Heaviside–Lorentz units (or Lorentz–Heaviside units) constitute a system of units and quantities that extends the CGS with a particular set of equations that defines electromagnetic quantities, named for Oliver Heaviside and Hendrik Antoon Lorentz.
They share with the CGS-Gaussian system that the electric constant ε0 and magnetic constant µ0 do not appear in the defining equations for electromagnetism, having been incorporated implicitly into the electromagnetic quantities.
Heaviside–Lorentz units may be thought of as normalizing ε0 = 1 and µ0 = 1, while at the same time revising Maxwell's equations to use the speed of light c instead.
[2] That this system is rationalized partly explains its appeal in quantum field theory: the Lagrangian underlying the theory does not have any factors of 4π when this system is used.
It is 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.
In the mid-late 19th century, electromagnetic measurements were frequently made in either the so-named electrostatic (ESU) or electromagnetic (EMU) systems of units.
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, including those without any circular or spherical symmetry.
For example, in the CGS-Gaussian system, the capacitance of sphere of radius r is r while that of a parallel plate capacitor is A/4πd, where A is the area of the smaller plate and d is their separation.
Heaviside, who was an important, though somewhat isolated,[citation needed] early theorist of electromagnetism, suggested in 1882 that the irrational appearance of 4π in these sorts of relations could be removed by redefining the units for charges and fields.
[4][5] In his 1893 book Electromagnetic Theory,[6] Heaviside wrote in the introduction: 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.
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.
Now, to make a mistake is easy and natural to man.
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.
This means that all of the units of electric and magnetic quantities are expressible in terms of the units of the base quantities length, time and mass.
These can be used to convert SI units to their corresponding Heaviside–Lorentz values, as detailed below.
When one puts ε0 = 8.854 pF/m, L = 1 cm, M = 1 g, and T = 1 s, this evaluates to 9.409669×10−11 C, the SI-equivalent of the Heaviside–Lorentz unit of charge.
This section has a list of the basic formulas of electromagnetism, given in the SI, Heaviside–Lorentz, and Gaussian systems.
Here E and D are the electric field and displacement field, respectively, B and H are the magnetic fields, P is the polarization density, M is the magnetization, ρ is charge density, J is current density, c is the speed of light in vacuum, ϕ is the electric potential, A is the magnetic vector potential, F is the Lorentz force acting on a body of charge q and velocity v, ε is the permittivity, χe is the electric susceptibility, μ is the magnetic permeability, and χm is the magnetic susceptibility.
The electric and magnetic fields can be written in terms of the potentials A and ϕ.
It is assumed here for simplicity that the medium is homogeneous, linear, isotropic, and nondispersive, so that the susceptibilities are constants.
is dimensionless in all the systems, but has different numeric values for the same material:
The below points are true in both Heaviside–Lorentz and Gaussian systems, but not SI.
Textbooks in theoretical physics use Heaviside–Lorentz units nearly exclusively, frequently in their natural form (see below), 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 c and ε0.
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.
[note 2] Outside of these contexts, including for example magazine articles on electric circuits, Heaviside–Lorentz and Gaussian CGS units are rarely encountered.
Moving the factor across in the latter identities and substituting, the result is