Lorentz–Heaviside 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 ε0 and magnetic constant µ0 do not appear, having been incorporated implicitly into the electromagnetic quantities by the way they are defined. Heaviside-Lorentz units may be regarded as normalizing ε0 = 1 and µ0 = 1, while at the same time revising Maxwell's equations to use the speed of light c instead.[1]

Heaviside–Lorentz units, like SI units but unlike Gaussian units, are rationalized, meaning that there are no factors of 4π 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 4π in these units.[3] Consequently, Heaviside-Lorentz 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 used in particle physics. They are particularly convenient when performing calculations in spatial dimensions greater than three such as in string theory.

MotivationEdit

In the mid-late 19th Century, electromagnetic measurements were frequently made in either the so-called Electrostatic (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 4π 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/(4πs), 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 4πs in these sorts of relations could be removed by redefining the unit of the charges and fields.[4][5]

In 1893 Heaviside wrote[6]

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. 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 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. 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.

Length–mass–time frameworkEdit

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
1
qG
2
/r2 in the Gaussian system, and F = qHL
1
qHL
2
/(4π r2) in the HL system. The unit of charge then connects to 1 dyn⋅cm2 = 1 ESU2 = 4π HLC2 , where HLC is the HL unit of charge. The HL quantity qHL describing a charge is then 4 π  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 (ε0) and the vacuum permeability (μ0). These can be used to convert SI units to their corresponding Heaviside–Lorentz values, as detailed below. For example, charge is ε0L3M / T2 . When one puts ε0 = 8.854 pF/m, L = 0.01 m = 1 cm, M = 0.001 kg = 1 g, and T = 1 s, this evaluates to 9.409669×10−11Coulomb, which size of the Heaviside-Lorentz unit of charge.

Comparison of Heaviside-Lorentz with other systems of unitsEdit

This section has a list of the basic formulas of electromagnetism, given in SI, Heaviside–Lorentz, and Gaussian units. Here   and   are the electric field and displacement field, respectively,   and   are the magnetic fields,   is the polarization density,   is the magnetization,   is charge density,   is current density,   is the speed of light in vacuum,   is the electric potential,   is the magnetic vector potential,   is the Lorentz force acting on a body of charge   and velocity  ,   is the permittivity,   is the electric susceptibility.   is the magnetic permeability, and   is the magnetic susceptibility.

Maxwell's equationsEdit

Name SI quantities Heaviside–Lorentz 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):
     

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.

Other basic lawsEdit

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

Dielectric and magnetic materialsEdit

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.

SI quantities Heaviside–Lorentz quantities Gaussian quantities
           
           
           
           

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 unitsEdit

AdvantagesEdit

  • The formulas above are clearly simpler in HL units compared to either SI or G units. As Heaviside proposed, removing the 4π from the Gauss law and putting it in the Force law considerably reduces the number of places the π appears compared to Gaussian CGS units.
  • Removing the explicit 4π 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 ε0 and μ0 that are present in the SI system. (In addition ε0 and μ0 are overdetermined, because ε0μ0 = 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  .

DisadvantagesEdit

  • 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..."[6]. 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+12, 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 systemsEdit

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.

Name SI units Heaviside–Lorentz units Gaussian units
electric field, electric potential      
displacement field      
charge, charge density,
current, 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      

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

 

Replacing Heaviside-Lorentz with natural unitsEdit

When one takes standard SI textbook equations, and sets ε0 = 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 4π, unlike for the Gaussian equations. Coulomb's inverse-square law equation in SI is F = q1q2 / 4πε0r2. Set ε0 = 1 to get the HL form: F = q1q2 / 4πr2. The Gaussian form does not have the 4π 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 ε0 = µ0 = c = 1.

 
 
 
 
 

In quantum mechanicsEdit

Additionally setting ε0 = µ0 = c = ħ = 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 = ħ / mc, and the time scale from t = ħ / mc2, etc.

Heaviside-Lorentz Planck unitsEdit

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

Key equations of physics in Heaviside–Lorentz 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 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  

 
 
 

 

 
 
 

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    
BekensteinHawking black hole entropy[9]    

NotesEdit

  1. ^ 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.
  2. ^ For example, the first and second editions of J. D. Jacksons's Classical Electrodynamics[7] 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,[8] a commonly-used textbook for introductory studies, was originally written in G units; the third edition was rewritten in SI.

ReferencesEdit

  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 2008-05-06.
  4. ^ Heaviside, O. (1882-11-18). "The Relations between Magnetic Force and Electric Current". The Electrician, London. London.
  5. ^ "System of Measurement Units". Engineering and Technology History Wiki. Retrieved 2021-12-23.
  6. ^ a b Heaviside, Oliver (1893). Electromagnetic Theory, Vol. 1. The D. van Nostrand Company, London. p. xi. (See also https://wiki.opensourceecology.de/Heaviside_1893)
  7. ^ Jackson, J.D. (1973). Classical Electrodynamics, Second Edition. John Wiley & Sons, New York. p. 811-821.
  8. ^ Purcell, E.M. (1965). electricity and magnetism, Berkeley physics course, vol 2 (first ed.). McGraw Hill, New York. p. 449-452.
  9. ^ Also see Roger Penrose (1989) The Road to Reality. Oxford Univ. Press: 714-17. Knopf.

External linksEdit