Gaussian units

One difference between Gaussian and SI units is in the factors of 4π in various formulas. SI electromagnetic units are called "rationalized",[5][6] because Maxwell's equations have no explicit factors of 4π 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 4π attached to the r². In unrationalized Gaussian units (not Lorentz–Heaviside units) the situation is reversed: two of Maxwell's equations have factors of 4π in the formulas, while both of the inverse-square force laws, Coulomb's law and the Biot–Savart law, have no factor of 4π attached to r² in the denominator.

(The quantity 4π appears because 4πr² 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.)

A major difference between Gaussian and SI units is in the definition of the unit of charge. In SI, a separate base unit (the ampere) is associated with electromagnetic phenomena, with the consequence that something like electrical charge (1 coulomb = 1 ampere × 1 second) is a unique dimension of physical quantity and is not expressed purely in terms of the mechanical units (kilogram, metre, second). On the other hand, in the Gaussian system, the unit of electrical charge (the statcoulomb, statC) can be written entirely as a dimensional combination of the mechanical units (gram, centimetre, second), as:

1 statC = 1 g^1/2⋅cm^3/2⋅s⁻¹

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

${\displaystyle F={\frac {Q_{1}^{\text{G}}Q_{2}^{\text{G}}}{r^{2}}}}$

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. If Q^G
₁ and Q^G
₂ are expressed in statC and r in cm, then F will come out expressed in dyne.

The same law in SI units is:

${\displaystyle F={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q_{1}^{\text{SI}}Q_{2}^{\text{SI}}}{r^{2}}}}$

where ε₀ is the vacuum permittivity, a quantity with dimension, namely (charge)² (time)² (mass)⁻¹ (length)⁻³. Without ε₀, the two sides would not have consistent dimensions in SI, whereas the quantity ε₀ 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/ε₀, 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 ${\displaystyle \mu _{0}\varepsilon _{0}=1/c^{2}}$.

In Gaussian units, unlike SI units, the electric field E^G and the magnetic field B^G have the same dimension. This amounts to a factor of c between how B is defined in the two unit systems, on top of the other differences.[5] (The same factor applies to other magnetic quantities such as H and M.) For example, in a planar light wave in vacuum, |E^G(r, t)| = |B^G(r, t)| in Gaussian units, while |E^SI(r, t)| = c |B^SI(r, t)| in SI units.

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.)

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.

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

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.

Gaussian quantities	SI quantities
${\displaystyle \mathbf {D} ^{\text{G}}=\mathbf {E} ^{\text{G}}+4\pi \mathbf {P} ^{\text{G}}}$	${\displaystyle \mathbf {D} ^{\text{SI}}=\varepsilon _{0}\mathbf {E} ^{\text{SI}}+\mathbf {P} ^{\text{SI}}}$
${\displaystyle \mathbf {P} ^{\text{G}}=\chi _{\text{e}}^{\text{G}}\mathbf {E} ^{\text{G}}}$	${\displaystyle \mathbf {P} ^{\text{SI}}=\chi _{\text{e}}^{\text{SI}}\varepsilon _{0}\mathbf {E} ^{\text{SI}}}$
${\displaystyle \mathbf {D} ^{\text{G}}=\varepsilon ^{\text{G}}\mathbf {E} ^{\text{G}}}$	${\displaystyle \mathbf {D} ^{\text{SI}}=\varepsilon ^{\text{SI}}\mathbf {E} ^{\text{SI}}}$
${\displaystyle \varepsilon ^{\text{G}}=1+4\pi \chi _{\text{e}}^{\text{G}}}$	${\displaystyle \varepsilon ^{\text{SI}}/\varepsilon _{0}=1+\chi _{\text{e}}^{\text{SI}}}$

where

• E and D are the electric field and displacement field, respectively;
• P is the polarization density;
• ${\displaystyle \varepsilon }$ is the permittivity;
• ${\displaystyle \varepsilon _{0}}$ is the permittivity of vacuum (used in the SI system, but meaningless in Gaussian units);
• ${\displaystyle \chi _{\text{e}}}$ is the electric susceptibility

The quantities ${\displaystyle \varepsilon ^{\text{G}}}$ and ${\displaystyle \varepsilon ^{\text{SI}}/\varepsilon _{0}}$ are both dimensionless, and they have the same numeric value. By contrast, the electric susceptibility ${\displaystyle \chi _{e}^{\text{G}}}$ and ${\displaystyle \chi _{e}^{\text{SI}}}$ are both unitless, but have different numeric values for the same material:

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

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.

Gaussian quantities	SI quantities
${\displaystyle \mathbf {B} ^{\text{G}}=\mathbf {H} ^{\text{G}}+4\pi \mathbf {M} ^{\text{G}}}$	${\displaystyle \mathbf {B} ^{\text{SI}}=\mu _{0}(\mathbf {H} ^{\text{SI}}+\mathbf {M} ^{\text{SI}})}$
${\displaystyle \mathbf {M} ^{\text{G}}=\chi _{\text{m}}^{\text{G}}\mathbf {H} ^{\text{G}}}$	${\displaystyle \mathbf {M} ^{\text{SI}}=\chi _{\text{m}}^{\text{SI}}\mathbf {H} ^{\text{SI}}}$
${\displaystyle \mathbf {B} ^{\text{G}}=\mu ^{\text{G}}\mathbf {H} ^{\text{G}}}$	${\displaystyle \mathbf {B} ^{\text{SI}}=\mu ^{\text{SI}}\mathbf {H} ^{\text{SI}}}$
${\displaystyle \mu ^{\text{G}}=1+4\pi \chi _{\text{m}}^{\text{G}}}$	${\displaystyle \mu ^{\text{SI}}/\mu _{0}=1+\chi _{\text{m}}^{\text{SI}}}$

where

• B and H are the magnetic fields
• M is magnetization
• ${\displaystyle \mu }$ is magnetic permeability
• ${\displaystyle \mu _{0}}$ is the permeability of vacuum (used in the SI system, but meaningless in Gaussian units);
• ${\displaystyle \chi _{\text{m}}}$ is the magnetic susceptibility

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

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

The electric and magnetic fields can be written in terms of a vector potential A and a scalar potential φ:

Name	Gaussian units	SI units
Electric field	${\displaystyle \mathbf {E} ^{\text{G}}=-\nabla \phi ^{\text{G}}-{\frac {1}{c}}{\frac {\partial \mathbf {A} ^{\text{G}}}{\partial t}}}$	${\displaystyle \mathbf {E} ^{\text{SI}}=-\nabla \phi ^{\text{SI}}-{\frac {\partial \mathbf {A} ^{\text{SI}}}{\partial t}}}$
Magnetic B field	${\displaystyle \mathbf {B} ^{\text{G}}=\nabla \times \mathbf {A} ^{\text{G}}}$	${\displaystyle \mathbf {B} ^{\text{SI}}=\nabla \times \mathbf {A} ^{\text{SI}}}$

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:[10]