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# electric and magnetic units

electric and magnetic units, units used to express the magnitudes of various quantities in electricity and magnetism. Three systems of such units, all based on the metric system, are commonly used. One of these, the mksa-practical system, is defined in terms of the units of the mks system and has the ampere of electric current as its basic unit. The units of this system—the volt, ohm, watt, and farad—are those commonly used by scientists and engineers to make practical measurements. The two other systems are both based on the cgs system. Electrostatic units (cgs-esu) are defined in a way that simplifies the description of interactions between static electric charges; there are no corresponding magnetic units in this system. Electromagnetic units (cgs-emu), on the other hand, are defined especially for the description of phenomena associated with moving electric charges, i.e., electric currents and magnetic poles. The two cgs systems have been widely used in the past and are still found in many texts and papers. The official body for maintaining such units in the United States is the National Institute of Standards and Technology.
The centimetre-gram-second system (CGS) is a system of physical units. It is always the same for mechanical units, but there are several variants of electric additions. It was replaced by the MKS, or metre-kilogram-second system, which in turn was replaced by the International System of Units (SI), which has the three base units of MKS plus the ampere, mole, candela and kelvin.

## History

The system goes back to a proposal made in 1833 by the German mathematician Carl Friedrich Gauss and was in 1874 extended by the British physicists James Clerk Maxwell and William Thomson with a set of electromagnetic units. The sizes (order of magnitude) of many CGS units turned out to be inconvenient for practical purposes, therefore the CGS system never gained wide general use outside the field of electrodynamics and was gradually superseded internationally starting in the 1880s but not to a significant extent until the mid-20th century by the more practical MKS (metre-kilogram-second) system, which led eventually to the modern SI standard units.

CGS units are still occasionally encountered in technical literature, especially in the United States in the fields of electrodynamics and astronomy. SI units were chosen such that electromagnetic equations concerning spheres contain 4π, those concerning coils contain 2π and those dealing with straight wires lack π entirely, which was the most convenient choice for electrical-engineering applications. In those fields where formulas concerning spheres dominate (for example, astronomy), it has been argued that the CGS system can be notationally slightly more convenient. it can also sometimes be a willy.

Starting from the international adoption of the MKS standard in the 1940s and the SI standard in the 1960s, the technical use of CGS units has gradually disappeared worldwide, in the United States more slowly than in the rest of the world. CGS units are today no longer accepted by the house styles of most scientific journals, textbook publishers and standards bodies, although they are commonly used in astronomical journals such as the Astrophysical Journal.

The units gram and centimetre remain useful within the SI, especially for instructional physics and chemistry experiments, where they match well the small scales of table-top setups. In these uses, they are occasionally referred to as the system of “LAB” units. However, where derived units are needed, the SI ones are generally used and taught today instead of the CGS ones.

## CGS units in mechanics

In mechanics, both CGS and SI systems are built in an identical way. The only difference between the two systems is the scale of two out of the three base units needed in mechanics (centimetre versus metre and gram versus kilogram), while the third unit (measure of time: second) is the same in both systems . The laws and definitions of mechanics that are used to obtain all derived units from the three base units are the same in both systems, for example:

$v=frac\left\{x\right\}\left\{t\right\}$  (deifnition of velocity)

$F=mfrac\left\{d^2x\right\}\left\{dt^2\right\}$  (Newton's first law of motion)

$E = Fcdot dx$  (energy defined in terms of work)

$p = frac\left\{F\right\}\left\{L^2\right\}$  (pressure defined as force per unit area)

$eta = tau/frac\left\{dv\right\}\left\{dx\right\}$  (dynamic viscosity defined as shear stress per unit velocity gradient).

This explains why, for example, the CGS unit of pressure, barye, is related to the CGS base units of length, mass, and time in the same way as the SI unit of pressure, Pascal, is related to the SI base units of length, mass, and time:

1 Ba = 1 g/(cm·s2)

1 Pa = 1 kg/(m·s2).

However, expressing a CGS derived unit in terms of the SI base units involves a combination of the scale factors that relate the two systems:

1 Ba = 1 g/(cm·s2) = 10-3 kg/(10-2 m·s2) = 10-1 kg/(m·s2) = 10-1 Pa.

Definitions and conversion factors of CGS unis in mechanics
Quantity Symbol CGS unit CGS unit
abbreviation
Definition Equivalent
in SI units
length, position L, x centimetre cm 1/100 of metre = 10−2 m
mass m gram g 1/1000 of kilogram = 10−3 kg
time t second s 1 second = 1 s
velocity v centimetre per second cm/s cm/s = 10−2 m/s
force F dyne dyn g cm / s2 = 10−5 N
energy E erg erg g cm2 / s2 = 10−7 J
power P erg per second erg/s g cm2 / s3 = 10−7 W
pressure p barye Ba g / (cm s2) = 10−1 Pa
dynamic viscosity η poise P g / (cm s) = 10−1 Pa·s

## CGS units in electromagnetism

The conversion factors relating electromagnetic units in the CGS and SI systems are much more involved — so much so that formulas for physical laws of electromagnetism are adjusted depending on what system of units one uses. This illustrates the fundamental difference in the ways the two systems are built:

• In SI, the unit of electric current is arbitrarily chosen to be 1 ampere (A). It is set as a base unit of the SI system along with metre, kilogram, and second. All other electric and magnetic untis are derived from these four base units using the most basic common definitions: for example, electric charge q is defined as current I multiplied by time t,

$q=Icdot t$,
therefore unit of electric charge, coulomb (C), is defined as 1 C = 1 A·s.

• CGS system takes a rather different, and a much more direct approach to the definition of electromagnetic units: it avoids introducing new base units and instead derives all electric and magnetic units from centimetre, gram, and second based on the physics laws that relate electromagnetic phenomena to mechanics.

### Alternative ways of deriving electromagnetic units in CGS

Relating electromagnetic quantities to length, time and mass, however, can be done in a variety of equally appealing ways. Two of them rely on the forces observed on charges. There are two fundamental laws that relate (independently of each other) the electric charge or its rate of change (electric current) to a mechanical quantity such as force. They can be written in system-independent form as follows:

• The first is Coulomb's law, $F = k_C frac\left\{q cdot q^prime\right\}\left\{d^2\right\}$, which describes the electrostatic force F between electric charges $q$ and $q^prime$, separated by distance d. Here $k_C$ is a constant which depends on how exactly the unit of charge is derived from the CGS base units.
• The second is Ampère's force law, $frac\left\{dF\right\}\left\{dL\right\} = 2 k_Afrac\left\{I , I^prime\right\}\left\{d\right\}$, which describes the magnetic force F per unit length L between currents I and I' flowing in two long parallel wires, separated by distance d. Since $I=q/t$ and $I^prime=q^prime/t$, the constant $k_A$ also depends on how the unit of charge is derived from the CGS base units.

Maxwell's theory of electromagnetism relates these two laws to each other. It states that the ratio of proportionality constants $k_C$ and $k_A$ must obey $k_C / k_A = c^2$, where c is the speed of light. Therefore, if one derives the unit of charge from the Coulomb's law by setting $k_C=1$, it is obvious that the Ampère's force law will contain a prefactor $2/c^2$. Alternatively, deriving the unit of current, and therefore the unit of charge, from the Ampère's force law by setting $k_A = 1$ or $k_A = 1/2$, will lead to a constant prefactor in the Coulomb's law.

Indeed, both of these mutually-exclusive approaches have been practiced by the users of CGS system, leading to the two independent and mutually-exclusive branches of CGS, described in the subsections below. However, the freedom of choice in deriving electromagnetic units from the units of length, mass, and time is not limited to the definition of charge. While the electric field can be related to the work performed by it on a moving electric charge, the magnetic force is always perpendicular to the velocity of the moving charge, and thus the work performed by the magnetic field on any charge is always zero. This leads to a choice between two laws of magnetism, each relating magnetic field to mechanical quantities and electric charge:

• The first law describes the Lorentz force produced by a magnetic field B on a charge q moving with velocity v:

$mathbf\left\{F\right\} = alpha_L q;mathbf\left\{v\right\} times mathbf\left\{B\right\};.$

• The second describes the creation of a static magnetic field B by an electric current I of finite length dl at a point displaced by a vector r, known as Biot-Savart law:

$dmathbf\left\{B\right\} = alpha_Bfrac\left\{I dmathbf\left\{l\right\} times mathbf\left\{hat r\right\}\right\}\left\{r^2\right\};,$ where r and $mathbf\left\{hat r\right\}$ are the length and the unit vector in the direction of vector r.
These two laws can be used to derive Ampère's force law, resulting in the relationship: $k_A = alpha_L cdot alpha_B;$. Therefore, if the unit of charge is based on the Ampère's force law such that $k_A = 1$, it is natural to derive the unit of magnetic field by setting $alpha_L = alpha_B=1;$. However, if it is not the case, a choice has to be made as to which of the two laws above is a more convenient basis for deriving the unit of magnetic field.

Furthermore, if we wish to describe the electric displacement field $vec D$ and the magnetic field $vec H$ in a medium other than a vacuum, we need to also define the constants ε0 and μ0, which are the vacuum permittivity and permeability, respectively. Then we have (generally) $vec D = epsilon_0 vec E + lambda vec P$ and $vec H = vec B / mu_0 - lambda^prime vec M$. The factors λ and λ′ are rationalization constants, which are usually chosen to be 4πkCε0, which is dimensionless. If this quantity equals 1, the system is said to be rationalized. The original CGS system, however, used λ = λ′ = 4π, or, equivalently, kCε0 = 1.

### Overview of various CGS systems

While the absence of some explicit prefactors in CGS simplifies theoretical calculations, it has the disadvantage that the units in CGS are hard to define through experiment. SI on the other hand starts with a unit of current, the ampere, which is easy to determine through experiment, but which requires that the constants in the electromagnetic equations take on extra prefactors.

The table below shows the constant values used in some common systems:

system $k_C$ $alpha_B$ $epsilon_0$ $mu_0$ $k_A=frac\left\{k_C\right\}\left\{c^2\right\}$ $alpha_L=frac\left\{k_C\right\}\left\{alpha_Bc^2\right\}$ $lambda=4pi k_Ccdotepsilon_0$ $lambda\text{'}$
Gaussian 1 c−1 1 1 c−2 c−1
electrostatic (esu) 1 c−1 1 $frac\left\{1\right\}\left\{c^2\right\}$ c−2 c−1
rationalized
electrostatic
1 c−1 $frac\left\{1\right\}\left\{4pi\right\}$ $frac\left\{4pi\right\}\left\{c^2\right\}$ c−2 c−1 1 1
electromagnetic (emu) c2 1 c−2 1 1 1
Heaviside-Lorentz $frac\left\{1\right\}\left\{4pi\right\}$ c−1 1 $frac\left\{1\right\}\left\{c^2\right\}$ $frac\left\{1\right\}\left\{4pi c^2\right\}$ $frac\left\{1\right\}\left\{4pi c\right\}$ 1 1
Heaviside $frac\left\{1\right\}\left\{4pi\right\}$ $frac\left\{1\right\}\left\{4pi c\right\}$ 1 1 $frac\left\{1\right\}\left\{4pi c^2\right\}$ c−1 1 1
SI $frac\left\{c^2\right\}\left\{b\right\}$ $frac\left\{1\right\}\left\{b\right\}$ $frac\left\{b\right\}\left\{4pi c^2\right\}$ $frac\left\{4pi\right\}\left\{b\right\}$ $frac\left\{1\right\}\left\{b\right\}$ 1 1 1

(The constant b in SI system is a unit-based scaling factor defined as: $b=10^7,mathrm\left\{A\right\}^2/mathrm\left\{N\right\} = 10^7,mathrm\left\{m/H\right\};$.)

In system-independent form, Maxwell's equations in vacuum can be written as:

$begin\left\{array\right\}\left\{ccl\right\} vec nabla cdot vec E & = & 4 pi k_C rho vec nabla cdot vec B & = & 0 vec nabla times vec E & = & displaystyle\left\{- alpha_L frac\left\{partial vec B\right\}\left\{partial t\right\}\right\} vec nabla times vec B & = & displaystyle\left\{4 pi alpha_B vec J + frac\left\{alpha_B\right\}\left\{k_C\right\}frac\left\{partial vec E\right\}\left\{partial t\right\}\right\} end\left\{array\right\}$

Dimension Unit Definition SI
charge electrostatic unit of charge, franklin, statcoulomb 1 esu = 1 statC = 1 Fr = √(g·cm³/s²) = 3.33564 × 10−10 C
electric current biot 1 esu/s = 3.33564 × 10−10 A
electric potential statvolt 1 statV = 1 erg/esu = 299.792458 V
electric field 1 statV/cm = 1 dyn/esu = 2.99792458 × 104 V/m
magnetic field strength H oersted 1 Oe = 1000/(4π) A/m = 79.577 A/m
magnetic dipole moment emu 1 emu = 4π × 10-6 Oe = 10−3 Am²
magnetic flux maxwell 1 Mw = 1 G·cm² = 10−8 Wb
magnetic induction B gauss 1 G = 1 Mw/cm² = 10−4 T
resistance 1 s/cm = 8.988 × 1011 Ω
resistivity 1 s = 8.988 × 109 Ω·m
capacitance 1 cm = 1.113 × 10−12 F
inductance statH = 8.988 × 1011 H
wavenumber kayser 1 /cm = 100 /m

The mantissas derived from the speed of light are more precisely 299792458, 333564095198152, 1112650056, and 89875517873681764.

A centimetre of capacitance is the capacitance between a sphere of radius 1 cm in vacuum and infinity. The capacitance C between two concentric spheres of radii R and r is

$frac\left\{1\right\}\left\{frac\left\{1\right\}\left\{r\right\}-frac\left\{1\right\}\left\{R\right\}\right\}$.
By taking the limit as R goes to infinity we see C equals r.

### Electrostatic units (ESU)

In one variant of the CGS system, Electrostatic units (ESU), charge is defined via the force it exerts on other charges, and current is then defined as charge per time. It is done by setting the Coulomb force constant $k_C = 1$, so that Coulomb’s law does not contain an explicit prefactor.

The ESU unit of charge, statcoulomb or esu charge, is therefore defined as follows: In CGS electrostatic units, a statcoulomb is equal to a centimetre times square root of dyne:

$\left(mathrm\left\{1,statcoulomb = 1,esu, charge = 1,cmsqrt\left\{dyne\right\}=1,g^\left\{1/2\right\} cdot cm^\left\{3/2\right\} cdot s^\left\{-1\right\}\right\}\right)$.
Dimensionally in the CGS ESU system, charge q is therefore equivalent to m1/2L3/2t−1 and is not an independent dimension of physical quantity. This reduction of units is an application of the Buckingham π theorem.

### Other variants

There were at various points in time about half a dozen systems of electromagnetic units in use, most based on the CGS system. These include electromagnetic units (emu, chosen such that the Biot-Savart law has no explicit prefactor), Gaussian units, and Heaviside-Lorentz units.

Further complicating matters is the fact that some physicists and engineers in the United States use hybrid units, such as volts per centimetre for electric field. In fact, this is essentially the same as the SI unit system, by the variant to translate all lengths used into cm, e.g. 1 m = 100 cm. More difficult is to translate electromagnetic quantities from SI to cgs, which is also not hard, e.g. by using the three relations $q\text{'}=q/sqrt\left\{4pi epsilon_0\right\}$,   $mathbf E\text{'}=mathbf Ecdot sqrt\left\{4pi epsilon_0\right\}$, and $mathbf B\text{'}=mathbf Bcdotsqrt\left\{4pi/ mu_0\right\}$, where $epsilon_0\left(,,equiv 1/\left(c^2mu_0\right)\right)$ and $mu_0$ are the well-known vacuum permittivities and c the corresponding light velocity, whereas $q, ,,mathbf E$ and $mathbf B$ are the electrical charge, electric field, and magnetic induction, respectively, without primes in a SI system and with primes in a CGS system.

However, the above-mentioned example of hybrid units can also simply be seen as a practical example of the previously described "LAB" units usage since electric fields near small circuit devices would be measured across distances on the order of magnitude of one centimetre.

## Physical constants in CGS units

Constant Symbol Value
Atomic mass unit u 1.660 × 10−24 g
Bohr magneton μB 9.274 × 10−21 erg/G
Bohr radius a0 5.291 × 10−9 cm
Boltzmann constant k 1.380 × 10−16 erg/K
Electron mass me 9.109 × 10−28 g
Elementary charge e 4.803 × 10−10 esu of charge
1.602 × 10−19 emu of charge
Fine-structure constant α 7.297 × 10−3
Gravitational constant G 6.674 × 10−8 cm3/(g·s2)
Planck constant h 6.626 × 10−27 erg·s
Speed of light in vacuum c 2.998 × 1010 cm/s

## Pro and contra

A key virtue of the Gaussian CGS system is that electric and magnetic fields have the same units, $4piepsilon_0$ is replaced by $1$, and the only dimensional constant appearing in the equations is $c$, the speed of light. The Heaviside-Lorentz system has these desirable properties as well (with $epsilon_0$ equalling 1), but is a "rationalized" system (as is SI) in which the charges and fields are defined in such a way that there are many fewer factors of $4 pi$ appearing in the formulas, and it is in Heaviside-Lorentz units that the Maxwell equations take their simplest possible form.

At the same time, the elimination of $epsilon_0$ and $mu_0$ can also be viewed as a major disadvantage of all the variants of the CGS system. Within classical electrodynamics, this elimination makes sense because it greatly simplifies the Maxwell equations. In quantum electrodynamics, however, the vacuum is no longer just empty space, but it is filled with virtual particles that interact in complicated ways. The fine structure constant in Gaussian CGS is given as $alpha=e^2/hbar c$ and it has been cause to much mystification how its numerical value $alpha approx 1/137.036$ should be explained. In SI units with $alpha = e^2/4 pi epsilon_0hbar c$ it may be clearer that it is in fact the complicated quantum structure of the vacuum that gives rise to a non-trivial vacuum permittivity. However, the advantage would be purely pedagogical, and in practice, SI units are essentially never used in quantum electrodynamics calculations. In fact the high energy community uses a system where every quantity is expressed by only one unit, namely by eV, i.e. lengths L by the corresponding reciprocal quantity $frac\left\{hbar \right\}\left\{m_Lcdot c\right\} equiv L=frac\left\{hbar c\right\}\left\{E_L\right\}$, where the Einstein expression corresponding to $m_L$, $E_L=m_L,,c^2$, is an energy, which thus can naturally be expressed in eV  ($hbar$ is Planck's constant divided by $2pi$).