Gaussian units

Carl Gauss

Gaussian units comprise 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: cgs contains within it several conflicting sets of electromagnetism units, not just Gaussian units, as described below.

The most common alternative to Gaussian units are SI units. SI units are predominant in most fields, and continue to increase in popularity at the expense of Gaussian units. (Other alternative unit systems also exist, as discussed below.) Conversions between Gaussian units and SI units are not as simple as normal unit conversions. For example, the formulas for physical laws of electromagnetism (such as Maxwell's equations) need to be adjusted depending on what system of units one uses. As another example, quantities that are unitless in one system may have dimensions in another.

Alternative unit systems

Main article: Alternative CGS units in electromagnetism

The main alternative to the Gaussian unit system is SI units, historically also called the MKS system of units for metre-kilogram-second.

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 atomic units, Planck units, and others. These are closer to CGS units than to SI units in some respects.

SI units are by far the most common today. In engineering and practical areas, SI is near-universal and has been for decades, while in technical, scientific literature (such as theoretical physics and astronomy), Gaussian units were predominant until recent decades, but are now getting progressively less so.

Natural units are most common 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 4π in various formulas. SI is called "rationalized", because Maxwell's equations have no explicit factors of 4π in the formulas. On the other hand, the force laws, Coulomb's law and the Biot–Savart law, do have factors of 4π in them. In Gaussian units, which are not "rationalized", the situation is reversed: Two of Maxwell's equations have factors of 4π in the formulas, while both of the force laws, Coulomb's law and the Biot–Savart law, have no factors of 4π.

Unit of charge

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 electrical phenomena, with the consequence that something like electrical charge (1 coulomb=1 ampere × 1 second) cannot be expressed purely in terms of the mechanical units (kilogram, metre, second). On the other hand, in Gaussian units, 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 cgs units is particularly simple:

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

where F is the repulsive force between two electrical charges, Q₁ and Q₂ are the two charges in question, and r is the distance separating them. If Q₁ and Q₂ are expressed in statC and r in cm, then F will come out expressed in dyne.

By contrast, the same law in SI units is:

$F={\frac {1}{4\pi \epsilon _{0}}}{\frac {Q_{1}Q_{2}}{r^{2}}}$

where ε₀ is the vacuum permitivity, a quantity with dimensions, namely A² s⁴ kg⁻¹ m⁻³. Without ε₀, the two sides could not have consistent dimensions in SI, and in fact the quantity ε₀ does not even exist in Gaussian units.

Units for magnetism

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

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, D, P, B, H, and M. 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. All formulas except otherwise noted are from Ref.

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 Kelvin–Stokes theorem.

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

Other basic laws

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

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.

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

where

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

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

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

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.

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

where

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

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

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

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 φ:

Name	Gaussian units	SI units
Electric field (static)	$\mathbf {E} =-\nabla \phi$	$\mathbf {E} =-\nabla \phi$
Electric field (general)	$\mathbf {E} =-\nabla \phi -{\frac {1}{c}}{\frac {\partial \mathbf {A} }{\partial t}}$	$\mathbf {E} =-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}$
Magnetic B field	$\mathbf {B} =\nabla \times \mathbf {A}$	$\mathbf {B} =\nabla \times \mathbf {A}$

Electromagnetic unit names

(For non-electromagnetic units, see main cgs article.)

Conversion of SI units in electromagnetism to Gaussian subsystem of CGS
c = 29,979,245,800 ≈ 3·10¹⁰

Quantity	Symbol	SI unit	Gaussian unit
electric charge	q	1 C	↔ (10−1 c) Fr
electric current	I	1 A	↔ (10−1 c) Fr·s−1
electric potential voltage	φ V	1 V	↔ (108 c−1) statV
electric field	E	1 V/m	↔ (106 c−1) statV/cm
magnetic induction	B	1 T	↔ (104) G
magnetic field strength	H	1 A/m	↔ (4π 10−3) Oe
magnetic dipole moment	μ	1 A·m²	↔ (103) erg/G
magnetic flux	Φm	1 Wb	↔ (108) G·cm²
resistance	R	1 Ω	↔ (109 c−2) s/cm
resistivity	ρ	1 Ω·m	↔ (1011 c−2) s
capacitance	C	1 F	↔ (10−9 c2) cm
inductance	L	1 H	↔ (109 c−2) s2·cm−1

In this table, the letter c represents the number 29,979,245,800 ≈ 3·10^{10, the numerical value of the speed of light expressed in cm/s. The symbol "↔" was used instead of "=" as a reminder that the SI and Gaussian units are corresponding but not equal because they have incompatible dimensions. For example, according to the top row of the table, something with a charge of 1 C also has a charge of (10−1 c) Fr, but it is usually incorrect to replace "1 C" with "(10−1 c) Fr" within an equation or formula.}

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., the 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:

Quantity	In Gaussian base units	Gaussian unit of measure
E	cm-1/2 g1/2 s−1	statV/cm
D	cm-1/2 g1/2 s−1	statC/cm2
P	cm-1/2 g1/2 s−1	statC/cm2
B	cm-1/2 g1/2 s−1	G
H	cm-1/2 g1/2 s−1	Oe
M	cm-1/2 g1/2 s−1	Mx/cm2 or emu/cm3

General rules to translate a formula

To convert any formula from Gaussian units to SI units, replace the quantity in the Gaussian column by the quantity 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. It may also be necessary to use the relation $\mu _{0}\epsilon _{0}=1/c^{2}$ to simplify. For some examples of how to use this table, see:

Name	Gaussian units	SI units
Electric field, Electric potential	$\mathbf {E} ,\varphi$	${\sqrt {4\pi \varepsilon _{0}}}(\mathbf {E} ,\varphi )$
Electric displacement field	$\mathbf {D}$	${\sqrt {4\pi /\varepsilon _{0}}}\mathbf {D}$
Charge, Charge density, Current, Current density, Polarization density, Electric dipole moment	$q,\rho ,I,\mathbf {j} ,\mathbf {P} ,\mathbf {p}$	${\frac {1}{\sqrt {4\pi \varepsilon _{0}}}}(q,\rho ,I,\mathbf {j} ,\mathbf {P} ,\mathbf {p} )$
Magnetic B field, Magnetic flux, Magnetic vector potential	$\mathbf {B} ,\Phi ,\mathbf {A}$	${\sqrt {4\pi /\mu _{0}}}(\mathbf {B} ,\Phi ,\mathbf {A} )$
Magnetic H field	$\mathbf {H}$	${\sqrt {4\pi \mu _{0}}}\mathbf {H}$
Magnetic moment, Magnetization	$\mathbf {m} ,\mathbf {M}$	${\sqrt {\mu _{0}/4\pi }}(\mathbf {m} ,\mathbf {M} )$
Relative permittivity, Relative permeability	$\varepsilon ,\mu$	$(\varepsilon _{r},\mu _{r})$, or equivalently, $(\varepsilon /\varepsilon _{0},\mu /\mu _{0})$
Electric susceptibility, Magnetic susceptibility	$\chi _{e},\chi _{m}$	${\frac {1}{4\pi }}(\chi _{e},\chi _{m})$
Conductivity, Conductance, Capacitance	$\sigma ,S,C$	${\frac {1}{4\pi \varepsilon _{0}}}(\sigma ,S,C)$
Resistivity, Resistance, Inductance	$\rho ,R,L$	$4\pi \varepsilon _{0}(\rho ,R,L)$

External links

• Comprehensive list of Gaussian unit names, and their expressions in base units