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 $ε 0$ and magnetic constant $ε 0$ do not appear, having been incorporated implicitly into the unit system and electromagnetic equations. Lorentz–Heaviside units may be regarded as normalizing $ε 0 = 1$ and $ε 0 = 1$ , while at the same time revising Maxwell's equations to use the speed of light $c$ instead.^[1]^[2]

Lorentz–Heaviside units, like SI units but unlike Gaussian units, are rationalized, meaning that there are no factors of $4 π$ appearing explicitly in Maxwell's equations.^[1]^[3] The fact that these units are rationalized partly explains their appeal in quantum field theory: the Lagrangian underlying the theory does not have any factors of $4 π$ in these units.^[1] Consequently, Lorentz–Heaviside units differ by factors of $\sqrt 4 π$ in the definitions of the electric and magnetic fields and of electric charge. They are particularly convenient when performing calculations in spatial dimensions greater than three such as in string theory. They are often used in relativistic calculations.

Length–Mass–Time Framework

As in the Gaussian units, the Heaviside–Lorentz units use the length–mass–time dimensions. This means that all of the electric and magnetic units are derived units, dependent on the sizes of length and force.

Coulomb's equation, used to derive the unit of charge, is $F = QQ / r 2$ in the Gaussian system, and $F = qq /4 πr 2$ in the HLU. The unit of charge then connects to $1 dyn cm 2 = 1 esu 2 = 4 π hlu$ . The HLU charge is then $\sqrt 4 π$ larger than the Gaussian (see below), and the rest follow.

When the dimensional analysis for the Gaussian units are used, including ε and μ are used to convert units, the result gives the conversion to and from the Heaviside–Lorentz units. For example, charge is $\sqrt ε L 3 MT -2$ . When one puts $ε = 8.854 pF/m$ , $L = 0.01 m$ , $M = 0.001 kg$ , and $T = 1$ second, this evaluates as 6989940966899999999♠9.409669×10⁻¹¹ C. This is the size of the HLU unit of charge.

Because the Heaviside–Lorentz units continue to use separate electric and magnetic units, an additional constant is needed when electric and magnetic quantities appear in the same formula. As in the Gaussian system, this constant appears as the electromagnetic velocity $c$ .

Rationalization

In system-independent form, the Maxwell equations are

{\begin{aligned}\nabla \cdot \mathbf {D} &=\rho /\beta ,\\\quad \nabla \cdot \mathbf {B} &=0,\\\quad \kappa \nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}},\\\quad \kappa \nabla \times \mathbf {H} &={\frac {\partial \mathbf {D} }{\partial t}}+\mathbf {J} /\beta ,\end{aligned}}

along with $D = ε 0 E$ and $B = μ 0 H$ . The constants $β$ and $κ$ vary from system to system. One can show that $ε 0 μ 0 c 2 = κ 2$ .

The Gaussian system puts

β = 1/4 π

κ = c

The HLU system puts

β = 1

κ = c

The SI system puts

β = 1

κ = 1

What rationalisation does is to replace the radiance constant ( $γ$ = intensity at radius² / source) with the gaussian divergence constant ( $β$ = flux through a surface / enclosed sources). One can easily show that $γ = 4 πβ$ , by considering the case of a sphere around a point, and intensity as density of flux. The older models set $γ = 1$ , while the rationalised systems have $β = 1$ . Rationalized equations in physics generally have a factor related to the effective spatial symmetry: $1$ for planar symmetry, $2 π$ for cylindrical symmetry and $4 π$ for spherical symmetry.

The constant $κ$ connects the electric and magnetic units through $Q = Iκt$ . When electric and magnetic systems are defined as in the Gaussian or Heaviside–Lorentz systems, $κ = c$ derives from the electromagnetic wave equations. Most systems have $κ = 1$ , where the electric and magnetic systems are connected by $Q = It$ . Therefore, most books use $Q = It$ instead of $Q = Iκt$ .

Maxwell's equations with sources

With Lorentz–Heaviside units, Maxwell's equations in free space with sources take the following form:

\nabla \cdot \mathbf {E} =\rho \,

\nabla \cdot \mathbf {B} =0\,

\nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}\,

\nabla \times \mathbf {B} ={\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}+{\frac {1}{c}}\mathbf {J} \,

where $c$ is the speed of light in vacuum. Here $E = D$ is the electric field, $H = B$ is the magnetic field, $ρ$ is charge density, and $J$ is current density.

The Lorentz force equation is:

\mathbf {F} _{q}=q\left(\mathbf {E} +{\frac {\mathbf {v} _{q}}{c}}\times \mathbf {B} \right)\,

here $q$ is the charge of a test particle with vector velocity $v q$ and $F q$ is the combined electric and magnetic force acting on that test particle.

In both the Gaussian and Heaviside–Lorentz systems, the electrical and magnetic units are derived from the mechanical systems. Charge is defined through Coulomb's equation, with $ε = 1$ . In the gaussian system, Coulomb's equation is $F = QQ / R 2$ . In the Heaviside Lorentz system, $F = qq /4 πR 2$ . From this, one sees that $QQ = qq /4 π$ , that the Gaussian units are larger by a factor of $\sqrt 4 π$ . Other quantities follow as follows.

q_{\mathrm {LH} }\ =\ {\sqrt {4\pi }}\ q_{\mathrm {G} }

\mathbf {E} _{\mathrm {LH} }\ =\ {\mathbf {E} _{\mathrm {G} } \over {\sqrt {4\pi }}}

\mathbf {B} _{\mathrm {LH} }\ =\ {\mathbf {B} _{\mathrm {G} } \over {\sqrt {4\pi }}}

List of equations and comparison with other systems of units

This section has a list of the basic formulae of electromagnetism, given in Lorentz-Heaviside, Gaussian and SI units. Most symbol names are not given; for complete explanations and definitions, please click to the appropriate dedicated article for each equation.

Maxwell's equations

Main article: Maxwell's equations

Here are Maxwell's equations, both in macroscopic and microscopic forms. Only the "differential form" of the equations is given, not the "integral form"; to get the integral forms apply the divergence theorem or Kelvin–Stokes theorem.

Name	Lorentz–Heaviside units	Gaussian units	SI units
Gauss's law (macroscopic)	$\nabla \cdot \mathbf {D} =\rho _{\text{f}}$	$\nabla \cdot \mathbf {D} =4\pi \rho _{\text{f}}$	$\nabla \cdot \mathbf {D} =\rho _{\text{f}}$
Gauss's law (microscopic)	$\nabla \cdot \mathbf {E} =\rho$	$\nabla \cdot \mathbf {E} =4\pi \rho$	$\nabla \cdot \mathbf {E} =\rho /\epsilon _{0}$
Gauss's law for magnetism:	$\nabla \cdot \mathbf {B} =0$	$\nabla \cdot \mathbf {B} =0$	$\nabla \cdot \mathbf {B} =0$
Maxwell–Faraday equation (Faraday's law of induction):	$\nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}$	$\nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}$	$\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}$
Ampère–Maxwell equation (macroscopic):	$\nabla \times \mathbf {H} ={\frac {1}{c}}\mathbf {J} _{\text{f}}+{\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}$	$\nabla \times \mathbf {H} ={\frac {4\pi }{c}}\mathbf {J} _{\text{f}}+{\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}$	$\nabla \times \mathbf {H} =\mathbf {J} _{\text{f}}+{\frac {\partial \mathbf {D} }{\partial t}}$
Ampère–Maxwell equation (microscopic):	$\nabla \times \mathbf {B} ={\frac {1}{c}}\mathbf {J} +{\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}$	$\nabla \times \mathbf {B} ={\frac {4\pi }{c}}\mathbf {J} +{\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}$	$\nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}$

Other basic laws

Name	Lorentz–Heaviside units	Gaussian units	SI units
Lorentz force	$\mathbf {F} =q\left(\mathbf {E} +{\frac {1}{c}}\mathbf {v} \times \mathbf {B} \right)$	$\mathbf {F} =q\left(\mathbf {E} +{\frac {1}{c}}\mathbf {v} \times \mathbf {B} \right)$	$\mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)$
Coulomb's law	$\mathbf {F} ={\frac {1}{4\pi }}{\frac {q_{1}q_{2}}{r^{2}}}\mathbf {\hat {r}}$	$\mathbf {F} ={\frac {q_{1}q_{2}}{r^{2}}}\mathbf {\hat {r}}$	$\mathbf {F} ={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}\mathbf {\hat {r}}$
Electric field of stationary point charge	$\mathbf {E} ={\frac {1}{4\pi }}{\frac {q}{r^{2}}}\mathbf {\hat {r}}$	$\mathbf {E} ={\frac {q}{r^{2}}}\mathbf {\hat {r}}$	$\mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}{\frac {q}{r^{2}}}\mathbf {\hat {r}}$
Biot–Savart law	$\mathbf {B} ={\frac {1}{4\pi }}{\frac {1}{c}}\oint {\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}$	$\mathbf {B} ={\frac {1}{c}}\oint {\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}$	$\mathbf {B} ={\frac {\mu _{0}}{4\pi }}\oint {\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}$

Dielectric and magnetic materials

Below are the expressions for the various fields in a dielectric medium. It is assumed here for simplicity that the medium is homogeneous, linear, isotropic, and nondispersive, so that the permittivity is a simple constant.

Lorentz–Heaviside units	Gaussian units	SI units
$\mathbf {D} =\mathbf {E} +\mathbf {P}$	$\mathbf {D} =\mathbf {E} +4\pi \mathbf {P}$	$\mathbf {D} =\epsilon _{0}\mathbf {E} +\mathbf {P}$
$\mathbf {P} =\chi _{\text{e}}\mathbf {E}$	$\mathbf {P} =\chi _{\text{e}}\mathbf {E}$	$\mathbf {P} =\chi _{\text{e}}\epsilon _{0}\mathbf {E}$
$\mathbf {D} =\epsilon \mathbf {E}$	$\mathbf {D} =\epsilon \mathbf {E}$	$\mathbf {D} =\epsilon \mathbf {E}$
$\epsilon =1+\chi _{\text{e}}$	$\epsilon =1+4\pi \chi _{\text{e}}$	$\epsilon /\epsilon _{0}=1+\chi _{\text{e}}$

where

E and D are the electric field and displacement field, respectively;
P is the polarization density;
$\epsilon$ is the permittivity;
$\epsilon _{0}$ is the permittivity of vacuum (used in the SI system, but takes on a numeric value of 1 in Gaussian and Lorentz–Heaviside units and so is negligible);
$\chi _{\text{e}}$ is the electric susceptibility

The quantities $\epsilon$ in both Lorentz-Heaviside and Gaussian units and $\epsilon /\epsilon _{0}$ in SI are dimensionless, and they have the same numeric value. By contrast, the electric susceptibility $\chi _{e}$ is unitless in all the systems, but has different numeric values for the same material:

\chi _{\text{e}}^{\text{SI}}=\chi _{\text{e}}^{\text{LH}}=4\pi \chi _{\text{e}}^{\text{G}}

Next, here are the expressions for the various fields in a magnetic medium. Again, it is assumed that the medium is homogeneous, linear, isotropic, and nondispersive, so that the permeability is a simple constant.

Lorentz–Heaviside units	Gaussian units	SI units
$\mathbf {B} =\mathbf {H} +\mathbf {M}$	$\mathbf {B} =\mathbf {H} +4\pi \mathbf {M}$	$\mathbf {B} =\mu _{0}(\mathbf {H} +\mathbf {M} )$
$\mathbf {M} =\chi _{\text{m}}\mathbf {H}$	$\mathbf {M} =\chi _{\text{m}}\mathbf {H}$	$\mathbf {M} =\chi _{\text{m}}\mathbf {H}$
$\mathbf {B} =\mu \mathbf {H}$	$\mathbf {B} =\mu \mathbf {H}$	$\mathbf {B} =\mu \mathbf {H}$
$\mu =1+\chi _{\text{m}}$	$\mu =1+4\pi \chi _{\text{m}}$	$\mu /\mu _{0}=1+\chi _{\text{m}}$

where

B and H are the magnetic fields
M is magnetization
$\mu$ is magnetic permeability
$\mu _{0}$ is the permeability of vacuum (used in the SI system, but takes on a numeric value of 1 in Gaussian and Lorentz–Heaviside units and so is negligible);
$\chi _{\text{m}}$ is the magnetic susceptibility

The quantities $\mu$ in both Lorentz-Heaviside and Gaussian units and $\mu /\mu _{0}$ in SI are dimensionless, and they have the same numeric value. By contrast, the magnetic susceptibility $\chi _{\text{m}}$ is unitless in all the systems, but has different numeric values for the same material:

\chi _{\text{m}}^{\text{SI}}=\chi _{\text{m}}^{\text{LH}}=4\pi \chi _{\text{m}}^{\text{G}}

Vector and scalar potentials

Main articles: Magnetic vector potential and Electric potential

The electric and magnetic fields can be written in terms of a vector potential A and a scalar potential φ:

Name	Lorentz–Heaviside units	Gaussian units	SI units
Electric field (static)	$\mathbf {E} =-\nabla \phi$	$\mathbf {E} =-\nabla \phi$	$\mathbf {E} =-\nabla \phi$
Electric field (general)	$\mathbf {E} =-\nabla \phi -{\frac {1}{c}}{\frac {\partial \mathbf {A} }{\partial t}}$	$\mathbf {E} =-\nabla \phi -{\frac {1}{c}}{\frac {\partial \mathbf {A} }{\partial t}}$	$\mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}$
Magnetic B field	$\mathbf {B} =\nabla \times \mathbf {A}$	$\mathbf {B} =\nabla \times \mathbf {A}$	$\mathbf {B} =\nabla \times \mathbf {A}$

General rules to translate a formula

To convert any formula from Lorentz–Heaviside units to Gaussian or to SI units, replace each symbol in the Lorentz-Heaviside column by the corresponding expression in the Gaussian column or in the SI column (vice versa to convert the other way). This will reproduce any of the specific formulas given in the list above, such as Maxwell's equations.

Name	Lorentz–Heaviside units	Gaussian units	SI units
Speed of light	$c$	$c$	${\frac {1}{\sqrt {\epsilon _{0}\mu _{0}}}}$
Electric field, Electric potential	$\left(\mathbf {E} ,\varphi \right)$	${\frac {1}{\sqrt {4\pi }}}\left(\mathbf {E} ,\varphi \right)$	${\sqrt {\epsilon _{0}}}\left(\mathbf {E} ,\varphi \right)$
Electric displacement field	$\mathbf {D}$	${\frac {1}{\sqrt {4\pi }}}\mathbf {D}$	${\frac {1}{\sqrt {\epsilon _{0}}}}\mathbf {D}$
Charge, Charge density, Current, Current density, Polarization density, Electric dipole moment	$\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)$	${\sqrt {4\pi }}\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)$	${\frac {1}{\sqrt {\epsilon _{0}}}}\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)$
Magnetic B field, Magnetic flux, Magnetic vector potential	$\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)$	${\frac {1}{\sqrt {4\pi }}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)$	${\frac {1}{\sqrt {\mu _{0}}}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)$
Magnetic H field	$\mathbf {H}$	${\frac {1}{\sqrt {4\pi }}}\mathbf {H}$	${\sqrt {\mu _{0}}}\mathbf {H}$
Magnetic moment, Magnetization	$\left(\mathbf {m} ,\mathbf {M} \right)$	${\sqrt {4\pi }}\left(\mathbf {m} ,\mathbf {M} \right)$	${\sqrt {\mu _{0}}}\left(\mathbf {m} ,\mathbf {M} \right)$
Relative permittivity, Relative permeability	$\left(\epsilon ,\mu \right)$	$\left(\epsilon ,\mu \right)$	$\left({\frac {\epsilon }{\epsilon _{0}}},{\frac {\mu }{\mu _{0}}}\right)$
Electric susceptibility, Magnetic susceptibility	$\left(\chi _{\text{e}},\chi _{\text{m}}\right)$	$4\pi \left(\chi _{\text{e}},\chi _{\text{m}}\right)$	$\left(\chi _{\text{e}},\chi _{\text{m}}\right)$
Conductivity, Conductance, Capacitance	$\left(\sigma ,S,C\right)$	$4\pi \left(\sigma ,S,C\right)$	${\frac {1}{\epsilon _{0}}}\left(\sigma ,S,C\right)$
Resistivity, Resistance, Inductance	$\left(\rho ,R,L\right)$	${\frac {1}{4\pi }}\left(\rho ,R,L\right)$	$\epsilon _{0}\left(\rho ,R,L\right)$

Replacing CGS with natural units

When one takes standard SI textbook equations, and sets $ε = μ = c = 1$ to get natural units, the resulting equations follow the Heaviside–Lorentz formulation and sizes. The conversion requires no changes to the factor $4 π$ , unlike for the Gaussian equations. Coulomb's inverse-square law equation in SI is $F = q 1 q 2 /4 πεr 2$ . Set $ε = 1$ to get the HLU form: $F = q 1 q 2 /4 πr 2$ . The Gaussian form does not have the $4 π$ in the denominator.

By setting $c = 1$ with HLU, Maxwell's equations and the Lorentz equation become the same as the SI example with $ε = μ = c = 1$ .

\nabla \cdot \mathbf {E} =\rho \,

\nabla \cdot \mathbf {B} =0\,

\nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\,

\nabla \times \mathbf {B} ={\frac {\partial \mathbf {E} }{\partial t}}+\mathbf {J} \,

\mathbf {F} _{q}=q(\mathbf {E} +\mathbf {v} _{q}\times \mathbf {B} )\,

Because these equations can be easily related to SI work, HLU-style (i.e. rationalized) systems are becoming more fashionable.

References

1 2 3 Littlejohn, Robert (Fall 2011). "Gaussian, SI and Other Systems of Units in Electromagnetic Theory" (pdf). Physics 221A, University of California, Berkeley lecture notes [Warning: this document contains a highly opinionated and in some places factually incorrect invective against the SI system.] Retrieved 2008-05-06.
↑ Silsbee, Francis (April–June 1962). "Systems of Electrical Units" (PDF). JOURNAL OF RESEARCH of the Nationol Bureau of Standards- C. Engineering and Instrumentation. 66C (2): pp. 137–183. doi:10.6028/jres.066C.014.
↑ Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity," The Physics Teacher 24(2): 97–99. Alternate web link (subscription required)

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