Heaviside–Lorentz units

Length–mass–time framework

As in the Gaussian system (G), the Heaviside–Lorentz system (HL) uses the length–mass–time dimensions. 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.

Coulomb's equation, used to define charge in these systems, is F = q^G
₁q^G
₂ / r² in the Gaussian system, and F = q^HL
₁q^HL
₂ / (4πr²) in the HL system. The unit of charge then connects to 1 dyn⋅cm² = 1 statC² = 4π HLC², where 'HLC' is the HL unit of charge. The HL quantity q^HL describing a charge is then √4π times 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, which defines two constants, 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, SI charge is √ε₀L³M / T². 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.

Comparison of Heaviside–Lorentz with other systems of units

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 the displacement field, respectively; B and H are the magnetic fields; P is the polarization density, M the magnetization, ρ the charge density, J the current density, c the speed of light in vacuum, ϕ the electric potential, A the magnetic vector potential, F the Lorentz force acting on a body of charge q and velocity v, ε the permittivity, χ_e the electric susceptibility, μ the magnetic permeability, and χ_m the magnetic susceptibility.

Maxwell's equations

The electric and magnetic fields can be written in terms of the potentials A and ϕ. The definition of the magnetic field in terms of A, B = ∇ × A, is the same in all systems of units, but the electric field is $\mathbf{E}=-\mathrm{\nabla }\varphi -\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}$ in the SI system, but $\mathbf{E}=-\mathrm{\nabla }\varphi -\frac{1}{c}\frac{\mathrm{\partial }\mathbf{A}}{\mathrm{\partial }t}$ in the HL or Gaussian systems.

Other basic laws

Dielectric and magnetic materials

Below are the expressions for the macroscopic fields ${\displaystyle \mathbf{D}}$, ${\displaystyle \mathbf{P}}$, ${\displaystyle \mathbf{H}}$ and ${\displaystyle \mathbf{M}}$ 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 ${\displaystyle {\epsilon }^{\mathsf{\text{SI}}}/{\epsilon }_0}$, ${\displaystyle {\epsilon }^{\mathsf{\text{HL}}}}$ and ${\displaystyle {\epsilon }^{\mathsf{\text{G}}}}$ are dimensionless, and they have the same numeric value. By contrast, the electric susceptibility ${\displaystyle {\chi }_{\text{e}}}$ is dimensionless in all the systems, but has different numeric values for the same material: $${\displaystyle {\chi }_{\text{e}}^{\mathsf{\text{SI}}}={\chi }_{\text{e}}^{\mathsf{\text{HL}}}=4\pi {\chi }_{\text{e}}^{\mathsf{\text{G}}}}$$ The same statements apply for the corresponding magnetic quantities.

Advantages and disadvantages of Heaviside–Lorentz units

Advantages

• The formulas above are clearly simpler in HL units compared to either SI or Gaussian 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 E 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 constants ε₀ and μ₀ that are present in the SI system. (In addition ε₀ and μ₀ are overdetermined, because ε₀ μ₀ = 1 / c².)

The below points are true in both Heaviside–Lorentz and Gaussian systems, but not SI.

• The electric and magnetic fields E and B have the same dimensions in the Heaviside–Lorentz 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 [E] / [B] is [c].
• Giving the E and B 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, ϕ and A have the same dimensions and are the four components of the 4-potential.
• The fields D, H, P, and M also have the same dimensions as E and B. For vacuum, any expression involving D can simply be recast as the same expression with E. In SI units, D and P have the same units, as do H and M, but they have different units from each other and from E and B.

Disadvantages

• 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 3.5, are frequently of rather inconvenient sizes. The ampere (coulomb/second) is reasonable unit for measuring currents commonly encountered, but the ESU/s, 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 Gaussian 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), 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. Outside of these contexts, including for example magazine articles on electric circuits, Heaviside–Lorentz and Gaussian CGS units are rarely encountered.

Translating formulas between systems

To convert any formula between the SI, Heaviside–Lorentz system or Gaussian system, the corresponding expressions shown in the table below can be equated and hence substituted for each other. Replace ${\displaystyle 1/c^2}$ by ${\displaystyle {\epsilon }_0{\mu }_0}$ or vice versa. This will reproduce any of the specific formulas given in the list above.

As an example, starting with the equation $${\displaystyle \mathrm{\nabla }\cdot {\mathbf{E}}^{\mathsf{\text{SI}}}={\rho }^{\mathsf{\text{SI}}}/{\epsilon }_0,}$$ and the equations from the table $${\displaystyle \begin{array}{cc}\sqrt{{\epsilon }_0}{\mathbf{E}}^{\mathsf{\text{SI}}}& ={\mathbf{E}}^{\mathsf{\text{HL}}}\\ \frac{1}{\sqrt{{\epsilon }_0}}{\rho }^{\mathsf{\text{SI}}}& ={\rho }^{\mathsf{\text{HL}}}.\end{array}}$$

Moving the factor across in the latter identities and substituting, the result is $${\displaystyle \mathrm{\nabla }\cdot \left(\frac{1}{\sqrt{{\epsilon }_0}}{\mathbf{E}}^{\mathsf{\text{HL}}}\right)=\left(\sqrt{{\epsilon }_0}{\rho }^{\mathsf{\text{HL}}}\right)/{\epsilon }_0,}$$ which then simplifies to $${\displaystyle \mathrm{\nabla }\cdot {\mathbf{E}}^{\mathsf{\text{HL}}}={\rho }^{\mathsf{\text{HL}}}.}$$