Lorentz–Heaviside units
Lorentz–Heaviside units constitute a system of units within CGS, named for Hendrik Antoon Lorentz and Oliver Heaviside. They share with CGS-Gaussian units the property that the electric constant and magnetic constant 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 and, while at the same time revising Maxwell's equations to use the speed of light instead.
Lorentz–Heaviside units, like SI units but unlike Gaussian units, are rationalized, meaning that there are no factors of 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 in these units. 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, 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 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 in the Gaussian system, and in the HLU. The unit of charge then connects to. The HLU quantity LH describing a charge is then larger than the corresponding Gaussian quantity, and the rest follows.
When dimensional analysis for SI units is used, including and are used to convert units, the result gives the conversion to and from the Heaviside–Lorentz units. For example, charge is. When one puts,,, and second, this evaluates as. 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.
Maxwell's equations with sources
With Lorentz–Heaviside units, Maxwell's equations in free space with sources take the following form:where is the speed of light in vacuum. Here LHLH is the electric field, LHLH is the magnetic field, LH is charge density, and LH is current density.
The Lorentz force equation is:
here LH is the charge of a test particle with vector velocity and 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. In the Gaussian system, Coulomb's equation is. In the Lorentz–Heaviside system,. From this, one sees that, 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 | |||
| Gauss's law | |||
| Gauss's law for magnetism: | |||
| Maxwell–Faraday equation : | |||
| Ampère–Maxwell equation : | |||
| Ampère–Maxwell equation : |
Other basic laws
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 superscript indicates in which system the quantity is defined
- E and D are the electric field and displacement field, respectively;
- P is the polarization density;
- is the permittivity;
- is the permittivity of vacuum ;
- is the electric susceptibility
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 superscript indicates in which system the quantity is defined
- B and H are the magnetic fields
- M is the magnetization
- is the magnetic permeability
- is the permeability of vacuum ;
- is the magnetic susceptibility
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 | |||
| Electric field | |||
| 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 to get natural units, the resulting equations follow the Heaviside–Lorentz formulation and sizes. The conversion requires no changes to the factor, unlike for the Gaussian equations. Coulomb's inverse-square law equation in SI is. Set to get the HLU form:. The Gaussian form does not have the in the denominator.By setting with HLU, Maxwell's equations and the Lorentz equation become the same as the SI example with.
Because these equations can be easily related to SI work, rationalized systems are becoming more fashionable.