Lorentz–Heaviside units
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Lorentz–Heaviside units (or Heaviside–Lorentz units) constitute a system of units (particularly electromagnetic units) within CGS, named from Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGSGaussian units the property that the electric constant ε_{0} and magnetic constant µ_{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 µ_{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 4π 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 4π in these units.^{[1]} Consequently, Lorentz–Heaviside 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^{[note 1]}, and are the unit of choice in High Energy Physics (particle physics). They are particularly convenient when performing calculations in spatial dimensions greater than three such as in string theory.
Contents
Length–Mass–Time Framework[edit]
As in the Gaussian units, the Heaviside–Lorentz units (HLU in this article) 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/4πr^{2} in the HLU. The unit of charge then connects to 1 dyn cm^{2} = 1 esu^{2} = 4π hlu. The HLU charge is then √4π larger than the Gaussian (see below), and the rest follows.
When dimensional analysis for the SI units is used, including ε_{0} and μ_{0} are used to convert units, the result gives the conversion to and from the Heaviside–Lorentz units. For example, charge is √ε_{0}L^{3}MT^{−2}. When one puts ε_{0} = 8.854 pF/m, L = 0.01 m, M = 0.001 kg, and T = 1 second, this evaluates as 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[edit]
In systemindependent form, the Maxwell equations are
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 = 1/4π, 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 radius^{2} / source) with the gaussian divergence constant (β = flux through a surface / enclosed sources). One can easily show that γ = 4πβ, 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: 2 for planar symmetry, 2π for cylindrical symmetry and 4π 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[edit]
With Lorentz–Heaviside units, Maxwell's equations in free space with sources take the following form:
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:
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/4πR^{2}. From this, one sees that QQ = qq/4π, that the Gaussian units are larger by a factor of √4π. Other quantities follow as follows.
 .
List of equations and comparison with other systems of units[edit]
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[edit]
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 the Kelvin–Stokes theorem.
Name  SI units  Gaussian units  Lorentz–Heaviside units 

Gauss's law (macroscopic) 

Gauss's law (microscopic) 

Gauss's law for magnetism:  
Maxwell–Faraday equation (Faraday's law of induction): 

Ampère–Maxwell equation (macroscopic): 

Ampère–Maxwell equation (microscopic): 
Other basic laws[edit]
Name  SI units  Gaussian units  Lorentz–Heaviside units 

Lorentz force  
Coulomb's law  

Electric field of stationary point charge 

Biot–Savart law 
Dielectric and magnetic materials[edit]
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 

where
 E and D are the electric field and displacement field, respectively;
 P is the polarization density;
 is the permittivity;
 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 may be omitted);
 is the electric susceptibility
The quantities in both Lorentz–Heaviside and Gaussian units and in SI are dimensionless, and they have the same numeric value. By contrast, the electric susceptibility is unitless in all the systems, but has different numeric values for the same material:
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 

where
 B and H are the magnetic fields
 M is the magnetization
 is the magnetic permeability
 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 may be omitted);
 is the magnetic susceptibility
The quantities in both Lorentz–Heaviside and Gaussian units and in SI are dimensionless, and they have the same numeric value. By contrast, the magnetic susceptibility is unitless in all the systems, but has different numeric values for the same material:
Vector and scalar potentials[edit]
The electric and magnetic fields can be written in terms of a vector potential A and a scalar potential :
Name  Lorentz–Heaviside units  Gaussian units  SI units 

Electric field (static) 

Electric field (general) 

Magnetic B field 
General rules to translate a formula[edit]
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.
Replacing CGS with natural units[edit]
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 4π, unlike for the Gaussian equations. Coulomb's inversesquare law equation in SI is F = q_{1}q_{2}/4πεr^{2}. Set ε = 1 to get the HLU form: F = q_{1}q_{2}/4πr^{2}. The Gaussian form does not have the 4π 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.
Because these equations can be easily related to SI work, HLUstyle (i.e. rationalized) systems are becoming more fashionable.
In quantum mechanics[edit]
Additionally setting ε = μ = c = ħ = k_{B} = 1 yields a natural unit system parameterized by a single value, the mass scale m. Choosing m determines the length scale via the reduced Compton wavelength ƛ = ħ / mc, and the time scale from ƛ / c.
Rationalized Planck units[edit]
Setting ε = μ = c = ħ = k_{B} = 4πG = 1 yields the LorentzHeaviside Planck units, in which the mass scale is chosen such that the gravitational coupling constant is 1.
SI form  Nondimensionalized form  

Mass–energy equivalence in special relativity  
Energy–momentum relation  
Ideal gas law  
Thermal energy per particle per degree of freedom  
Boltzmann's entropy formula  
Planck–Einstein relation for energy and angular frequency  
Planck's law (surface intensity per unit solid angle per unit angular frequency) for black body at temperature T.  
Stefan–Boltzmann constant σ defined  
Schrödinger's equation  
Hamiltonian form of Schrödinger's equation  
Covariant form of the Dirac equation  
Unruh temperature  
Coulomb's law  
Maxwell's equations 


Newton's law of universal gravitation  
Einstein field equations in general relativity  
Schwarzschild radius  
Hawking temperature of a black hole  
Bekenstein–Hawking black hole entropy^{[4]} 
Notes[edit]
 ^ By Einstein himself, as in his book: Einstein, Albert. "The Meaning of Relativity (1956, 5th Edition)". Princeton University Press (2005)., pag 21 et seq.
References[edit]
 ^ ^{a} ^{b} ^{c} Littlejohn, Robert (Fall 2011). "Gaussian, SI and Other Systems of Units in Electromagnetic Theory" (PDF). Physics 221A, University of California, Berkeley lecture notes. Retrieved 20080506.
 ^ Silsbee, Francis (April–June 1962). "Systems of Electrical Units". JOURNAL OF RESEARCH of the National Bureau of Standards – C. Engineering and Instrumentation. 66C (2): 137–183. doi:10.6028/jres.066C.014.
 ^ Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity, Archived 20090429 at the Wayback Machine" The Physics Teacher 24(2): 97–99. Alternate web link (subscription required)
 ^ Also see Roger Penrose (1989) The Road to Reality. Oxford Univ. Press: 71417. Knopf.