Lorentz–Heaviside units

As in the Gaussian units, the Heaviside–Lorentz units (HLU in this article) 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 = q^G
₁q^G
₂/r² in the Gaussian system, and F = q^LH
₁q^LH
₂/4πr² in the HLU. The unit of charge then connects to 1 dyn⋅cm² = 1 esu² = 4π hlu. The HLU quantity q^LH describing a charge is then √4π larger than the corresponding Gaussian quantity (see below), and the rest follows.

When dimensional analysis for SI units is used, including ε₀ and μ₀ are used to convert units, the result gives the conversion to and from the Heaviside–Lorentz units. For example, charge is √ε₀L³MT⁻². When one puts ε₀ = 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.

In system-independent form, the Maxwell equations are

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

along with D = ε₀ E and B = μ₀ H. The constants β and κ vary from system to system. One can show that ε₀ μ₀ c² = 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 = 4 π β, 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: 2 for planar symmetry, 2 π for cylindrical symmetry and 4 π for spherical symmetry.

The constant κ connects the electric and magnetic units through Q = I κ t. 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 = I t. Therefore, most books use Q = I t instead of Q = I κ t.

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

${\displaystyle \nabla \cdot \mathbf {E} ^{\mathrm {LH} }=\rho ^{\mathrm {LH} }}$

${\displaystyle \nabla \cdot \mathbf {B} ^{\mathrm {LH} }=0}$

${\displaystyle \nabla \times \mathbf {E} ^{\mathrm {LH} }=-{\frac {1}{c}}{\frac {\partial \mathbf {B^{\mathrm {LH} }} }{\partial t}}}$

${\displaystyle \nabla \times \mathbf {B} ^{\mathrm {LH} }={\frac {1}{c}}{\frac {\partial \mathbf {E} ^{\mathrm {LH} }}{\partial t}}+{\frac {1}{c}}\mathbf {J} ^{\mathrm {LH} }}$

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

The Lorentz force equation is:

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

here q^LH is the charge of a test particle with vector velocity v and F 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 = q^G
₁q^G
₂/r². In the Lorentz–Heaviside system, F = q^LH
₁q^LH
₂/4πr². From this, one sees that q^G
₁q^G
₂ = q^LH
₁q^LH
₂/4π, that the Gaussian charge quantities are smaller than the corresponding Lorentz–Heaviside quantities by a factor of √4π. Other quantities are related as follows.

${\displaystyle q^{\mathrm {LH} }\ =\ {\sqrt {4\pi }}\ q^{\mathrm {G} }}$

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

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

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

Main article: 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 the Kelvin–Stokes theorem.

Name	SI units	Gaussian units	Lorentz–Heaviside units
Gauss's law (macroscopic)	${\displaystyle \nabla \cdot \mathbf {D} =\rho _{\text{f}}}$	${\displaystyle \nabla \cdot \mathbf {D} =4\pi \rho _{\text{f}}}$	${\displaystyle \nabla \cdot \mathbf {D} =\rho _{\text{f}}}$
Gauss's law (microscopic)	${\displaystyle \nabla \cdot \mathbf {E} =\rho /\epsilon _{0}}$	${\displaystyle \nabla \cdot \mathbf {E} =4\pi \rho }$	${\displaystyle \nabla \cdot \mathbf {E} =\rho }$
Gauss's law for magnetism:	${\displaystyle \nabla \cdot \mathbf {B} =0}$	${\displaystyle \nabla \cdot \mathbf {B} =0}$	${\displaystyle \nabla \cdot \mathbf {B} =0}$
Maxwell–Faraday equation (Faraday's law of induction):	${\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$	${\displaystyle \nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}}$	${\displaystyle \nabla \times \mathbf {E} =-{\frac {1}{c}}{\frac {\partial \mathbf {B} }{\partial t}}}$
Ampère–Maxwell equation (macroscopic):	${\displaystyle \nabla \times \mathbf {H} =\mathbf {J} _{\text{f}}+{\frac {\partial \mathbf {D} }{\partial t}}}$	${\displaystyle \nabla \times \mathbf {H} ={\frac {4\pi }{c}}\mathbf {J} _{\text{f}}+{\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}}$	${\displaystyle \nabla \times \mathbf {H} ={\frac {1}{c}}\mathbf {J} _{\text{f}}+{\frac {1}{c}}{\frac {\partial \mathbf {D} }{\partial t}}}$
Ampère–Maxwell equation (microscopic):	${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} +{\frac {1}{c^{2}}}{\frac {\partial \mathbf {E} }{\partial t}}}$	${\displaystyle \nabla \times \mathbf {B} ={\frac {4\pi }{c}}\mathbf {J} +{\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}}$	${\displaystyle \nabla \times \mathbf {B} ={\frac {1}{c}}\mathbf {J} +{\frac {1}{c}}{\frac {\partial \mathbf {E} }{\partial t}}}$

Other basic laws

Name	SI units	Gaussian units	Lorentz–Heaviside units
Lorentz force	${\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}$	${\displaystyle \mathbf {F} =q\left(\mathbf {E} +{\frac {1}{c}}\mathbf {v} \times \mathbf {B} \right)}$	${\displaystyle \mathbf {F} =q\left(\mathbf {E} +{\frac {1}{c}}\mathbf {v} \times \mathbf {B} \right)}$
Coulomb's law	${\displaystyle \mathbf {F} ={\frac {1}{4\pi \epsilon _{0}}}{\frac {q_{1}q_{2}}{r^{2}}}\mathbf {\hat {r}} }$	${\displaystyle \mathbf {F} ={\frac {q_{1}q_{2}}{r^{2}}}\mathbf {\hat {r}} }$	${\displaystyle \mathbf {F} ={\frac {1}{4\pi }}{\frac {q_{1}q_{2}}{r^{2}}}\mathbf {\hat {r}} }$
Electric field of stationary point charge	${\displaystyle \mathbf {E} ={\frac {1}{4\pi \epsilon _{0}}}{\frac {q}{r^{2}}}\mathbf {\hat {r}} }$	${\displaystyle \mathbf {E} ={\frac {q}{r^{2}}}\mathbf {\hat {r}} }$	${\displaystyle \mathbf {E} ={\frac {1}{4\pi }}{\frac {q}{r^{2}}}\mathbf {\hat {r}} }$
Biot–Savart law	${\displaystyle \mathbf {B} ={\frac {\mu _{0}}{4\pi }}\oint {\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}}$	${\displaystyle \mathbf {B} ={\frac {1}{c}}\oint {\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}}$	${\displaystyle \mathbf {B} ={\frac {1}{4\pi c}}\oint {\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}}$

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.

Lorentz–Heaviside units	Gaussian units	SI units
${\displaystyle \mathbf {D} =\mathbf {E} +\mathbf {P} }$	${\displaystyle \mathbf {D} =\mathbf {E} +4\pi \mathbf {P} }$	${\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} +\mathbf {P} }$
${\displaystyle \mathbf {P} =\chi _{\text{e}}\mathbf {E} }$	${\displaystyle \mathbf {P} =\chi _{\text{e}}\mathbf {E} }$	${\displaystyle \mathbf {P} =\chi _{\text{e}}\epsilon _{0}\mathbf {E} }$
${\displaystyle \mathbf {D} =\epsilon \mathbf {E} }$	${\displaystyle \mathbf {D} =\epsilon \mathbf {E} }$	${\displaystyle \mathbf {D} =\epsilon \mathbf {E} }$
${\displaystyle \epsilon =1+\chi _{\text{e}}}$	${\displaystyle \epsilon =1+4\pi \chi _{\text{e}}}$	${\displaystyle \epsilon /\epsilon _{0}=1+\chi _{\text{e}}}$

where

• E and D are the electric field and displacement field, respectively;
• P is the polarization density;
• ${\displaystyle \epsilon }$ is the permittivity;
• ${\displaystyle \epsilon _{0}}$ 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 may be omitted);
• ${\displaystyle \chi _{\text{e}}}$ is the electric susceptibility

The quantities ${\displaystyle \epsilon }$ in both Lorentz–Heaviside and Gaussian units and ${\displaystyle \epsilon /\epsilon _{0}}$ in SI are dimensionless, and they have the same numeric value. By contrast, the electric susceptibility ${\displaystyle \chi _{e}}$ is unitless in all the systems, but has different numeric values for the same material:

${\displaystyle \chi _{\text{e}}^{\text{SI}}=\chi _{\text{e}}^{\text{LH}}=4\pi \chi _{\text{e}}^{\text{G}}}$

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.

Lorentz–Heaviside units	Gaussian units	SI units
${\displaystyle \mathbf {B} =\mathbf {H} +\mathbf {M} }$	${\displaystyle \mathbf {B} =\mathbf {H} +4\pi \mathbf {M} }$	${\displaystyle \mathbf {B} =\mu _{0}(\mathbf {H} +\mathbf {M} )}$
${\displaystyle \mathbf {M} =\chi _{\text{m}}\mathbf {H} }$	${\displaystyle \mathbf {M} =\chi _{\text{m}}\mathbf {H} }$	${\displaystyle \mathbf {M} =\chi _{\text{m}}\mathbf {H} }$
${\displaystyle \mathbf {B} =\mu \mathbf {H} }$	${\displaystyle \mathbf {B} =\mu \mathbf {H} }$	${\displaystyle \mathbf {B} =\mu \mathbf {H} }$
${\displaystyle \mu =1+\chi _{\text{m}}}$	${\displaystyle \mu =1+4\pi \chi _{\text{m}}}$	${\displaystyle \mu /\mu _{0}=1+\chi _{\text{m}}}$

where

• B and H are the magnetic fields
• M is the magnetization
• ${\displaystyle \mu }$ is the magnetic permeability
• ${\displaystyle \mu _{0}}$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 may be omitted);
• ${\displaystyle \chi _{\text{m}}}$ is the magnetic susceptibility

The quantities ${\displaystyle \mu }$ in both Lorentz–Heaviside and Gaussian units and ${\displaystyle \mu /\mu _{0}}$in SI are dimensionless, and they have the same numeric value. By contrast, the magnetic susceptibility ${\displaystyle \chi _{\text{m}}}$ is unitless in all the systems, but has different numeric values for the same material:

${\displaystyle \chi _{\text{m}}^{\text{SI}}=\chi _{\text{m}}^{\text{LH}}=4\pi \chi _{\text{m}}^{\text{G}}}$

Vector and scalar potentials

Main articles: Magnetic vector potential and Electric potential

The electric and magnetic fields can be written in terms of a vector potential A and a scalar potential ${\displaystyle \phi }$:

Name	Lorentz–Heaviside quantities	Gaussian quantities	SI quantities
Electric field (static)	${\displaystyle \mathbf {E} ^{\text{LH}}=-\nabla \phi ^{\text{LH}}}$	${\displaystyle \mathbf {E} ^{\text{G}}=-\nabla \phi ^{\text{G}}}$	${\displaystyle \mathbf {E} ^{\text{SI}}=-\nabla \phi ^{\text{SI}}}$
Electric field (general)	${\displaystyle \mathbf {E} ^{\text{LH}}=-\nabla \phi ^{\text{LH}}-{\frac {1}{c}}{\frac {\partial \mathbf {A} ^{\text{LH}}}{\partial t}}}$	${\displaystyle \mathbf {E} ^{\text{G}}=-\nabla \phi ^{\text{G}}-{\frac {1}{c}}{\frac {\partial \mathbf {A} ^{\text{G}}}{\partial t}}}$	${\displaystyle \mathbf {E} ^{\text{SI}}=-\nabla \phi ^{\text{SI}}-{\frac {\partial \mathbf {A} ^{\text{SI}}}{\partial t}}}$
Magnetic B field	${\displaystyle \mathbf {B} ^{\text{LH}}=\nabla \times \mathbf {A} ^{\text{LH}}}$	${\displaystyle \mathbf {B} ^{\text{G}}=\nabla \times \mathbf {A} ^{\text{G}}}$	${\displaystyle \mathbf {B} ^{\text{SI}}=\nabla \times \mathbf {A} ^{\text{SI}}}$

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.

Name	Lorentz–Heaviside units	Gaussian units	SI units
speed of light	${\displaystyle c}$	${\displaystyle c}$	${\displaystyle {\frac {1}{\sqrt {\epsilon _{0}\mu _{0}}}}}$
electric field, electric potential	${\displaystyle \left(\mathbf {E} ,\varphi \right)}$	${\displaystyle {\frac {1}{\sqrt {4\pi }}}\left(\mathbf {E} ,\varphi \right)}$	${\displaystyle {\sqrt {\epsilon _{0}}}\left(\mathbf {E} ,\varphi \right)}$
electric displacement field	${\displaystyle \mathbf {D} }$	${\displaystyle {\frac {1}{\sqrt {4\pi }}}\mathbf {D} }$	${\displaystyle {\frac {1}{\sqrt {\epsilon _{0}}}}\mathbf {D} }$
electric charge, electric charge density, electric current, electric current density, polarization density, electric dipole moment	${\displaystyle \left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$	${\displaystyle {\sqrt {4\pi }}\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$	${\displaystyle {\frac {1}{\sqrt {\epsilon _{0}}}}\left(q,\rho ,I,\mathbf {J} ,\mathbf {P} ,\mathbf {p} \right)}$
magnetic B field, magnetic flux, magnetic vector potential	${\displaystyle \left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$	${\displaystyle {\frac {1}{\sqrt {4\pi }}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$	${\displaystyle {\frac {1}{\sqrt {\mu _{0}}}}\left(\mathbf {B} ,\Phi _{\text{m}},\mathbf {A} \right)}$
magnetic H field	${\displaystyle \mathbf {H} }$	${\displaystyle {\frac {1}{\sqrt {4\pi }}}\mathbf {H} }$	${\displaystyle {\sqrt {\mu _{0}}}\mathbf {H} }$
magnetic moment, magnetization	${\displaystyle \left(\mathbf {m} ,\mathbf {M} \right)}$	${\displaystyle {\sqrt {4\pi }}\left(\mathbf {m} ,\mathbf {M} \right)}$	${\displaystyle {\sqrt {\mu _{0}}}\left(\mathbf {m} ,\mathbf {M} \right)}$
relative permittivity, relative permeability	${\displaystyle \left(\epsilon ,\mu \right)}$	${\displaystyle \left(\epsilon ,\mu \right)}$	${\displaystyle \left({\frac {\epsilon }{\epsilon _{0}}},{\frac {\mu }{\mu _{0}}}\right)}$
electric susceptibility, magnetic susceptibility	${\displaystyle \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$	${\displaystyle 4\pi \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$	${\displaystyle \left(\chi _{\text{e}},\chi _{\text{m}}\right)}$
conductivity, conductance, capacitance	${\displaystyle \left(\sigma ,S,C\right)}$	${\displaystyle 4\pi \left(\sigma ,S,C\right)}$	${\displaystyle {\frac {1}{\epsilon _{0}}}\left(\sigma ,S,C\right)}$
resistivity, resistance, inductance	${\displaystyle \left(\rho ,R,L\right)}$	${\displaystyle {\frac {1}{4\pi }}\left(\rho ,R,L\right)}$	${\displaystyle \epsilon _{0}\left(\rho ,R,L\right)}$

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 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 ε₀ = µ₀ = c = 1.

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

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

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

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

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

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

• Heaviside–Lorentz units