Lorentz–Heaviside units

q

{\displaystyle q}

) should have units of force*
*Electric field (

E

{\displaystyle E}

) should have units of force per area (the same as stress)*
*Electric constant (

ϵ
0

{\displaystyle \epsilon\_0}

) should have units of force per area (the same as bulk modulus)*
*Magnetic field (

B

{\displaystyle B}

) should have units of force per velocity per time^2*
*Magnetic constant (

μ
0

{\displaystyle \mu\_0}

) should have units of 1/(force per time^2)*
See this for more details

**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 *ε*0 and magnetic constant *µ*0 do not appear, having been incorporated implicitly into the electromagnetic quantities by the way they are defined. Lorentz–Heaviside units may be regarded as normalizing *ε*0 = 1 and *µ*0 = 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 4*π* appearing explicitly in Maxwell's equations. That these units are rationalized partly explains their appeal in quantum field theory: the Lagrangian underlying the theory does not have any factors of 4*π* in these units. Consequently, Lorentz–Heaviside units differ by factors of √4*π* in the definitions of the electric and magnetic fields and of electric charge. They are often used in relativistic calculations, and are used in particle physics. They are particularly convenient when performing calculations in spatial dimensions greater than three such as in string theory.

## Length–mass–time framework

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
1*q*G
2/*r*2 in the Gaussian system, and *F* = *q*LH
1*q*LH
2/4*πr*2 in the HLU. The unit of charge then connects to 1 dyn⋅cm2 = 1 esu2 = 4*π* hlu. The HLU quantity *q*LH describing a charge is then √4*π* larger than the corresponding Gaussian quantity (see below), and the rest follows.

When dimensional analysis for SI units is used, including *ε*0 and *μ*0 are used to convert units, the result gives the conversion to and from the Heaviside–Lorentz units. For example, charge is √*ε*0*L*3*MT*−2. When one puts *ε*0 = 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.

## Maxwell's equations with sources

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

∇
⋅

E

LH

=

ρ
LH

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

∇
⋅

B

LH

=
0

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

∇
×

E

LH

=
−

1
c

∂

B
LH

∂
t

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

∇
×

B

LH

=

1
c

∂

E

LH

∂
t

1
c

J

LH

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

F

=

q
LH

(

E

LH

v

c

×

B

LH

)

{\displaystyle \mathbf{F} = q^\text{LH} \left(\mathbf{E}^\text{LH} + \frac{\mathbf{v}}{c} \times \mathbf{B}^\text{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
1*q*G
2/*r*2. In the Lorentz–Heaviside system, *F* = *q*LH
1*q*LH
2/4*πr*2. From this, one sees that *q*G
1*q*G
2 = *q*LH
1*q*LH
2/4*π*, that the Gaussian charge quantities are smaller than the corresponding Lorentz–Heaviside quantities by a factor of √4*π*. Other quantities are related as follows.

q
LH

=

4
π

q
G

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

E

LH

=

E

G

4
π

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

B

LH

=

B

G

4
π

{\displaystyle \mathbf{B}^\text{LH} \ = \ {\mathbf{B}^\text{G} \over \sqrt{4\pi}} }

.
## 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

Main articles: 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.

### Other basic laws

F

=
q

(

E

SI

v

×

B

SI

)

{\displaystyle \mathbf{F} = q\left(\mathbf{E}^\text{SI}+\mathbf{v}\times\mathbf{B}^\text{SI}\right)}

F

=
q

(

E

LH

1
c

v

×

B

LH

)

{\displaystyle \mathbf{F} = q\left(\mathbf{E}^\text{LH}+\frac{1}{c}\mathbf{v}\times\mathbf{B}^\text{LH}\right)}

F

=
q

(

E

G

1
c

v

×

B

G

)

{\displaystyle \mathbf{F} = q\left(\mathbf{E}^\text{G}+\frac{1}{c}\mathbf{v}\times\mathbf{B}^\text{G}\right)}

F

=

1

4
π

ϵ
0

q
1
SI

q
2
SI

r
2

r
^

{\displaystyle \mathbf{F} = \frac{1}{4\pi\epsilon\_0}\frac{q\_1^\text{SI} q\_2^\text{SI}}{r^2} \mathbf{\hat r}}

F

=

1

4
π

q
1
LH

q
2
LH

r
2

r
^

{\displaystyle \mathbf{F} = \frac{1}{4\pi}\frac{q\_1^\text{LH} q\_2^\text{LH}}{r^2} \mathbf{\hat r}}

F

=

q
1
G

q
2
G

r
2

r
^

{\displaystyle \mathbf{F} = \frac{q\_1^\text{G} q\_2^\text{G}}{r^2} \mathbf{\hat r}}

E

SI

=

1

4
π

ϵ
0

q
SI

r
2

r
^

{\displaystyle \mathbf{E}^\text{SI} = \frac{1}{4\pi\epsilon\_0}\frac{q^\text{SI}}{r^2} \mathbf{\hat r}}

E

LH

=

1

4
π

q
LH

r
2

r
^

{\displaystyle \mathbf{E}^\text{LH} = \frac{1}{4\pi}\frac{q^\text{LH}}{r^2} \mathbf{\hat r}}

E

G

=

q
G

r
2

r
^

{\displaystyle \mathbf{E}^\text{G} = \frac{q^\text{G}}{r^2} \mathbf{\hat r}}

B

SI

=

μ
0

4
π

∮

I
SI

d

l

×

r
^

r
2

{\displaystyle \mathbf{B}^\text{SI} = \frac{\mu\_0}{4\pi} \oint\frac{I^\text{SI} d\mathbf{l} \times \mathbf{\hat r}}{r^2}}

B

LH

=

1

4
π
c

∮

I
LH

d

l

×

r
^

r
2

{\displaystyle \mathbf{B}^\text{LH} = \frac{1}{4\pi c} \oint\frac{I^\text{LH} d\mathbf{l} \times \mathbf{\hat r}}{r^2}}

B

G

=

1
c

∮

I
G

d

l

×

r
^

r
2

{\displaystyle \mathbf{B}^\text{G} = \frac{1}{c} \oint\frac{I^\text{G} d\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.

D

SI

=

ϵ
0

E

SI

P

SI

{\displaystyle \mathbf{D}^\text{SI} = \epsilon\_0 \mathbf{E}^\text{SI}+\mathbf{P}^\text{SI}}

D

LH

=

E

LH

P

LH

{\displaystyle \mathbf{D}^\text{LH} = \mathbf{E}^\text{LH}+\mathbf{P}^\text{LH}}

D

G

=

E

G

4
π

P

G

{\displaystyle \mathbf{D}^\text{G} = \mathbf{E}^\text{G}+4\pi\mathbf{P}^\text{G}}

P

SI

=

χ
e
SI

ϵ
0

E

SI

{\displaystyle \mathbf{P}^\text{SI} = \chi\_\text{e}^\text{SI}\epsilon\_0\mathbf{E}^\text{SI}}

P

LH

=

χ
e
LH

E

LH

{\displaystyle \mathbf{P}^\text{LH} = \chi\_\text{e}^\text{LH}\mathbf{E}^\text{LH}}

P

G

=

χ
e
G

E

G

{\displaystyle \mathbf{P}^\text{G} = \chi\_\text{e}^\text{G}\mathbf{E}^\text{G}}

D

SI

=
ϵ

E

SI

{\displaystyle \mathbf{D}^\text{SI} = \epsilon\mathbf{E}^\text{SI}}

D

LH

=
ϵ

E

LH

{\displaystyle \mathbf{D}^\text{LH} = \epsilon\mathbf{E}^\text{LH}}

D

G

=
ϵ

E

G

{\displaystyle \mathbf{D}^\text{G} = \epsilon\mathbf{E}^\text{G}}

ϵ
SI

/

ϵ
0

=
1

χ
e
SI

{\displaystyle \epsilon^\text{SI}/\epsilon\_0 = 1+\chi\_\text{e}^\text{SI}}

ϵ
LH

=
1

χ
e
LH

{\displaystyle \epsilon^\text{LH} = 1+\chi\_\text{e}^\text{LH}}

ϵ
G

=
1
4
π

χ
e
G

{\displaystyle \epsilon^\text{G} = 1+4\pi\chi\_\text{e}^\text{G}}

where

* the superscript (SI, LH, G) indicates in which system the quantity is defined
* **E** and **D** are the electric field and displacement field, respectively;
* **P** is the polarization density;
* ϵ

{\displaystyle \epsilon}

 is the permittivity;
* ϵ
0

{\displaystyle \epsilon\_0}

 is the permittivity of vacuum (used in the SI system, but meaningless in the Gaussian and Lorentz–Heaviside systems);
* χ
e

{\displaystyle \chi\_\text{e}}

 is the electric susceptibility

The quantities

ϵ
SI

/

ϵ
0

{\displaystyle \epsilon^\text{SI}/\epsilon\_0}

,

ϵ
LH

{\displaystyle \epsilon^\text{LH}}

 and

ϵ
G

{\displaystyle \epsilon^\text{G}}

 are dimensionless, and they have the same numeric value. By contrast, the electric susceptibility

χ
e

{\displaystyle \chi\_e}

 is dimensionless in all the systems, but has *different numeric values* for the same material:

χ
e
SI

=

χ
e
LH

=
4
π

χ
e
G

{\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 can be expressed as a scalar constant.

B

SI

=

μ
0

(

H

SI

M

SI

)

{\displaystyle \mathbf{B}^\text{SI} = \mu\_0 (\mathbf{H}^\text{SI}+\mathbf{M}^\text{SI})}

B

LH

=

H

LH

M

LH

{\displaystyle \mathbf{B}^\text{LH} = \mathbf{H}^\text{LH}+\mathbf{M}^\text{LH}}

B

G

=

H

G

4
π

M

G

{\displaystyle \mathbf{B}^\text{G} = \mathbf{H}^\text{G}+4\pi\mathbf{M}^\text{G}}

M

SI

=

χ
m
SI

H

SI

{\displaystyle \mathbf{M}^\text{SI} = \chi\_\text{m}^\text{SI}\mathbf{H}^\text{SI}}

M

LH

=

χ
m
LH

H

LH

{\displaystyle \mathbf{M}^\text{LH} = \chi\_\text{m}^\text{LH}\mathbf{H}^\text{LH}}

M

G

=

χ
m
G

H

G

{\displaystyle \mathbf{M}^\text{G} = \chi\_\text{m}^\text{G}\mathbf{H}^\text{G}}

B

SI

=

μ
SI

H

SI

{\displaystyle \mathbf{B}^\text{SI} = \mu^\text{SI}\mathbf{H}^\text{SI}}

B

LH

=

μ
LH

H

LH

{\displaystyle \mathbf{B}^\text{LH} = \mu^\text{LH}\mathbf{H}^\text{LH}}

B

G

=

μ
G

H

G

{\displaystyle \mathbf{B}^\text{G} = \mu^\text{G}\mathbf{H}^\text{G}}

μ
SI

/

μ
0

=
1

χ
m
SI

{\displaystyle \mu^\text{SI}/\mu\_0 = 1+\chi\_\text{m}^\text{SI}}

μ
LH

=
1

χ
m
LH

{\displaystyle \mu^\text{LH} = 1+\chi\_\text{m}^\text{LH}}

μ
G

=
1
4
π

χ
m
G

{\displaystyle \mu^\text{G} = 1+4\pi\chi\_\text{m}^\text{G}}

where

* the superscript (LH, G, SI) indicates in which system the quantity is defined
* **B** and **H** are the magnetic fields
* **M** is the magnetization
* μ

{\displaystyle \mu}

 is the magnetic permeability
* μ
0

{\displaystyle \mu\_0}

is the permeability of vacuum (used in the SI system, but meaningless in the Gaussian and Lorentz–Heaviside systems);
* χ
m

{\displaystyle \chi\_\text{m}}

 is the magnetic susceptibility

The quantities

μ
SI

/

μ
0

{\displaystyle \mu^\text{SI}/\mu\_0}

,

μ
LH

{\displaystyle \mu^\text{LH}}

 and

μ
G

{\displaystyle \mu^\text{G}}

 are dimensionless, and they have the same numeric value. By contrast, the magnetic susceptibility

χ
m

{\displaystyle \chi\_\text{m}}

 is dimensionless in all the systems, but has *different numeric values* for the same material:

χ
m
SI

=

χ
m
LH

=
4
π

χ
m
G

{\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, Electric potential
The electric and magnetic fields can be written in terms of a vector potential **A** and a scalar potential

ϕ

{\displaystyle \phi}

:

E

SI

=
−
∇

ϕ
SI

{\displaystyle \mathbf{E}^\text{SI} = -\nabla\phi^\text{SI}}

E

LH

=
−
∇

ϕ
LH

{\displaystyle \mathbf{E}^\text{LH} = -\nabla\phi^\text{LH}}

E

G

=
−
∇

ϕ
G

{\displaystyle \mathbf{E}^\text{G} = -\nabla\phi^\text{G}}
 Electric field(general)

E

SI

=
−
∇

ϕ
SI

−

∂

A

SI

∂
t

{\displaystyle \mathbf{E}^\text{SI} = -\nabla\phi^\text{SI}-\frac{\partial \mathbf{A}^\text{SI}}{\partial t}}

E

LH

=
−
∇

ϕ
LH

−

1
c

∂

A

LH

∂
t

{\displaystyle \mathbf{E}^\text{LH} = -\nabla\phi^\text{LH}-\frac{1}{c}\frac{\partial \mathbf{A}^\text{LH}}{\partial t}}

E

G

=
−
∇

ϕ
G

−

1
c

∂

A

G

∂
t

{\displaystyle \mathbf{E}^\text{G} = -\nabla\phi^\text{G}-\frac{1}{c}\frac{\partial \mathbf{A}^\text{G}}{\partial t}}

B

SI

=
∇
×

A

SI

{\displaystyle \mathbf{B}^\text{SI} = \nabla \times \mathbf{A}^\text{SI}}

B

LH

=
∇
×

A

LH

{\displaystyle \mathbf{B}^\text{LH} = \nabla \times \mathbf{A}^\text{LH}}

B

G

=
∇
×

A

G

{\displaystyle \mathbf{B}^\text{G} = \nabla \times \mathbf{A}^\text{G}}

## Translating expressions and formulae between systems

To convert any expression or formula between SI, Lorentz–Heaviside 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, such as Maxwell's equations.

As an example, starting with the equation

∇
⋅

E

SI

=

ρ
SI

/

ϵ
0

,

{\displaystyle \nabla \cdot \mathbf{E}^\text{SI} = \rho^\text{SI}/\epsilon\_0 ,}

and the equations from the table

ϵ
0

E

SI

=

E

LH

{\displaystyle \sqrt{\epsilon\_0} \ \mathbf{E}^\text{SI} = \mathbf{E}^\text{LH} }

1

ϵ
0

ρ
SI

=

ρ
LH

,

{\displaystyle \frac{1}{\sqrt{\epsilon\_0}} \rho^\text{SI} = \rho^\text{LH} ,}

moving the factor across in the latter identities and substituting, the result is

∇
⋅

(

1

ϵ
0

E

LH

)

=

(

ϵ
0

ρ
LH

)

/

ϵ
0

,

{\displaystyle \nabla \cdot \left(\frac{1}{\sqrt{\epsilon\_0}} \mathbf{E}^\text{LH}\right) = \left(\sqrt{\epsilon\_0} \rho^\text{LH}\right)/\epsilon\_0 ,}

which then simplifies to

∇
⋅

E

LH

=

ρ
LH

.

{\displaystyle \nabla \cdot \mathbf{E}^\text{LH} = \rho^\text{LH} .}

## Replacing CGS with natural units

When one takes standard SI textbook equations, and sets *ε*0 = *µ*0 = *c* = 1 to get natural units, the resulting equations follow the Heaviside–Lorentz formulation and sizes. The conversion requires no changes to the factor 4*π*, unlike for the Gaussian equations. Coulomb's inverse-square law equation in SI is **F** = *q*1*q*2/4*πε*0*r*2. Set *ε*0 = 1 to get the HLU form: **F** = *q*1*q*2/4*πr*2. The Gaussian form does not have the 4*π* 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*0 = *µ*0 = *c* = 1.

∇
⋅

E

=
ρ

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

∇
⋅

B

=
0

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

∇
×

E

=
−

∂

B

∂
t

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

∇
×

B

=

∂

E

∂
t

J

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

F

=
q
(

E

v

×

B

)

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

Because these equations can be easily related to SI work, rationalized systems are becoming more fashionable.

### In quantum mechanics

Additionally setting *ε*0 = *µ*0 = *c* = *ħ* = *k*B = 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* / *mc*2, etc.

### Lorentz–Heaviside Planck units

Setting

ϵ
0

=

μ
0

=
# ℏ

k
B

=
4
π
# G
1

{\displaystyle \epsilon\_0=\mu\_0=c=\hbar=k\_\text{B}=4\pi G=1}

 yields the Lorentz–Heaviside Planck units, or **rationalized Planck units**. The mass scale is chosen such that the gravitational constant is

1

4
π

{\displaystyle \frac{1}{4\pi}}

, equal to the Coulomb constant. (By constrast, Gaussian Planck units set

4
π

ϵ
0

=

μ
0

4
π

=
# ℏ

k
B

=
# G
1

{\displaystyle 4\pi \epsilon\_0=\frac{\mu\_0}{4\pi}=c=\hbar=k\_\text{B}=G=1}

.)