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Heaviside-Lorentz units, like SI units but unlike Gaussian units, are rationalized, meaning that there are no factors of 4? 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 4? in these units. Consequently, Heaviside-Lorentz 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.
In the mid-late 19th Century, electromagnetic measurements were frequently made in either the so-calledElectrostatic (ESU) or Electromagnetic (EMU) systems of units. These were based respectively on Coulomb's and Ampere's Law. Use of these systems, as with to the subsequently-developed Gaussian CGS units, resulted in many factors of 4? appearing in formulas for electromagnetic results, even in examples without circular or spherical symmetry. For example, in Gaussian CGS units, the capacitance of sphere of radius r is r while that of a parallel plate capacitor is A/(4?s), where A is the area of the plates and s is their separation.
Heaviside, who was an important, though somewhat isolated, early theorist of electromagnetism, suggested in 1882 that the irrational appearance of 4?s in these sorts of relations could be removed by redefining the unit of the charges and fields.
It is not long since it was taken for granted that the common electrical units were correct. That curious and obtrusive constant 4? 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 4? was an unfortunate and mischievous mistake, the source of many evils. In plain English, the common system of electrical units involves an irrationality of the same kind as would be brought into the metric system of weights and measures, were we to define the unit area to be the area, not of a square with unit side, but of a circle of unit diameter. The constant ? 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. So it is in the common electrical units, which are truly irrational. Now, to make a mistake is easy and natural to man. But that is not enough. 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.
As in the Gaussian units (G), the Heaviside-Lorentz (HL) 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 F = qG 1qG 2/r2 in the Gaussian system, and F = qHL 1qHL 2/(4?r2) in the HL system. The unit of charge then connects to 1 dyn?cm2 = 1 ESU2 = 4?HLC2, where HLC is the HL unit of charge. The HL quantity qHL describing a charge is then larger than the corresponding Gaussian quantity. There are comparable relationships for the other electromagnetic quantities (see below).
The commonly used set of units is the called the SI system, and it defines two constants, called the vacuum permittivity (?0) and the vacuum permeability (?0). These can be used to convert SI units to their corresponding Heaviside-Lorentz values, as detailed below. For example, charge is . When one puts ?0 = 8.854 pF/m, L = 0.01 m = 1 cm, M = 0.001 kg = 1 g, and T = 1 s, this evaluates to Coulomb, which size of the Heaviside-Lorentz unit of charge.
Comparison of Heaviside-Lorentz with other systems of units
The electric and magnetic fields can be written in terms of the potentials and .
The definition of the magnetic field in terms of ,
, is the same in all systems of units, but the electric field is
in the SI system,
but in the HL or G systems.
Below are the expressions for the macroscopic fields , , and in a material medium. It is assumed here for simplicity that the medium is homogeneous, linear, isotropic, and nondispersive, so that the susceptibilities are constants.
Note that 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:
The same statements apply for the corresponding magnetic quantities.
Advantages and disadvantages of Heaviside-Lorentz units
The formulas above are clearly simpler in HL units compared to either SI or G units. As Heaviside proposed, removing the 4? from the Gauss law and putting it in the Force law considerably reduces the number of places the ? appears compared to Gaussian CGS units.
Removing the explicit 4? from the Gauss law makes it clear that the inverse-square force law arises by the field spreading out over the surface of a sphere. This allows a straightforward extension to other dimensions. For example the case of long, parallel wires extending straight in the z direction can be considered a two-dimensional system. Another example is in string theory, where more than three spatial dimensions often need to be considered.
The equations are free of the confusing and arbitrary constants ?0 and ?0 that are present in the SI system. (In addition ?0 and ?0 are overdetermined, because ?0?0 = c-2.)
The below points are true in both HL and G systems, but not SI.
The electric and magnetic fields and have the same dimensions in the HL system, meaning it is easy to recall where factors of c go in the Maxwell equation. Every time derivative comes with a 1/c, which makes it dimensionally the same as a space derivative. In contrast, in SI units is [c].
Giving the and fields the same dimension makes the assembly into the electromagnetic tensor more transparent. There are no factors of c that need to be inserted when assembling the tensor out of the three-dimensional fields. Similarly, and have the same dimensions and are the four components of the 4-potential.
The fields , , and also have the same dimensions as and . In a vacuum, any expression involving can simply be recast as the same expression with . In SI units, and have the same units, as do and , but they have different units from each other and from and .
Despite Heaviside's urgings, it proved difficult to persuade people to switch from the established units. He believed that if the units were changed, "[o]ld style instruments would very soon be in a minority, and then disappear...". Persuading people to switch was already difficult in 1893, and in the meanwhile there have been more than a century's worth of additional textbooks printed and voltmeters built.
Heaviside-Lorentz units, like the Gaussian CGS units by which they generally differ by a factor of about 3+1⁄2, are frequently of rather inconvenient sizes. The Ampere (Coulomb/sec) is reasonable unit for measuring currents commonly encountered, but the ESU/sec, as demonstrated above, is far too small. The Gaussian CGS unit of electric potential is named a statvolt. It is about 300 V, a value which is larger than most commonly encountered potentials. The Henry, the SI unit for inductance is already on the large side compared to most inductors, the G unit is 12 orders of magnitude larger.
A few of the Gaussian CGS units have names; none of the Heaviside-Lorentz units do.
Textbooks in theoretical physics use Heaviside-Lorentz units nearly exclusively, frequently in their natural form (see below), because the 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 and . 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, HL and G units are rarely encountered.
Translating expressions and formulas between systems
To convert any expression or formula between SI, Heaviside-Lorentz 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.
moving the factor across in the latter identities and substituting, the result is
which then simplifies to
Replacing Heaviside-Lorentz 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 HL form: F = q1q2 / 4?r2. The Gaussian form does not have the 4? in the denominator.
By setting c = 1 while using HL units, Maxwell's equations and the Lorentz equation become the same as the SI example with ?0 = µ0 = c = 1.
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.
^ As used by Einstein, such as in his book: Einstein, Albert (2005). The Meaning of Relativity (1956, 5th ed.). Princeton University Press (2005). pp. 21 ff.
^For example, the first and second editions of J. D. Jacksons's Classical Electrodynamics used G units exclusively, but in the third edition Jackson rewrote many of the chapters in SI units. Likewise, E. M. Purcell's Electricity and Magnetism, a commonly-used textbook for introductory studies, was originally written in G units; the third edition was rewritten in SI.