Gaussian units

In physics, in particular in electromagnetic theory, Gaussian units are a set of units for electric and magnetic quantities. The units are named for the German mathematician and physicist Carl Friedrich Gauss, who was the first to define magnetic units.

The most common and most elaborate set of units are the SI units (formerly known as metric or MKSA units). Their main advantage is that they are very widespread and well defined by international committees for all different engineering and science disciplines. The entire engineering world uses SI units, so almost any discussion of electrical equipment or experimental apparatus is in terms of SI units.

The main advantage of Gaussian units is that they simplify, more than the SI units do, the fundamental physical issues and theoretical relations involving electromagnetic phenomena. Especially, the theories of relativity and electrodynamics are simpler, more transparent and more elegant in Gaussian units than in SI units. In addition, the various formulas of electromagnetism are easier to remember in Gaussian units than in SI units. Because they are superior for fundamental physical questions, it is unlikely that Gaussian units will ever be completely abandoned.

The Gaussian system is based on cgs (centimeter-gram-second) units. The base mechanical units (length, mass, time) and some of the derived mechanical units (force, work, etc.) are given in Table 1.

In contrast to the SI units, the Gaussian units are unrationalized. This means that the factor 4π arises in the Maxwell equations, and is missing in other places, such as in Coulomb's law and the Biot-Savart law. In general, unrationalized systems of units give simpler formulas when we are dealing with problems of spherical symmetry, while the rationalized units give simpler formulas in problems with rectangular symmetry.

The Gaussian system is a mixed system, which means that it takes the unit of charge (the statC) from the esu system (electrostatic system of units), and the unit of magnetic flux (the maxwell) from the emu system (electromagnetic system of units). (The maxwell is a derived unit in the emu system; the abampere is an emu base unit). The electric units that the Gaussian system shares with the esu system are given in Table 2 and the magnetic units shared by the Gaussian system with the emu system in Table 3.

The Gaussian system does not know about the electric constant ε₀ or the magnetic constant μ₀, which are related to the speed of light c by

$\varepsilon _{0}\,\mu _{0}={\frac {1}{c^{2}}}.$

These constants do not represent physical properties of the vacuum, but are artifacts of the SI system. Instead, the Gaussian system uses c.

In Gaussian units the electric field E, the polarization P, the electric displacement D, the magnetic induction B, the magnetization M and the magnetic field H have the same dimensions, while in SI units the dimensions are all different. In addition, the scalar and vector potentials φ and A have the same dimensions in Gaussian units, but not in SI units. The uniform dimensions for fields in Gaussian units make it easy to remember formulas in that system, and it makes fundamental physical relations more transparent.

Conversion tables

Table 1:  Mechanical units

Symbol	Property	SI Unit	Factor	cgs
l	Length	meter (m)	100	centimeter(cm)
m	Mass	kilogram (kg)	1000	gram (g)
t	Time	second (s)	1	second (s)
a	Acceleration	m/s2	100	galileo (Gal)
F	Force	newton (N)	105	dyne (dyn)
W	Energy	joule (J)	107	erg (erg)
P	Power	watt (W)	107	erg/s

Example: 1 J = 10⁷ erg

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Table 2:  Electric units

Symbol	Property	SI Unit	Factor	Gaussian
I	Electric current	ampere (A)	10c	statampere (statA)
Q	Charge	coulomb (C)	10c	statcoulomb (statC)
V	Electric potential	volt (V)	106/c	statvolt (statV)
R	Resistance	ohm (Ω)	105/c2	statohm (statΩ)
G	Conductance	siemens (S)	10−5c2	statsiemens (statS)
L	Self-inductance	henry (H)	105/c2	abhenry (abH)
C	Capacitance	farad (F)	10−5c2	cm
E	Electric field	V/m	104/c	statV/cm
ρ	Electric charge density	C/m3	c/105	statC/cm3
D	Electric displacement	C/m2	4π10-3c	statV/cm

c is the speed of light in m/s (≈ 3⋅10⁸ m/s).
Example: 1 A = 10c statA.

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Table 3:  Magnetic units

Symbol	Property	Gaussian → SI
Φ	magnetic flux	1 Mx → 10−8 Wb = 10−8 V⋅s
B	magnetic flux density	1 G → 10−4 T = 10−4 Wb/m2
	magnetic induction
H	magnetic field	1 Oe → 103/(4π) A/m
m	magnetic moment	1 erg/G = 1 emu → 10−3 A⋅m2 = 10−3 J/T
M	magnetization	1 erg/(G⋅cm3) = 1 emu/cm3 → 103 A/m
4πM	magnetization	1 G → 103/(4π) A/m
σ	mass magnetization	1 erg/(G⋅g) = 1 emu/g → 1 A⋅m2/kg
	specific magnetization
j	magnetic dipole moment	1 erg/G = 1 emu → 4π ⋅ 10−10 Wb⋅m
J	magnetic polarization	1 erg/(G⋅cm3) = 1 emu/cm3 → 4π ⋅ 10−4 T
χ, κ	susceptibility	1 → 4π
χρ	mass susceptibility	1 cm3/g → 4π ⋅ 10−3 m3/kg
μ	permeability	1 → 4π ⋅ 10−7 H/m = 4π ⋅ 10−7 Wb/(A⋅m)
μr	relative permeability	μ → μr
w, W	energy density	1 erg/cm3 → 10−1 J/m3
N, D	demagnetizing factor	1 → 1/(4π)

Mx = maxwell, G = gauss, Oe = oersted ; Wb = weber, V = volt, s = second, T = tesla, m = meter, A = ampere, J = joule, kg = kilogram, H = henry

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