Physics:Lorentz–Heaviside units

Lorentz–Heaviside units (or Heaviside–Lorentz units) constitute a system of units (particularly electromagnetic units) within CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that the electric constant ε₀ and magnetic constant µ₀ 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 ε₀ = 1 and µ₀ = 1, while at the same time revising Maxwell's equations to use the speed of light c instead.

Heaviside–Lorentz units, like SI units but unlike Gaussian units, are rationalized, meaning that there are no factors of 4π appearing explicitly in Maxwell's equations. That these units are rationalized partly explains their appeal in quantum field theory: the Lagrangian underlying the theory does not have any factors of 4π in these units. Consequently, Heaviside-Lorentz units differ by factors of √4π in the definitions of the electric and magnetic fields and of electric charge. They are 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-called 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 4π 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 r is r while that of a parallel plate capacitor is A/(4πs), where A is the area of the plates and s is their separation.

Heaviside, who was an important, though somewhat isolated, early theorist of electromagnetism, suggested in 1882 that the irrational appearance of 4πs in these sorts of relations could be removed by redefining the unit of the charges and fields.

In 1893 Heaviside wrote

It is not long since it was taken for granted that the common electrical units were correct. That curious and obtrusive constant 4π 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 4π 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 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 inconvenience.

Length–mass–time framework

As in the Gaussian units (G), the Heaviside–Lorentz (HL) 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 F = qG1qG2/r² in the Gaussian system, and F = qHL1qHL2/(4π r²) in the HL system. The unit of charge then connects to 1 dyn⋅cm² = 1 ESU² = 4π HLC² , where HLC is the HL unit of charge. The HL quantity q^HL describing a charge is then √4 π  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 system, and it defines two constants, called the vacuum permittivity (ε₀) and the vacuum permeability (μ₀). These can be used to convert SI units to their corresponding Heaviside–Lorentz values, as detailed below. For example, charge is √ε₀L³M / T² . When one puts ε₀ = 8.854 pF/m, L = 0.01 m = 1 cm, M = 0.001 kg = 1 g, and T = 1 s, this evaluates to 9.409669×10⁻¹¹Coulomb, 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 [math]\displaystyle{ \mathbf{E} }[/math] and [math]\displaystyle{ \mathbf{D} }[/math] are the electric field and displacement field, respectively, [math]\displaystyle{ \mathbf{B} }[/math] and [math]\displaystyle{ \mathbf{H} }[/math] are the magnetic fields, [math]\displaystyle{ \mathbf{P} }[/math] is the polarization density, [math]\displaystyle{ \mathbf{M} }[/math] is the magnetization, [math]\displaystyle{ \mathbf{\rho} }[/math] is charge density, [math]\displaystyle{ \mathbf{J} }[/math] is current density, [math]\displaystyle{ c }[/math] is the speed of light in vacuum, [math]\displaystyle{ \phi }[/math] is the electric potential, [math]\displaystyle{ \mathbf{A} }[/math] is the magnetic vector potential, [math]\displaystyle{ \mathbf{F} }[/math] is the Lorentz force acting on a body of charge [math]\displaystyle{ q }[/math] and velocity [math]\displaystyle{ v }[/math], [math]\displaystyle{ \epsilon }[/math] is the permittivity, [math]\displaystyle{ \chi_\text{e} }[/math] is the electric susceptibility. [math]\displaystyle{ \mu }[/math] is the magnetic permeability, and [math]\displaystyle{ \chi_\text{m} }[/math] is the magnetic susceptibility.

Maxwell's equations

Name	SI quantities	Heaviside–Lorentz quantities	Gaussian quantities
Gauss's law (macroscopic)	[math]\displaystyle{ \nabla \cdot \mathbf{D}^\textsf{SI} = \rho_\text{f}^\textsf{SI} }[/math]	[math]\displaystyle{ \nabla \cdot \mathbf{D}^\textsf{HL} = \rho_\text{f}^\textsf{HL} }[/math]	[math]\displaystyle{ \nabla \cdot \mathbf{D}^\textsf{G} = 4\pi\rho_\text{f}^\textsf{G} }[/math]
Gauss's law (microscopic)	[math]\displaystyle{ \nabla \cdot \mathbf{E}^\textsf{SI} = \rho^\textsf{SI}/\epsilon_0 }[/math]	[math]\displaystyle{ \nabla \cdot \mathbf{E}^\textsf{HL} = \rho^\textsf{HL} }[/math]	[math]\displaystyle{ \nabla \cdot \mathbf{E}^\textsf{G} = 4\pi\rho^\textsf{G} }[/math]
Gauss's law for magnetism:	[math]\displaystyle{ \nabla \cdot \mathbf{B}^\textsf{SI} = 0 }[/math]	[math]\displaystyle{ \nabla \cdot \mathbf{B}^\textsf{HL} = 0 }[/math]	[math]\displaystyle{ \nabla \cdot \mathbf{B}^\textsf{G} = 0 }[/math]
Maxwell–Faraday equation (Faraday's law of induction):	[math]\displaystyle{ \nabla \times \mathbf{E}^\textsf{SI} = -\frac{\partial \mathbf{B}^\textsf{SI}} {\partial t} }[/math]	[math]\displaystyle{ \nabla \times \mathbf{E}^\textsf{HL} = -\frac{1}{c}\frac{\partial \mathbf{B}^\textsf{HL}} {\partial t} }[/math]	[math]\displaystyle{ \nabla \times \mathbf{E}^\textsf{G} = -\frac{1}{c}\frac{\partial \mathbf{B}^\textsf{G}} {\partial t} }[/math]
Ampère–Maxwell equation (macroscopic):	[math]\displaystyle{ \nabla \times \mathbf{H}^\textsf{SI} = \mathbf{J}_{\text{f}}^\textsf{SI} + \frac{\partial \mathbf{D}^\textsf{SI}} {\partial t} }[/math]	[math]\displaystyle{ \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]\displaystyle{ \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]
Ampère–Maxwell equation (microscopic):	[math]\displaystyle{ \nabla \times \mathbf{B}^\textsf{SI} = \mu_0 \left( \mathbf{J}^\textsf{SI} + \epsilon_0\frac{\partial \mathbf{E}^\textsf{SI}} {\partial t} \right) }[/math]	[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mathbf{A} }[/math] and [math]\displaystyle{ \phi }[/math]. The definition of the magnetic field in terms of [math]\displaystyle{ \mathbf{A} }[/math], [math]\displaystyle{ \mathbf{B} = \nabla \times \mathbf{A} }[/math], is the same in all systems of units, but the electric field is [math]\displaystyle{ \mathbf{E} = -\nabla\phi-\frac{\partial \mathbf{A}}{\partial t} }[/math] in the SI system, but [math]\displaystyle{ \mathbf{E} = -\nabla\phi-\frac{1}{c} \frac{\partial \mathbf{A}}{\partial t} }[/math] in the HL or G systems.

Other basic laws

Name	SI quantities	Heaviside–Lorentz quantities	Gaussian quantities
Lorentz force	[math]\displaystyle{ \mathbf{F} = q^\textsf{SI}\left(\mathbf{E}^\textsf{SI}+\mathbf{v}\times\mathbf{B}^\textsf{SI}\right) }[/math]	[math]\displaystyle{ \mathbf{F} = q^\textsf{HL}\left(\mathbf{E}^\textsf{HL}+\frac{1}{c}\mathbf{v}\times\mathbf{B}^\textsf{HL}\right) }[/math]	[math]\displaystyle{ \mathbf{F} = q^\textsf{G}\left(\mathbf{E}^\textsf{G}+\frac{1}{c}\mathbf{v}\times\mathbf{B}^\textsf{G}\right) }[/math]
Coulomb's law	[math]\displaystyle{ \mathbf{F} = \frac{1}{4\pi\epsilon_0}\frac{q_1^\textsf{SI} q_2^\textsf{SI}}{r^2} \mathbf{\hat r} }[/math]	[math]\displaystyle{ \mathbf{F} = \frac{1}{4\pi}\frac{q_1^\textsf{HL} q_2^\textsf{HL}}{r^2} \mathbf{\hat r} }[/math]	[math]\displaystyle{ \mathbf{F} = \frac{q_1^\textsf{G} q_2^\textsf{G}}{r^2} \mathbf{\hat r} }[/math]
Electric field of stationary point charge	[math]\displaystyle{ \mathbf{E}^\textsf{SI} = \frac{1}{4\pi\epsilon_0}\frac{q^\textsf{SI}}{r^2} \mathbf{\hat r} }[/math]	[math]\displaystyle{ \mathbf{E}^\textsf{HL} = \frac{1}{4\pi}\frac{q^\textsf{HL}}{r^2} \mathbf{\hat r} }[/math]	[math]\displaystyle{ \mathbf{E}^\textsf{G} = \frac{q^\textsf{G}}{r^2} \mathbf{\hat r} }[/math]
Biot–Savart law	[math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \mathbf{D} }[/math], [math]\displaystyle{ \mathbf{P} }[/math], [math]\displaystyle{ \mathbf{H} }[/math] and [math]\displaystyle{ \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.

SI quantities		Heaviside–Lorentz quantities		Gaussian quantities
[math]\displaystyle{ \mathbf{D}^\textsf{SI} = \epsilon_0 \mathbf{E}^\textsf{SI}+\mathbf{P}^\textsf{SI} }[/math]	[math]\displaystyle{ \mathbf{B}^\textsf{SI} = \mu_0 (\mathbf{H}^\textsf{SI}+\mathbf{M}^\textsf{SI}) }[/math]	[math]\displaystyle{ \mathbf{D}^\textsf{HL} = \mathbf{E}^\textsf{HL}+\mathbf{P}^\textsf{HL} }[/math]	[math]\displaystyle{ \mathbf{B}^\textsf{HL} = \mathbf{H}^\textsf{HL}+\mathbf{M}^\textsf{HL} }[/math]	[math]\displaystyle{ \mathbf{D}^\textsf{G} = \mathbf{E}^\textsf{G}+4\pi\mathbf{P}^\textsf{G} }[/math]	[math]\displaystyle{ \mathbf{B}^\textsf{G} = \mathbf{H}^\textsf{G}+4\pi\mathbf{M}^\textsf{G} }[/math]
[math]\displaystyle{ \mathbf{P}^\textsf{SI} = \chi_\text{e}^\textsf{SI}\epsilon_0\mathbf{E}^\textsf{SI} }[/math]	[math]\displaystyle{ \mathbf{M}^\textsf{SI} = \chi_\text{m}^\textsf{SI}\mathbf{H}^\textsf{SI} }[/math]	[math]\displaystyle{ \mathbf{P}^\textsf{HL} = \chi_\text{e}^\textsf{HL}\mathbf{E}^\textsf{HL} }[/math]	[math]\displaystyle{ \mathbf{M}^\textsf{HL} = \chi_\text{m}^\textsf{HL}\mathbf{H}^\textsf{HL} }[/math]	[math]\displaystyle{ \mathbf{P}^\textsf{G} = \chi_\text{e}^\textsf{G}\mathbf{E}^\textsf{G} }[/math]	[math]\displaystyle{ \mathbf{M}^\textsf{G} = \chi_\text{m}^\textsf{G}\mathbf{H}^\textsf{G} }[/math]
[math]\displaystyle{ \mathbf{D}^\textsf{SI} = \epsilon\mathbf{E}^\textsf{SI} }[/math]	[math]\displaystyle{ \mathbf{B}^\textsf{SI} = \mu^\textsf{SI}\mathbf{H}^\textsf{SI} }[/math]	[math]\displaystyle{ \mathbf{D}^\textsf{HL} = \epsilon\mathbf{E}^\textsf{HL} }[/math]	[math]\displaystyle{ \mathbf{B}^\textsf{HL} = \mu^\textsf{HL}\mathbf{H}^\textsf{HL} }[/math]	[math]\displaystyle{ \mathbf{D}^\textsf{G} = \epsilon\mathbf{E}^\textsf{G} }[/math]	[math]\displaystyle{ \mathbf{B}^\textsf{G} = \mu^\textsf{G}\mathbf{H}^\textsf{G} }[/math]
[math]\displaystyle{ \epsilon^\textsf{SI}/\epsilon_0 = 1+\chi_\text{e}^\textsf{SI} }[/math]	[math]\displaystyle{ \mu^\textsf{SI}/\mu_0 = 1+\chi_\text{m}^\textsf{SI} }[/math]	[math]\displaystyle{ \epsilon^\textsf{HL} = 1+\chi_\text{e}^\textsf{HL} }[/math]	[math]\displaystyle{ \mu^\textsf{HL} = 1+\chi_\text{m}^\textsf{HL} }[/math]	[math]\displaystyle{ \epsilon^\textsf{G} = 1+4\pi\chi_\text{e}^\textsf{G} }[/math]	[math]\displaystyle{ \mu^\textsf{G} = 1+4\pi\chi_\text{m}^\textsf{G} }[/math]

Note that The quantities [math]\displaystyle{ \epsilon^\textsf{SI}/\epsilon_0 }[/math], [math]\displaystyle{ \epsilon^\textsf{HL} }[/math] and [math]\displaystyle{ \epsilon^\textsf{G} }[/math] are dimensionless, and they have the same numeric value. By contrast, the electric susceptibility [math]\displaystyle{ \chi_e }[/math] is dimensionless in all the systems, but has different numeric values for the same material: [math]\displaystyle{ \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 HL units compared to either SI or G units. As Heaviside proposed, removing the 4π from the Gauss law and putting it in the Force law considerably reduces the number of places the π appears compared to Gaussian CGS units.
• Removing the explicit 4π from the Gauss law makes it clear that the inverse-square force law arises by the [math]\displaystyle{ \mathbf{E} }[/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 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 ε₀ and μ₀ that are present in the SI system. (In addition ε₀ and μ₀ are overdetermined, because ε₀μ₀ = c⁻².)

The below points are true in both HL and G systems, but not SI.

• The electric and magnetic fields [math]\displaystyle{ \mathbf{E} }[/math] and [math]\displaystyle{ \mathbf{B} }[/math] have the same dimensions in the HL system, meaning it is easy to recall where factors of c go in the Maxwell equation. Every time derivative comes with a 1/c, which makes it dimensionally the same as a space derivative. In contrast, in SI units [math]\displaystyle{ [\mathbf{E}] / [\mathbf{B}] }[/math] is [c].
• Giving the [math]\displaystyle{ \mathbf{E} }[/math] and [math]\displaystyle{ \mathbf{B} }[/math] fields the same dimension makes the assembly into the electromagnetic tensor more transparent. There are no factors of c that need to be inserted when assembling the tensor out of the three-dimensional fields. Similarly, [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ \mathbf{A} }[/math] have the same dimensions and are the four components of the 4-potential.
• The fields [math]\displaystyle{ \mathbf{D} }[/math], [math]\displaystyle{ \mathbf{H} }[/math], [math]\displaystyle{ \mathbf{P} }[/math] and [math]\displaystyle{ \mathbf{M} }[/math] also have the same dimensions as [math]\displaystyle{ \mathbf{E} }[/math] and [math]\displaystyle{ \mathbf{B} }[/math]. In a vacuum, any expression involving [math]\displaystyle{ \mathbf{D} }[/math] can simply be recast as the same expression with [math]\displaystyle{ \mathbf{E} }[/math]. In SI units, [math]\displaystyle{ \mathbf{D} }[/math] and [math]\displaystyle{ \mathbf{P} }[/math] have the same units, as do [math]\displaystyle{ \mathbf{H} }[/math] and [math]\displaystyle{ \mathbf{M} }[/math], but they have different units from each other and from [math]\displaystyle{ \mathbf{E} }[/math] and [math]\displaystyle{ \mathbf{B} }[/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 ¹⁄₂, 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 SI unit for inductance is already on the large side compared to most inductors, the G 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 HL 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 [math]\displaystyle{ c }[/math] and [math]\displaystyle{ \epsilon_0 }[/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. Outside of these contexts, including for example magazine articles on electric circuits, HL and G 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.

Name	SI units	Heaviside–Lorentz units	Gaussian units
electric field, electric potential	[math]\displaystyle{ \sqrt{\epsilon_0} \left(\mathbf{E}^\textsf{SI}, \varphi^\textsf{SI}\right) }[/math]	[math]\displaystyle{ \left(\mathbf{E}^\textsf{HL}, \varphi^\textsf{HL}\right) }[/math]	[math]\displaystyle{ \frac{1}{\sqrt{4\pi}} \left(\mathbf{E}^\textsf{G}, \varphi^\textsf{G}\right) }[/math]
displacement field	[math]\displaystyle{ \frac{1}{\sqrt{\epsilon_0}}\mathbf{D}^\textsf{SI} }[/math]	[math]\displaystyle{ \mathbf{D}^\textsf{HL} }[/math]	[math]\displaystyle{ \frac{1}{\sqrt{4\pi}}\mathbf{D}^\textsf{G} }[/math]
charge, charge density, current, current density, polarization density, electric dipole moment	[math]\displaystyle{ \frac{1}{\sqrt{\epsilon_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]\displaystyle{ \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]\displaystyle{ \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]
magnetic B field, magnetic flux, magnetic vector potential	[math]\displaystyle{ \frac{1}{\sqrt{\mu_0}}\left(\mathbf{B}^\textsf{SI}, \Phi_\text{m}^\textsf{SI},\mathbf{A}^\textsf{SI}\right) }[/math]	[math]\displaystyle{ \left(\mathbf{B}^\textsf{HL}, \Phi_\text{m}^\textsf{HL},\mathbf{A}^\textsf{HL}\right) }[/math]	[math]\displaystyle{ \frac{1}{\sqrt{4\pi}} \left(\mathbf{B}^\textsf{G}, \Phi_\text{m}^\textsf{G},\mathbf{A}^\textsf{G}\right) }[/math]
magnetic H field	[math]\displaystyle{ \sqrt{\mu_0}\ \mathbf{H}^\textsf{SI} }[/math]	[math]\displaystyle{ \mathbf{H}^\textsf{HL} }[/math]	[math]\displaystyle{ \frac{1}{\sqrt{4\pi}} \mathbf{H}^\textsf{G} }[/math]
magnetic moment, magnetization	[math]\displaystyle{ \sqrt{\mu_0}\left(\mathbf{m}^\textsf{SI}, \mathbf{M}^\textsf{SI}\right) }[/math]	[math]\displaystyle{ \left(\mathbf{m}^\textsf{HL}, \mathbf{M}^\textsf{HL}\right) }[/math]	[math]\displaystyle{ \sqrt{4\pi} \left(\mathbf{m}^\textsf{G}, \mathbf{M}^\textsf{G}\right) }[/math]
relative permittivity, relative permeability	[math]\displaystyle{ \left(\frac{\epsilon^\textsf{SI}}{\epsilon_0}, \frac{\mu^\textsf{SI}}{\mu_0}\right) }[/math]	[math]\displaystyle{ \left(\epsilon^\textsf{HL}, \mu^\textsf{HL}\right) }[/math]	[math]\displaystyle{ \left(\epsilon^\textsf{G}, \mu^\textsf{G}\right) }[/math]
electric susceptibility, magnetic susceptibility	[math]\displaystyle{ \left(\chi_\text{e}^\textsf{SI}, \chi_\text{m}^\textsf{SI}\right) }[/math]	[math]\displaystyle{ \left(\chi_\text{e}^\textsf{HL}, \chi_\text{m}^\textsf{HL}\right) }[/math]	[math]\displaystyle{ 4\pi \left(\chi_\text{e}^\textsf{G}, \chi_\text{m}^\textsf{G}\right) }[/math]
conductivity, conductance, capacitance	[math]\displaystyle{ \frac{1}{\epsilon_0}\left(\sigma^\textsf{SI},S^\textsf{SI},C^\textsf{SI}\right) }[/math]	[math]\displaystyle{ \left(\sigma^\textsf{HL}, S^\textsf{HL}, C^\textsf{HL}\right) }[/math]	[math]\displaystyle{ 4\pi \left(\sigma^\textsf{G}, S^\textsf{G}, C^\textsf{G}\right) }[/math]
resistivity, resistance, inductance	[math]\displaystyle{ \epsilon_0\left(\rho^\textsf{SI},R^\textsf{SI},L^\textsf{SI}\right) }[/math]	[math]\displaystyle{ \left(\rho^\textsf{HL},R^\textsf{HL},L^\textsf{HL}\right) }[/math]	[math]\displaystyle{ \frac{1}{4\pi} \left(\rho^\textsf{G},R^\textsf{G},L^\textsf{G}\right) }[/math]

As an example, starting with the equation

[math]\displaystyle{ \nabla \cdot \mathbf{E}^\textsf{SI} = \rho^\textsf{SI}/\epsilon_0 , }[/math]

and the equations from the table

[math]\displaystyle{ \sqrt{\epsilon_0} \ \mathbf{E}^\textsf{SI} = \mathbf{E}^\textsf{HL} }[/math]

[math]\displaystyle{ \frac{1}{\sqrt{\epsilon_0}} \rho^\textsf{SI} = \rho^\textsf{HL} , }[/math]

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

[math]\displaystyle{ \nabla \cdot \left(\frac{1}{\sqrt{\epsilon_0}} \mathbf{E}^\textsf{HL}\right) = \left(\sqrt{\epsilon_0} \rho^\textsf{HL}\right)/\epsilon_0 , }[/math]

which then simplifies to

[math]\displaystyle{ \nabla \cdot \mathbf{E}^\textsf{HL} = \rho^\textsf{HL} . }[/math]

Replacing Heaviside-Lorentz with natural units

When one takes standard SI textbook equations, and sets ε₀ = µ₀ = c = 1 to get natural units, the resulting equations follow the Heaviside–Lorentz formulation and sizes. The conversion requires no changes to the factor 4π, unlike for the Gaussian equations. Coulomb's inverse-square law equation in SI is F = q₁q₂ / 4πε₀r². Set ε₀ = 1 to get the HL form: F = q₁q₂ / 4πr². The Gaussian form does not have the 4π in the denominator.

By setting c = 1 while using HL units, Maxwell's equations and the Lorentz equation become the same as the SI example with ε₀ = µ₀ = c = 1.

[math]\displaystyle{ \nabla \cdot \mathbf{E} = \rho }[/math] [math]\displaystyle{ \nabla \cdot \mathbf{B} = 0 }[/math] [math]\displaystyle{ \nabla \times \mathbf{E} = -\frac{ \partial \mathbf{B}} {\partial t} }[/math] [math]\displaystyle{ \nabla \times \mathbf{B} = \frac{ \partial \mathbf{E}} {\partial t} + \mathbf{J} }[/math] [math]\displaystyle{ \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}) }[/math]

In quantum mechanics

Additionally setting ε₀ = µ₀ = c = ħ = k_B = 1 yields a natural unit system parameterized by a single scale value, which can be chosen to be a value for mass, time, energy, length, etc. Choosing one, for example a mass m, the others are determined by multiplying with these constants: the length scale via l = ħ / mc, and the time scale from t = ħ / mc², etc.

Heaviside-Lorentz Planck units

Setting [math]\displaystyle{ \epsilon_0 = \mu_0 = c = \hbar = k_\text{B}=4\pi G=1 }[/math] yields the Heaviside–Lorentz Planck units, or rationalized Planck units. The mass scale is chosen such that the gravitational constant is [math]\displaystyle{ \frac{1}{4\pi} }[/math], equal to the Coulomb constant. (By contrast, Gaussian Planck units set [math]\displaystyle{ 4\pi \epsilon_0 = \frac{\mu_0}{4\pi} = c = \hbar = k_\text{B} = G = 1 }[/math].)

Key equations of physics in Heaviside–Lorentz Planck units (rationalized Planck units)

	SI form	Nondimensionalized form
Mass–energy equivalence in special relativity	[math]\displaystyle{ { E = m c^2} \ }[/math]	[math]\displaystyle{ { E = m } \ }[/math]
Energy–momentum relation	[math]\displaystyle{ E^2 = m^2 c^4 + p^2 c^2 \; }[/math]	[math]\displaystyle{ E^2 = m^2 + p^2 \; }[/math]
Ideal gas law	[math]\displaystyle{ PV = nRT = Nk_\text{B}T }[/math]	[math]\displaystyle{ PV = N T }[/math]
Thermal energy per particle per degree of freedom	[math]\displaystyle{ { E = \tfrac12 k_\text{B} T} \ }[/math]	[math]\displaystyle{ { E = \tfrac12 T} \ }[/math]
Boltzmann's entropy formula	[math]\displaystyle{ { S = k_\text{B} \ln \Omega } \ }[/math]	[math]\displaystyle{ { S = \ln \Omega } \ }[/math]
Planck–Einstein relation for angular frequency	[math]\displaystyle{ { E = \hbar \omega } \ }[/math]	[math]\displaystyle{ { E = \omega } \ }[/math]
Planck's law for black body at temperature T	[math]\displaystyle{ I(\omega,T) = \frac{\hbar \omega^3 }{4 \pi^3 c^2}~\frac{1}{e^{\frac{\hbar \omega}{k_\text{B} T}}-1} }[/math]	[math]\displaystyle{ I(\omega,T) = \frac{\omega^3 }{4 \pi^3}~\frac{1}{e^{\omega/T}-1} }[/math]
Stefan–Boltzmann constant σ defined	[math]\displaystyle{ \sigma = \frac{\pi^2 k_\text{B}^4}{60 \hbar^3 c^2} }[/math]	[math]\displaystyle{ \ \sigma = \frac{\pi^2}{60} }[/math]
Schrödinger's equation	[math]\displaystyle{ - \frac{\hbar^2}{2m} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}, t) \psi(\mathbf{r}, t) = i \hbar \frac{\partial \psi(\mathbf{r}, t)}{\partial t} }[/math]	[math]\displaystyle{ - \frac{1}{2m} \nabla^2 \psi(\mathbf{r}, t) + V(\mathbf{r}, t) \psi(\mathbf{r}, t) = i \frac{\partial \psi(\mathbf{r}, t)}{\partial t} }[/math]
Hamiltonian form of Schrödinger's equation	[math]\displaystyle{ H \left| \psi_t \right\rangle = i \hbar \frac{\partial}{\partial t} \left| \psi_t \right\rangle }[/math]	[math]\displaystyle{ H \left| \psi_t \right\rangle = i \frac{\partial}{\partial t} \left| \psi_t \right\rangle }[/math]
Covariant form of the Dirac equation	[math]\displaystyle{ \ ( i\hbar \gamma^\mu \partial_\mu - mc) \psi = 0 }[/math]	[math]\displaystyle{ \ ( i\gamma^\mu \partial_\mu - m) \psi = 0 }[/math]
Unruh temperature	[math]\displaystyle{ T=\frac{\hbar a}{2\pi c k_B} }[/math]	[math]\displaystyle{ T=\frac{a}{2\pi} }[/math]
Coulomb's law	[math]\displaystyle{ F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} }[/math]	[math]\displaystyle{ F = \frac{q_1 q_2}{4 \pi r^2} }[/math]
Maxwell's equations	[math]\displaystyle{ \nabla \cdot \mathbf{E} = \frac{1}{\epsilon_0} \rho }[/math] [math]\displaystyle{ \nabla \cdot \mathbf{B} = 0 \ }[/math] [math]\displaystyle{ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} }[/math] [math]\displaystyle{ \nabla \times \mathbf{B} = \frac{1}{c^2} \left(\frac{1}{\epsilon_0} \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t} \right) }[/math]	[math]\displaystyle{ \nabla \cdot \mathbf{E} = \rho \ }[/math] [math]\displaystyle{ \nabla \cdot \mathbf{B} = 0 \ }[/math] [math]\displaystyle{ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t} }[/math] [math]\displaystyle{ \nabla \times \mathbf{B} = \mathbf{J} + \frac{\partial \mathbf{E}} {\partial t} }[/math]
Biot–Savart law	[math]\displaystyle{ \Delta B = \frac{\mu_0 I}{4\pi} \frac{\Delta L}{r^2} \sin \theta }[/math]	[math]\displaystyle{ \Delta B = \frac{I}{4\pi} \frac{\Delta L}{r^2} \sin \theta }[/math]
Biot–Savart law	[math]\displaystyle{ \mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int_C \frac{I \, d\boldsymbol \ell\times\mathbf{r'}}{|\mathbf{r'}|^3} }[/math]	[math]\displaystyle{ \mathbf{B}(\mathbf{r}) = \frac{1}{4\pi} \int_C \frac{I \, d\boldsymbol \ell\times\mathbf{r'}}{|\mathbf{r'}|^3} }[/math]
Electric field intensity and electric induction	[math]\displaystyle{ \mathbf{D}=\epsilon_0 \mathbf{E} }[/math]	[math]\displaystyle{ \mathbf{D}=\mathbf{E} }[/math]
Magnetic field intensity and magnetic induction	[math]\displaystyle{ \mathbf{B}=\mu_0 \mathbf{H} }[/math]	[math]\displaystyle{ \mathbf{B}=\mathbf{H} }[/math]
Newton's law of universal gravitation	[math]\displaystyle{ F = -G \frac{m_1 m_2}{r^2} }[/math]	[math]\displaystyle{ F = -\frac{m_1 m_2}{4 \pi r^2} }[/math]
Einstein field equations in general relativity	[math]\displaystyle{ { G_{\mu \nu} = 8 \pi {G \over c^4} T_{\mu \nu} } \ }[/math]	[math]\displaystyle{ { G_{\mu \nu} = 2 T_{\mu \nu} } \ }[/math]
Schwarzschild radius	[math]\displaystyle{ r_s = \frac{2 G M}{c^2} }[/math]	[math]\displaystyle{ r_s = \frac{M}{2\pi} }[/math]
Hawking temperature of a black hole	[math]\displaystyle{ T_H=\frac{\hbar c^3}{8\pi G M k_B} }[/math]	[math]\displaystyle{ T_H=\frac{1}{2 M} }[/math]
Bekenstein–Hawking black hole entropy	[math]\displaystyle{ S_\text{BH} = \frac{A_\text{BH} k_\text{B} c^3}{4 G \hbar} = \frac{4\pi G k_\text{B} m^2_\text{BH}}{\hbar c} }[/math]	[math]\displaystyle{ S_\text{BH} = \pi A_\text{BH} = m^2_\text{BH} }[/math]

External links

• Heaviside–Lorentz units