Lorentz-Heaviside Units
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Lorentz%E2%80%93Heaviside 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 ?0 and magnetic constant µ0 do not appear, having been incorporated implicitly into the electromagnetic quantities by the way they are defined. Lorentz-Heaviside units may be regarded as normalizing ?0 = 1 and µ0 = 1, while at the same time revising Maxwell's equations to use the speed of light c instead.[1]

Lorentz-Heaviside units, like SI units but unlike Gaussian units, are rationalized, meaning that there are no factors of 4? appearing explicitly in Maxwell's equations.[2] 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? in these units.[3] Consequently, Lorentz-Heaviside units differ by factors of in the definitions of the electric and magnetic fields and of electric charge. They are often used in relativistic calculations,[note 1] and are used in particle physics. They are particularly convenient when performing calculations in spatial dimensions greater than three such as in string theory.

Length-mass-time framework

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 expressible in terms of the base units of length, time and mass.

Coulomb's equation, used to define charge in these systems, is F = qG
1
qG
2
/r2 in the Gaussian system, and F = qLH
1
qLH
2
/4?r2 in the HLU. The unit of charge then connects to 1 dyn?cm2 = 1 esu2 = 4? hlu. The HLU quantity qLH describing a charge is then larger than the corresponding Gaussian quantity (see below), and the rest follows.

When dimensional analysis for SI units is used, including ?0 and ?0 are used to convert units, the result gives the conversion to and from the Heaviside-Lorentz units. For example, charge is . When one puts ?0 = 8.854 pF/m, L = 0.01 m, M = 0.001 kg, and T = 1 second, this evaluates as . This is the size of the HLU unit of charge.

Maxwell's equations with sources

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 ELH = DLH is the electric field, HLH = BLH is the magnetic field, ?LH is charge density, and JLH is current density.

The Lorentz force equation is:

here qLH is the charge of a test particle with vector velocity v and F 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 ? = 1. In the Gaussian system, Coulomb's equation is F = qG
1
qG
2
/r2. In the Lorentz-Heaviside system, F = qLH
1
qLH
2
/4?r2. From this, one sees that qG
1
qG
2
 = qLH
1
qLH
2
/4?, that the Gaussian charge quantities are smaller than the corresponding Lorentz-Heaviside quantities by a factor of . Other quantities are related as follows.

.

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

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 quantities Lorentz-Heaviside quantities Gaussian quantities
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

Name SI quantities Lorentz-Heaviside quantities Gaussian quantities
Lorentz force
Coulomb's law
Electric field of
stationary point charge
Biot-Savart law

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.

SI quantities Lorentz-Heaviside quantities Gaussian quantities

where

The quantities , and are dimensionless, and they have the same numeric value. By contrast, the electric susceptibility is dimensionless 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 can be expressed as a scalar constant.

SI quantities Lorentz-Heaviside quantities Gaussian quantities

where

The quantities , and are dimensionless, and they have the same numeric value. By contrast, the magnetic susceptibility is dimensionless in all the systems, but has different numeric values for the same material:

Vector and scalar potentials

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

Name SI quantities Lorentz-Heaviside quantities Gaussian quantities
Electric field
(static)
Electric field
(general)
Magnetic B field

Translating expressions and formulae between systems

To convert any expression or formula between SI, Lorentz-Heaviside 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, such as Maxwell's equations.

As an example, starting with the equation

and the equations from the table

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

which then simplifies to

Name SI units Lorentz-Heaviside units Gaussian units
electric field, electric potential
electric displacement field
electric charge, electric charge density,
electric current, electric current density,
polarization density, electric dipole moment
magnetic B field, magnetic flux,
magnetic vector potential
magnetic H field
magnetic moment, magnetization
relative permittivity,
relative permeability
electric susceptibility,
magnetic susceptibility
conductivity, conductance, capacitance
resistivity, resistance, inductance

Replacing CGS with natural units

When one takes standard SI textbook equations, and sets ?0 = 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 4?, unlike for the Gaussian equations. Coulomb's inverse-square law equation in SI is F = q1q2/40r2. Set ?0 = 1 to get the HLU form: F = q1q2/4?r2. The Gaussian form does not have the 4? in the denominator.

By setting c = 1 with HLU, Maxwell's equations and the Lorentz equation become the same as the SI example with ?0 = µ0 = c = 1.

Because these equations can be easily related to SI work, rationalized systems are becoming more fashionable.

In quantum mechanics

Additionally setting ?0 = µ0 = c = ? = kB = 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 = ? / mc, and the time scale from t = ? / mc2, etc.

Lorentz-Heaviside Planck units

Setting yields the Lorentz-Heaviside Planck units, or rationalized Planck units. The mass scale is chosen such that the gravitational constant is , equal to the Coulomb constant. (By constrast, Gaussian Planck units set .)

Key equations of physics in Lorentz-Heaviside Planck units (rationalized Planck units)
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 angular frequency
Planck's law for black body at temperature T
Stefan-Boltzmann constant ? defined
Schrödinger's equation
Hamiltonian form of Schrödinger's equation
Covariant form of the Dirac equation
Unruh temperature
Coulomb's law
Maxwell's equations





Biot-Savart law
Biot-Savart law
Electric field intensity and electric induction
Magnetic field intensity and magnetic induction
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

  1. ^ As used by Einstein, such as in his book: Einstein, Albert (2005). "The Meaning of Relativity (1956, 5th Edition)". Princeton University Press (2005)., pp. 21-

References

  1. ^ Silsbee, Francis (April-June 1962). "Systems of Electrical Units". Journal of Research of the National Bureau of Standards Section C. 66C (2): 137-183. doi:10.6028/jres.066C.014.
  2. ^ Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity, Archived 2009-04-29 at the Wayback Machine" The Physics Teacher 24(2): 97-99. Alternate web link (subscription required)
  3. ^ Littlejohn, Robert (Fall 2011). "Gaussian, SI and Other Systems of Units in Electromagnetic Theory" (PDF). Physics 221A, University of California, Berkeley lecture notes. Retrieved .
  4. ^ Also see Roger Penrose (1989) The Road to Reality. Oxford Univ. Press: 714-17. Knopf.

External links


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