\text{HL}}},{\mu }^{\mathsf{\text{HL}}})$	$({\epsilon }^{\mathsf{\text{G}}},{\mu }^{\mathsf{\text{G}}})$
electric susceptibility, magnetic susceptibility	$({\chi }_{\text{e}}^{\mathsf{\text{SI}}},{\chi }_{\text{m}}^{\mathsf{\text{SI}}})$	$({\chi }_{\text{e}}^{\mathsf{\text{HL}}},{\chi }_{\text{m}}^{\mathsf{\text{HL}}})$	$4\pi ({\chi }_{\text{e}}^{\mathsf{\text{G}}},{\chi }_{\text{m}}^{\mathsf{\text{G}}})$
conductivity, conductance, capacitance	$\frac{1}{{\epsilon }_0}({\sigma }^{\mathsf{\text{SI}}},S^{\mathsf{\text{SI}}},C^{\mathsf{\text{SI}}})$	$({\sigma }^{\mathsf{\text{HL}}},S^{\mathsf{\text{HL}}},C^{\mathsf{\text{HL}}})$	$4\pi ({\sigma }^{\mathsf{\text{G}}},S^{\mathsf{\text{G}}},C^{\mathsf{\text{G}}})$
resistivity, resistance, inductance	${\epsilon }_0({\rho }^{\mathsf{\text{SI}}},R^{\mathsf{\text{SI}}},L^{\mathsf{\text{SI}}})$	$({\rho }^{\mathsf{\text{HL}}},R^{\mathsf{\text{HL}}},L^{\mathsf{\text{HL}}})$	$\frac{1}{4\pi }({\rho }^{\mathsf{\text{G}}},R^{\mathsf{\text{G}}},L^{\mathsf{\text{G}}})$

As an example, starting with the equation $$\mathrm{\nabla }\cdot {𝐄}^{\mathsf{\text{SI}}}={\rho }^{\mathsf{\text{SI}}}/{\epsilon }_0,$$ and the equations from the table $$\begin{array}{cc}\sqrt{{\epsilon }_0}{𝐄}^{\mathsf{\text{SI}}}& ={𝐄}^{\mathsf{\text{HL}}}\\ \frac{1}{\sqrt{{\epsilon }_0}}{\rho }^{\mathsf{\text{SI}}}& ={\rho }^{\mathsf{\text{HL}}}.\end{array}$$

Moving the factor across in the latter identities and substituting, the result is $$\mathrm{\nabla }\cdot \left(\frac{1}{\sqrt{{\epsilon }_0}}{𝐄}^{\mathsf{\text{HL}}}\right)=\left(\sqrt{{\epsilon }_0}{\rho }^{\mathsf{\text{HL}}}\right)/{\epsilon }_0,$$ which then simplifies to $$\mathrm{\nabla }\cdot {𝐄}^{\mathsf{\text{HL}}}={\rho }^{\mathsf{\text{HL}}}.$$