Gaussian units

Carl Friedrich Gauss

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.^[1] 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.^[2][3] (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 dimensionless (loosely "unitless") 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

The main alternative to the Gaussian unit system is SI units, historically also called the MKSA system of units for metre–kilogram–second–ampere.^[2]

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.

SI units are by far the most common today. In engineering and practical areas, SI is nearly universal and has been for decades,^[2] 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.^[2][3]

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 electromagnetic units are called "rationalized",^[4][5] 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. 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. 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 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 Gaussian units is simple:

$F = \frac{Q_1Q_2}{r^2}$

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:

$F = \frac{1}{4\pi\epsilon_0} \frac{Q_1Q_2}{r^2} = k_\text{e} \frac{Q_1Q_2}{r^2}$

where ε₀ is the vacuum permittivity, a quantity with dimension, namely (charge)² (time)² (mass)⁻¹ (length)⁻³, and k_e is Coulomb's constant. 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. 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, flux density is the very same as electric field in free space, not just a scaled copy.

Since the unit of charge is built out of mechanical units (mass, length, time), the relation between mechanical units and electromagnetic phenomena is clearer in Gaussian units than in SI. In particular, in Gaussian units, the speed of light c shows up directly in electromagnetic formulas like Maxwell's equations (see below), whereas in SI it only shows up implicitly via the relation $\mu_0 \epsilon_0=1/c^2$.

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.^[4] (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.^[4]

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)	$\nabla \cdot \mathbf{D} = 4\pi\rho_\text{f}$	$\nabla \cdot \mathbf{D} = \rho_\text{f}$
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):	$\nabla \times \mathbf{H} = \frac{4\pi}{c}\mathbf{J}_\text{f} + \frac{1}{c}\frac{\partial \mathbf{D}} {\partial t}$	$\nabla \times \mathbf{H} = \mathbf{J}_\text{f} + \frac{\partial \mathbf{D}} {\partial t}$
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	$\mathbf{F} = \frac{q_1 q_2}{r^2} \mathbf{\hat r}$	$\mathbf{F} = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r^2} \mathbf{\hat r}$
Electric field of stationary point charge	$\mathbf{E} = \frac{q}{r^2} \mathbf{\hat r}$	$\mathbf{E} = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2} \mathbf{\hat r}$
Biot–Savart law	$\mathbf{B} = \frac{1}{c} \oint\frac{I d\mathbf{l} \times \mathbf{\hat r}}{r^2}$[6]	$\mathbf{B} = \frac{\mu_0}{4\pi} \oint\frac{I d\mathbf{l} \times \mathbf{\hat r}}{r^2}$

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.

Gaussian units	SI units
$\mathbf{D} = \mathbf{E}+4\pi\mathbf{P}$	$\mathbf{D} = \epsilon_0 \mathbf{E}+\mathbf{P}$
$\mathbf{P} = \chi_\text{e}\mathbf{E}$	$\mathbf{P} = \chi_\text{e}\epsilon_0\mathbf{E}$
$\mathbf{D} = \epsilon\mathbf{E}$	$\mathbf{D} = \epsilon\mathbf{E}$
$\epsilon = 1+4\pi\chi_\text{e}$	$\epsilon/\epsilon_0 = 1+\chi_\text{e}$

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);
• $\chi_\text{e}$ 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:

$\chi_\text{e}^\text{SI} = 4\pi \chi_\text{e}^\text{G}$

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.

Gaussian units	SI units
$\mathbf{B} = \mathbf{H}+4\pi\mathbf{M}$	$\mathbf{B} = \mu_0 (\mathbf{H}+\mathbf{M})$
$\mathbf{M} = \chi_\text{m}\mathbf{H}$	$\mathbf{M} = \chi_\text{m}\mathbf{H}$
$\mathbf{B} = \mu\mathbf{H}$	$\mathbf{B} = \mu\mathbf{H}$
$\mu = 1+4\pi\chi_\text{m}$	$\mu/\mu_0 = 1+\chi_\text{m}$

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);
• $\chi_\text{m}$ 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 $\chi_\text{m}$ is unitless in both systems, but has different numeric values in the two systems for the same material:

$\chi_\text{m}^\text{SI} = 4\pi \chi_\text{m}^\text{G}$

Vector and scalar potentials

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^[7]
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
electric potential voltage	φ V	1 V	↔ (10^8 c^−1) statV
electric field	E	1 V/m	↔ (10^6 c^−1) statV/cm
magnetic induction	B	1 T	↔ (10^4) Gs
magnetic field strength	H	1 A/m	↔ (4π 10^−3) Oe
magnetic dipole moment	μ	1 A·m²	↔ (10^3) erg/Gs
magnetic flux	Φm	1 Wb	↔ (10^8) Gs·cm²
resistance	R	1 Ω	↔ (10^9 c^−2) s/cm
resistivity	ρ	1 Ω·m	↔ (10^11 c^−2) s
capacitance	C	1 F	↔ (10^−9 c^2) cm
inductance	L	1 H	↔ (10^9 c^−2) s^2/cm

In this table, the letter c represents the number 29,979,245,800 ≈ 3·10¹⁰, 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, unless all other units in the formula are also replaced by their Gaussian equivalents.

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:^[8]

Quantity	In Gaussian base units	Gaussian unit of measure
E	cm^−1/2 g^1/2 s^−1	statV/cm
D	cm^−1/2 g^1/2 s^−1	statC/cm^2
P	cm^−1/2 g^1/2 s^−1	statC/cm^2
B	cm^−1/2 g^1/2 s^−1	Gs
H	cm^−1/2 g^1/2 s^−1	Oe
M	cm^−1/2 g^1/2 s^−1	Mx/cm^2 or emu/cm^3

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.^[9] For some examples of how to use this table, see:^[10]

Name	Gaussian units	SI units
Speed of light	$c$	$\frac{1}{\sqrt{\epsilon_0\mu_0}}$
Electric field, Electric potential	$\left(\mathbf{E}, \varphi\right)$	$\sqrt{4\pi\epsilon_0}\left(\mathbf{E}, \varphi\right)$
Electric displacement field	$\mathbf{D}$	$\sqrt{\frac{4\pi}{\epsilon_0}}\mathbf{D}$
Charge, Charge density, Current, Current density, Polarization density, Electric dipole moment	$\left(q, \rho, I, \mathbf{J},\mathbf{P}, \mathbf{p}\right)$	$\frac{1}{\sqrt{4\pi\epsilon_0}}\left(q, \rho, I, \mathbf{J},\mathbf{P},\mathbf{p}\right)$
Magnetic B field, Magnetic flux, Magnetic vector potential	$\left(\mathbf{B}, \Phi_\text{m},\mathbf{A}\right)$	$\sqrt{\frac{4\pi}{\mu_0}}\left(\mathbf{B}, \Phi_\text{m},\mathbf{A}\right)$
Magnetic H field	$\mathbf{H}$	$\sqrt{4\pi\mu_0}\mathbf{H}$
Magnetic moment, Magnetization	$\left(\mathbf{m}, \mathbf{M}\right)$	$\sqrt{\frac{\mu_0}{4\pi}}\left(\mathbf{m}, \mathbf{M}\right)$
Relative permittivity, Relative permeability	$\left(\epsilon, \mu\right)$	$\left(\frac{\epsilon}{\epsilon_0}, \frac{\mu}{\mu_0}\right)$
Electric susceptibility, Magnetic susceptibility	$\left(\chi_\text{e}, \chi_\text{m}\right)$	$\frac{1}{4\pi}\left(\chi_\text{e}, \chi_\text{m}\right)$
Conductivity, Conductance, Capacitance	$\left(\sigma, S, C\right)$	$\frac{1}{4\pi\epsilon_0}\left(\sigma,S,C\right)$
Resistivity, Resistance, Inductance	$\left(\rho,R,L\right)$	$4\pi\epsilon_0\left(\rho,R,L\right)$

Notes and references

1. One of many examples of using the term "cgs units" to refer to Gaussian units is: Lecture notes from Stanford University
2. "CGS", in How Many? A Dictionary of Units of Measurement, by Russ Rowlett and the University of North Carolina at Chapel Hill
3. For example, one widely used graduate electromagnetism textbook is Classical Electrodynamics by J.D. Jackson. The second edition, published in 1975, used Gaussian units exclusively, but the third edition, published in 1998, uses mostly SI units. Similarly, Electricity and Magnetism by Edward Purcell is a popular undergraduate textbook. The second edition, published in 1984, used Gaussian units, while the third edition, published in 2013, switched to SI units.
4. Littlejohn, Robert (Fall 2011). "Gaussian, SI and Other Systems of Units in Electromagnetic Theory" (PDF). Physics 221A, University of California, Berkeley lecture notes. Retrieved 2008-05-06.
5. Kowalski, Ludwik, 1986, "A Short History of the SI Units in Electricity," The Physics Teacher 24(2): 97–99. Alternate web link (subscription required)
6. Introduction to Electrodynamics by Capri and Panat, p180
7. Cardarelli, F. (2004). Encyclopaedia of Scientific Units, Weights and Measures: Their SI Equivalences and Origins (2nd ed.). Springer. pp. 20–25. ISBN 1-85233-682-X.
8. ''Demystifying Electromagnetic Equations''. Books.google.com. p. 155. Retrieved 2012-12-25.
9. Бредов М.М., Румянцев В.В., Топтыгин И.Н. (1985). "Appendix 5: Units transform (p.385)". Классическая электродинамика. Nauka.
10. Units in Electricity and Magnetism. See the section "Conversion of Gaussian formulae into SI" and the subsequent text.

External links

• Comprehensive list of Gaussian unit names, and their expressions in base units
• The evolution of the Gaussian Units by Dan Petru Danescu