Gaussian 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

**Gaussian units** constitute a metric system of physical units. This system is the most common of the several electromagnetic unit systems based on cgs (centimetre–gram–second) units. It is also called the **Gaussian unit system**, **Gaussian-cgs units**, or often just **cgs units**. The term "cgs units" is ambiguous and therefore to be avoided if possible: there are several variants of cgs with conflicting definitions of electromagnetic quantities and units.

SI units predominate in most fields, and continue to increase in popularity at the expense of Gaussian units. Alternative unit systems also exist. Conversions between quantities in the Gaussian unit system and the SI unit system are not as straightforward as direct unit conversions because the quantities themselves are defined differently in the different systems, which has the effect that the equations expressing physical laws of electromagnetism (such as Maxwell's equations) change depending on what system of units is being used. As an example, quantities that are dimensionless in one system may have dimension in another.

## History

Gaussian units existed before the CGS system. The British Association report of 1873 that proposed the CGS contains gaussian units derived from the foot–grain–second and metre–gram–second as well. There are also references to foot–pound–second gaussian units.

## Alternative unit systems

Main articles: Centimetre gram second system of units#Alternate derivations of CGS units in electromagnetism
The Gaussian unit system is just one of several electromagnetic unit systems within CGS. Others include "electrostatic units", "electromagnetic units", and Lorentz–Heaviside units.

Some other unit systems are called "natural units", a category that includes Hartree atomic units, Planck units, and others.

SI units are by far the most common system of units today. In engineering and practical areas, SI is nearly universal and has been for decades. In technical, scientific literature (such as theoretical physics and astronomy), Gaussian units were predominant until recent decades, but are now getting progressively less so. The 8th SI Brochure acknowledges that the CGS-Gaussian unit system has advantages in classical and relativistic electrodynamics, but the 9th SI Brochure makes no mention of CGS systems.

Natural units may be used in more theoretical and abstract fields of physics, particularly particle physics and string theory.

## Major differences between Gaussian and SI units

### "Rationalized" unit systems

One difference between Gaussian and SI units is in the factors of 4*π* in various formulas. SI electromagnetic units are called "rationalized", because Maxwell's equations have no explicit factors of 4*π* in the formulae. On the other hand, the inverse-square force laws – Coulomb's law and the Biot–Savart law – *do* have a factor of 4*π* attached to the *r*2.

* In *unrationalized* Gaussian units (not Lorentz–Heaviside units) the situation is reversed: two of Maxwell's equations have factors of 4*π* in the formulas, while both of the inverse-square force laws, Coulomb's law and the Biot–Savart law, have no factor of 4*π* attached to *r*2 in the denominator.

(The quantity 4*π* appears because 4*πr*2 is the surface area of the sphere of radius *r*, which reflects the geometry of the configuration. For details, see the articles Relation between Gauss's law and Coulomb's law and Inverse-square law.)

### Unit of charge

A major difference between Gaussian and SI units is in the definition of the unit of charge.

For example, Coulomb's law in Gaussian units has no constant:

# F

Q
1
G

Q
2
G

r
2

{\displaystyle F = \frac{Q^\text{G}\_1Q^\text{G}\_2}{r^2}}

where *F* is the repulsive force between two electrical charges, *Q*G
1 and *Q*G
2 are the two charges in question, and *r* is the distance separating them.

The unit of gaussian charge, the statcoulomb, is such that two units, a distance of 1 centimeter apart, repel each other with a force equal to the cgs unit of force, the dyne. Thus the unit of gaussian charge can also be written 1 dyne1/2 cm. "Gaussian electric charge" is not the same mathematical quantity as modern (MKS and subsequently the SI) electric charge and is not measured in coulombs.

The same law in SI units is:

# F

1

4
π

ε
0

Q
1
SI

Q
2
SI

r
2

{\displaystyle F = \frac{1}{4\pi\varepsilon\_0} \frac{Q^\text{SI}\_1 Q^\text{SI}\_2}{r^2}}

where *ε*0 is the vacuum permittivity, a quantity with *dimension*, namely (charge)2 (time)2 (mass)−1 (length)−3. Without *ε*0, the two sides would not have consistent dimensions in SI, whereas the quantity *ε*0 does not appear in Gaussian equations. This is an example of how some dimensional physical constants can be eliminated from the expressions of physical law simply by the judicious choice of units. In SI, 1/*ε*0, converts or scales flux density, **D**, to electric field, **E** (the latter has dimension of force per charge), while in *rationalized* Gaussian units, electric flux density is the same quantity as electric field strength in free space.

In Gaussian units, the speed of light *c* appears explicitly in electromagnetic formulas like Maxwell's equations (see below), whereas in SI it appears only via the product

μ
0

ε
0

=
1

/

c
2

{\displaystyle \mu\_0 \varepsilon\_0=1/c^2}

.

### Polarization, magnetization

There are further differences between Gaussian and SI units in how quantities related to polarization and magnetization are defined. For one thing, in Gaussian units, *all* of the following quantities have the same dimension: **E**G, **D**G, **P**G, **B**G, **H**G, and **M**G.

Another important point is that the electric and magnetic susceptibility of a material is dimensionless in both Gaussian and SI units, but a given material will have a different numerical susceptibility in the two systems. (Equation is given below.)

## List of equations

This section has a list of the basic formulae of electromagnetism, given in both 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. A simple conversion scheme for use when tables are not available may be found in
Ref.
All formulas except otherwise noted are from Ref.

### 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 Kelvin–Stokes theorem.

∇
⋅

D

G

=
4
π

ρ
f
G

{\displaystyle \nabla \cdot \mathbf{D}^\text{G} = 4\pi\rho\_\text{f}^\text{G}}

∇
⋅

D

SI

=

ρ
f
SI

{\displaystyle \nabla \cdot \mathbf{D}^\text{SI} = \rho\_\text{f}^\text{SI}}
 Gauss's law(microscopic)

∇
⋅

E

G

=
4
π

ρ
G

{\displaystyle \nabla \cdot \mathbf{E}^\text{G} = 4\pi\rho^\text{G}}

∇
⋅

E

SI

=

ρ
SI

/

ϵ
0

{\displaystyle \nabla \cdot \mathbf{E}^\text{SI} = \rho^\text{SI}/\epsilon\_0}
 Gauss's law for magnetism:

∇
⋅

B

G

=
0

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

∇
⋅

B

SI

=
0

{\displaystyle \nabla \cdot \mathbf{B}^\text{SI} = 0}
 Maxwell–Faraday equation(Faraday's law of induction):

∇
×

E

G

=
−

1
c

∂

B

G

∂
t

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

∇
×

E

SI

=
−

∂

B

SI

∂
t

{\displaystyle \nabla \times \mathbf{E}^\text{SI} = -\frac{\partial \mathbf{B}^\text{SI}} {\partial t}}
 Ampère–Maxwell equation (macroscopic):

∇
×

H

G

=

4
π

c

J

f
G

1
c

∂

D

G

∂
t

{\displaystyle \nabla \times \mathbf{H}^\text{G} = \frac{4\pi}{c}\mathbf{J}\_\text{f}^\text{G} + \frac{1}{c}\frac{\partial \mathbf{D}^\text{G}} {\partial t}}

∇
×

H

SI

=

J

f

SI

∂

D

SI

∂
t

{\displaystyle \nabla \times \mathbf{H}^\text{SI} = \mathbf{J}\_{\text{f}}^\text{SI} + \frac{\partial \mathbf{D}^\text{SI}} {\partial t}}
 Ampère–Maxwell equation (microscopic):

∇
×

B

G

=

4
π

c

J

G

1
c

∂

E

G

∂
t

{\displaystyle \nabla \times \mathbf{B}^\text{G} = \frac{4\pi}{c}\mathbf{J}^\text{G} + \frac{1}{c}\frac{\partial \mathbf{E}^\text{G}} {\partial t}}

∇
×

B

SI

=

μ
0

J

SI

1

c
2

∂

E

SI

∂
t

{\displaystyle \nabla \times \mathbf{B}^\text{SI} = \mu\_0\mathbf{J}^\text{SI} + \frac{1}{c^2}\frac{\partial \mathbf{E}^\text{SI}} {\partial t}}

### Other basic laws

F

=

q

G

(

E

G

1
c

v

×

B

G

)

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

F

=

q

SI

(

E

SI

v

×

B

SI

)

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

F

=

q
1

G

q
2

G

r
2

r
^

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

F

=

1

4
π

ε
0

q
1

SI

q
2

SI

r
2

r
^

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

E

=

q

G

r
2

r
^

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

E

=

1

4
π

ε
0

q

SI

r
2

r
^

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

B

G

=

1
c

∮

I

G

×

r
^

r
2

d

ℓ

{\displaystyle \mathbf{B}^{\text{G}} = \frac{1}{c}\!\oint\frac{I^{\text{G}} \times \mathbf{\hat r}}{r^2}\,\operatorname{d}\!\mathbf{\text{ℓ}}}

B

SI

=

μ
0

4
π

∮

I

SI

×

r
^

r
2

d

ℓ

{\displaystyle \mathbf{B}^{\text{SI}} = \frac{\mu\_0}{4\pi}\!\oint\frac{I^{\text{SI}} \times \mathbf{\hat r}}{r^2}\,\operatorname{d}\!\mathbf{\text{ℓ}}}
 Poynting vector(microscopic)

S

=

c

4
π

E

G

×

B

G

{\displaystyle \mathbf{S} = \frac{c}{4\pi}\,\mathbf{E}^{\text{G}} \times \mathbf{B}^{\text{G}}}

S

=

1

μ
0

E

SI

×

B

SI

{\displaystyle \mathbf{S} = \frac{1}{\mu\_0}\,\mathbf{E}^{\text{SI}} \times \mathbf{B}^{\text{SI}}}

### 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

G

=

E

G

4
π

P

G

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

D

SI

=

ε
0

E

SI

P

SI

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

P

G

=

χ
e
G

E

G

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

P

SI

=

χ
e
SI

ε
0

E

SI

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

D

G

=

ε
G

E

G

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

D

SI

=

ε
SI

E

SI

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

ε
G

=
1
4
π

χ
e
G

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

ε
SI

/

ε
0

=
1

χ
e
SI

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

where

* **E** and **D** are the electric field and displacement field, respectively;
* **P** is the polarization density;
* ε

{\displaystyle \varepsilon}

 is the permittivity;
* ε
0

{\displaystyle \varepsilon\_0}

 is the permittivity of vacuum (used in the SI system, but meaningless in Gaussian units);
* χ
e

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

 is the electric susceptibility

The quantities

ε
G

{\displaystyle \varepsilon^\text{G}}

 and

ε
SI

/

ε
0

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

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

χ
e
G

{\displaystyle \chi\_e^\text{G}}

 and

χ
e
SI

{\displaystyle \chi\_e^\text{SI}}

 are both unitless, but have *different numeric values* for the same material:

4
π

χ
e
G

=

χ
e
SI

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

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.

B

G

=

H

G

4
π

M

G

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

B

SI

=

μ
0

(

H

SI

M

SI

)

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

M

G

=

χ
m
G

H

G

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

M

SI

=

χ
m
SI

H

SI

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

B

G

=

μ
G

H

G

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

B

SI

=

μ
SI

H

SI

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

μ
G

=
1
4
π

χ
m
G

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

μ
SI

/

μ
0

=
1

χ
m
SI

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

where

* **B** and **H** are the magnetic fields
* **M** is magnetization
* μ

{\displaystyle \mu}

 is magnetic permeability
* μ
0

{\displaystyle \mu\_0}

 is the permeability of vacuum (used in the SI system, but meaningless in Gaussian units);
* χ
m

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

 is the magnetic susceptibility

The quantities

μ
G

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

 and

μ
SI

/

μ
0

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

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

χ
m
G

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

 and

χ
m
SI

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

 are both unitless, but has *different numeric values* in the two systems for the same material:

4
π

χ
m
G

=

χ
m
SI

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

### 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*:

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}}

E

SI

=
−
∇

ϕ
SI

−

∂

A

SI

∂
t

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

B

G

=
∇
×

A

G

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

B

SI

=
∇
×

A

SI

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

## Electromagnetic unit names

(For non-electromagnetic units, see Centimetre–gram–second system of units.)

Note: The SI quantities

ϵ
0

{\displaystyle \epsilon\_0}

 and

μ
0

{\displaystyle \mu\_0}

 satisfy

ϵ
0

μ
0

=
1

/

c
2

{\displaystyle \epsilon\_0\mu\_0 = 1/c^2}

.
The conversion factors are written both symbolically and numerically. The numerical conversion factors can be derived from the symbolic conversion factors by dimensional analysis. For example, the top row says

1

4
π

ϵ
0

=

2.998
×

10
9

Fr

1

C

{\displaystyle \frac{1}{\sqrt{4\pi\epsilon\_0}} = \frac{2.998 \times 10^9 \, \text{Fr}}{1\, \text{C}}}

, a relation which can be verified with dimensional analysis, by expanding

ϵ
0

{\displaystyle \epsilon\_0}

 and C in SI base units, and expanding Fr in Gaussian base units.

It is surprising to think of measuring capacitance in centimetres. One useful example is that a centimetre of capacitance is the capacitance between a sphere of radius 1 cm in vacuum and infinity.

Another surprising unit is measuring resistivity in units of seconds. A physical example is: Take a parallel-plate capacitor, which has a "leaky" dielectric with permittivity 1 but a finite resistivity. After charging it up, the capacitor will discharge itself over time, due to current leaking through the dielectric. If the resistivity of the dielectric is "X" seconds, the half-life of the discharge is ~0.05X seconds. This result is independent of the size, shape, and charge of the capacitor, and therefore this example illuminates the fundamental connection between resistivity and time units.

### Dimensionally equivalent units

A number of the units defined by the table have different names but are in fact dimensionally equivalent – i.e., they have the same expression in terms of the base units cm, g, s. (This is analogous to the distinction in SI between becquerel and Hz, or between newton-metre and joule.) The different names help avoid ambiguities and misunderstandings as to what physical quantity is being measured. In particular, *all* of the following quantities are dimensionally equivalent in Gaussian units, but they are nevertheless given different unit names as follows:

## General rules to translate a formula

Any formula can be converted between Gaussian and SI units by using the symbolic conversion factors from Table 1 above.

For example, the electric field of a stationary point charge has the SI formula

E

SI

=

q

SI

4
π

ϵ
0

r
2

r

^

,

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

where *r* is distance, and the "SI" subscripts indicate that the electric field and charge are defined using SI definitions. If we want the formula to instead use the Gaussian definitions of electric field and charge, we look up how these are related using Table 1, which says:

E

G

E

SI

=

4
π

ϵ
0

,

q

G

q
SI

=

1

4
π

ϵ
0

.

{\displaystyle \frac{\mathbf{E}^{\text{G}}}{\mathbf{E}^{\text{SI}}} = \sqrt{4\pi\epsilon\_0} \quad , \quad \frac{q^{\text{G}}}{q^\text{SI}} = \frac{1}{\sqrt{4\pi\epsilon\_0}} \, .}

Therefore, after substituting and simplifying, we get the Gaussian-units formula:

E

G

=

q

G

r
2

r

^

,

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

which is the correct Gaussian-units formula, as mentioned in a previous section.

For convenience, the table below has a compilation of the symbolic conversion factors from Table 1. To convert any formula from Gaussian units to SI units using this table, replace each symbol in the Gaussian column by the corresponding expression 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, as well as any other formula not listed. For some examples of how to use this table, see:

Once all occurrences of the product

ϵ
0

μ
0

{\displaystyle \epsilon\_0 \mu\_0}

 have been replaced by

1

/

c
2

{\displaystyle 1/c^2}

, there should be no remaining quantities in the equation with an SI electromagnetic dimension remaining.