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 unit system and electromagnetic equations. Lorentz–Heaviside 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.

Lorentz–Heaviside units, like SI units but unlike Gaussian units, are rationalized, meaning that there are no factors of 4π appearing explicitly in Maxwell's equations. The fact 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. Consequently Lorentz–Heaviside units differ by factors of √4π in the definitions of the electric and magnetic fields and of electric charge. They are particularly convenient when performing calculations in spatial dimensions greater than three such as in string theory. They are often used in relativistic calculations.

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 Gaussian charge is then √4π larger than the HLU, 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–Lorents 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 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

$\begin{align} \nabla \cdot \mathbf{D} &= \rho / \beta, \\ \quad \nabla \cdot \mathbf{B} &= 0, \\ \quad \kappa \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t}, \\ \quad \kappa \nabla \times \mathbf{H} &= \frac{\partial \mathbf{D}}{\partial t} + \mathbf{J} / \beta, \end{align}$

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.

Maxwell's equations with sources

With Lorentz–Heaviside units, Maxwell's equations in free space with sources take the following form:

$\nabla \cdot \mathbf{E} = \rho \,$

$\nabla \cdot \mathbf{B} = 0 \,$

$\nabla \times \mathbf{E} = -\frac{1}{c} \frac{\partial \mathbf{B}} {\partial t} \,$

$\nabla \times \mathbf{B} = \frac{1}{c} \frac{ \partial \mathbf{E}} {\partial t} + \frac{1}{c} \mathbf{J} \,$

where c is the speed of light in vacuum. Here E = D is the electric field, H = B is the magnetic field, ρ is charge density, and J is current density.

The Lorentz force equation is:

$\mathbf{F}_q = q \left(\mathbf{E} + \frac{\mathbf{v}_q}{c} \times \mathbf{B}\right) \,$

here q is a the charge of a test particle with vector velocity v_q and F_q 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 ε = 1. In the gaussian system, Coulomb's equation is F = QQ/R². In the Heaviside Lorentz system, F = qq/4πR². From this, one sees that QQ = qq/4π, that the Gaussian units are larger by a factor of √4π. Other quantities follow as follows.

$q_\mathrm{LH} \ = \ \sqrt{4\pi} \ q_\mathrm{G}$

$\mathbf{E}_\mathrm{LH} \ = \ {\mathbf{E}_\mathrm{G} \over \sqrt{4\pi}}$

$\mathbf{B}_\mathrm{LH} \ = \ {\mathbf{B}_\mathrm{G} \over \sqrt{4\pi}}$.

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.

$\nabla \cdot \mathbf{E} = \rho \,$

$\nabla \cdot \mathbf{B} = 0 \,$

$\nabla \times \mathbf{E} = -\frac{ \partial \mathbf{B}} {\partial t} \,$

$\nabla \times \mathbf{B} = \frac{ \partial \mathbf{E}} {\partial t} + \mathbf{J} \,$

$\mathbf{F}_q = q (\mathbf{E} + \mathbf{v}_q \times \mathbf{B}) \,$

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

External links

• Heaviside–Lorentz units