Lorentz–Heaviside units

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 derived units, dependent on the sizes of length and force.

Coulomb's equation, used to derive the unit of charge, is F = QQ/r² in the Gaussian system, and F = qq/4πr² in the HLU. The unit of charge then connects to 1 dyn cm² = 1 esu² = 4π hlu. The HLU charge is then √4π larger than the Gaussian (see below), and the rest follow.

When the dimensional analysis for the Gaussian units are used, including ε and μ are used to convert units, the result gives the conversion to and from the Heaviside–Lorentz units. For example, charge is √ε L³MT⁻². When one puts ε = 8.854 pF/m, L = 0.01 m, M = 0.001 kg, and T = 1 second, this evaluates as 6989940966899999999♠9.409669×10⁻¹¹ C. 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 c.

Rationalization

In system-independent form, the Maxwell equations are

∇ ⋅ D = ρ / β , ∇ ⋅ B = 0 , κ ∇ × E = − ∂ B ∂ t , κ ∇ × H = ∂ D ∂ t + J / β ,

along with D = ε₀E and B = μ₀H. The constants β and κ vary from system to system. One can show that ε₀μ₀c² = κ².

The Gaussian system puts β = 1/4π, κ = c. The HLU system puts β = 1, κ = c. The SI system puts β = 1, κ = 1.

What rationalisation does is to replace the radiance constant (γ = intensity at radius² / source) with the gaussian divergence constant (β = flux through a surface / enclosed sources). One can easily show that γ = 4πβ, by considering the case of a sphere around a point, and intensity as density of flux. The older models set γ = 1, while the rationalised systems have β = 1. Rationalized equations in physics generally have a factor related to the effective spatial symmetry: 1 for planar symmetry, 2π for cylindrical symmetry and 4π for spherical symmetry.

The constant κ connects the electric and magnetic units through Q = Iκt. When electric and magnetic systems are defined as in the Gaussian or Heaviside–Lorentz systems, κ = c derives from the electromagnetic wave equations. Most systems have κ = 1, where the electric and magnetic systems are connected by Q = It. Therefore, most books use Q = It instead of Q = Iκt.

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 Kelvin–Stokes theorem.

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.

where

E and D are the electric field and displacement field, respectively;

P is the polarization density;

ϵ is the permittivity;

ϵ 0 is the permittivity of vacuum (used in the SI system, but takes on a numeric value of 1 in Gaussian and Lorentz–Heaviside units and so is negligible);

χ e is the electric susceptibility

The quantities ϵ in both Lorentz-Heaviside and Gaussian units and ϵ / ϵ 0 in SI are dimensionless, and they have the same numeric value. By contrast, the electric susceptibility χ e is unitless 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 is a simple constant.

where

B and H are the magnetic fields

M is magnetization

μ is magnetic permeability

μ 0 is the permeability of vacuum (used in the SI system, but takes on a numeric value of 1 in Gaussian and Lorentz–Heaviside units and so is negligible);

χ m is the magnetic susceptibility

The quantities μ in both Lorentz-Heaviside and Gaussian units and μ / μ 0 in SI are dimensionless, and they have the same numeric value. By contrast, the magnetic susceptibility χ m is unitless 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 φ:

General rules to translate a formula

To convert any formula from Lorentz–Heaviside units to Gaussian or to SI units, replace each symbol in the Lorentz-Heaviside column by the corresponding expression in the Gaussian column or in the SI column (vice versa to convert the other way). This will reproduce any of the specific formulas given in the list above, such as Maxwell's equations.

Replacing CGS with natural units

When one takes standard SI textbook equations, and sets ε = μ = 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 = q₁q₂/4πεr². Set ε = 1 to get the HLU form: F = q₁q₂/4πr². The Gaussian form does not have the 4π in the denominator.

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

∇ ⋅ E = ρ ∇ ⋅ B = 0 ∇ × E = − ∂ B ∂ t ∇ × B = ∂ E ∂ t + J

Because these equations can be easily related to SI work, HLU-style (i.e. rationalized) systems are becoming more fashionable.