Definitions and conversion factors of CGS units in mechanics

CGS approach to electromagnetic units

The conversion factors relating electromagnetic units in the CGS and SI systems are made more complex by the differences in the formulas expressing physical laws of electromagnetism as assumed by each system of units, specifically in the nature of the constants that appear in these formulas. This illustrates the fundamental difference in the ways the two systems are built:

• In SI, the unit of electric current, the ampere (A), was historically defined such that the magnetic force exerted by two infinitely long, thin, parallel wires 1 metre apart and carrying a current of 1 ampere is exactly 2×10⁻⁷ N/m. This definition results in all SI electromagnetic units being numerically consistent (subject to factors of some integer powers of 10) with those of the CGS-EMU system described in further sections. The ampere is a base unit of the SI system, with the same status as the metre, kilogram, and second. Thus the relationship in the definition of the ampere with the metre and newton is disregarded, and the ampere is not treated as dimensionally equivalent to any combination of other base units. As a result, electromagnetic laws in SI require an additional constant of proportionality (see Vacuum permeability) to relate electromagnetic units to kinematic units. (This constant of proportionality is derivable directly from the above definition of the ampere.) All other electric and magnetic units 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,
  $${\displaystyle q=I\,t,}$$

  
  resulting in the unit of electric charge, the coulomb (C), being defined as 1 C = 1 A⋅s.
• The CGS system variant avoids introducing new base quantities and units, and instead defines all electromagnetic quantities by expressing the physical laws that relate electromagnetic phenomena to mechanics with only dimensionless constants, and hence all units for these quantities are directly derived from the centimetre, gram, and second.

Alternative derivations of CGS units in electromagnetism

Electromagnetic relationships to length, time and mass may be derived by several equally appealing methods. Two of them rely on the forces observed on charges. Two fundamental laws relate (seemingly 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^[7] in system-independent form as follows:

• The first is Coulomb's law, ${\displaystyle F=k_{\rm {C}}{\frac {q\,q^{\prime }}{d^{2}}},}$ which describes the electrostatic force F between electric charges ${\displaystyle q}$ and ${\displaystyle q^{\prime },}$ separated by distance d. Here ${\displaystyle k_{\rm {C}}}$ is a constant which depends on how exactly the unit of charge is derived from the base units.
• The second is Ampère's force law, ${\displaystyle {\frac {dF}{dL}}=2k_{\rm {A}}{\frac {I\,I^{\prime }}{d}},}$ which describes the magnetic force F per unit length L between currents I and I′ flowing in two straight parallel wires of infinite length, separated by a distance d that is much greater than the wire diameters. Since ${\displaystyle I=q/t\,}$ and ${\displaystyle I^{\prime }=q^{\prime }/t,}$ the constant ${\displaystyle k_{\rm {A}}}$ also depends on how the unit of charge is derived from the base units.

Maxwell's theory of electromagnetism relates these two laws to each other. It states that the ratio of proportionality constants ${\displaystyle k_{\rm {C}}}$ and ${\displaystyle k_{\rm {A}}}$ must obey ${\displaystyle k_{\rm {C}}/k_{\rm {A}}=c^{2},}$ where c is the speed of light in vacuum. Therefore, if one derives the unit of charge from the Coulomb's law by setting ${\displaystyle k_{\rm {C}}=1}$ then Ampère's force law will contain a factor ${\displaystyle 2/c^{2}.}$ Alternatively, deriving the unit of current, and therefore the unit of charge, from the Ampère's force law by setting ${\displaystyle k_{\rm {A}}=1}$ or ${\displaystyle k_{\rm {A}}=1/2,}$ will lead to a constant factor 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:
  $${\displaystyle \mathbf {F} =\alpha _{\rm {L}}q\;\mathbf {v} \times \mathbf {B} \;.}$$

  
• 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:
  $${\displaystyle d\mathbf {B} =\alpha _{\rm {B}}{\frac {Id\mathbf {l} \times \mathbf {\hat {r}} }{r^{2}}}\;,}$$

  
  where r and ${\displaystyle \mathbf {\hat {r}} }$ are the length and the unit vector in the direction of vector r respectively.

These two laws can be used to derive Ampère's force law above, resulting in the relationship: ${\displaystyle k_{\rm {A}}=\alpha _{\rm {L}}\cdot \alpha _{\rm {B}}\;.}$ Therefore, if the unit of charge is based on the Ampère's force law such that ${\displaystyle k_{\rm {A}}=1,}$ it is natural to derive the unit of magnetic field by setting ${\displaystyle \alpha _{\rm {L}}=\alpha _{\rm {B}}=1\;.}$ 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 D and the magnetic field H in a medium other than vacuum, we need to also define the constants ε₀ and μ₀, which are the vacuum permittivity and permeability, respectively. Then we have^[7] (generally) ${\displaystyle \mathbf {D} =\epsilon _{0}\mathbf {E} +\lambda \mathbf {P} }$ and ${\displaystyle \mathbf {H} =\mathbf {B} /\mu _{0}-\lambda ^{\prime }\mathbf {M} ,}$ where P and M are polarization density and magnetization vectors. The units of P and M are usually so chosen that the factors λ and λ′ are equal to the "rationalization constants" ${\displaystyle 4\pi k_{\rm {C}}\epsilon _{0}}$ and ${\displaystyle 4\pi \alpha _{\rm {B}}/(\mu _{0}\alpha _{\rm {L}}),}$ respectively. If the rationalization constants are equal, then ${\displaystyle c^{2}=1/(\epsilon _{0}\mu _{0}\alpha _{\rm {L}}^{2}).}$ If they are equal to one, then the system is said to be "rationalized":^[11] the laws for systems of spherical geometry contain factors of 4π (for example, point charges), those of cylindrical geometry – factors of 2π (for example, wires), and those of planar geometry contain no factors of π (for example, parallel-plate capacitors). However, the original CGS system used λ = λ′ = 4π, or, equivalently, ${\displaystyle k_{\rm {C}}\epsilon _{0}=\alpha _{\rm {B}}/(\mu _{0}\alpha _{\rm {L}})=1.}$ Therefore, Gaussian, ESU, and EMU subsystems of CGS (described below) are not rationalized.

Various extensions of the CGS system to electromagnetism

The table below shows the values of the above constants used in some common CGS subsystems:

System	${\displaystyle k_{\rm {C}}}$	${\displaystyle \alpha _{\rm {B}}}$	${\displaystyle \epsilon _{0}}$	${\displaystyle \mu _{0}}$	${\displaystyle k_{\rm {A}}={\frac {k_{\rm {C}}}{c^{2}}}}$	${\displaystyle \alpha _{\rm {L}}={\frac {k_{\rm {C}}}{\alpha _{\rm {B}}c^{2}}}}$	${\displaystyle \lambda =4\pi k_{\rm {C}}\epsilon _{0}}$	${\displaystyle \lambda '={\frac {4\pi \alpha _{\rm {B}}}{\mu _{0}\alpha _{\rm {L}}}}}$
Electrostatic[7] CGS (ESU, esu, or stat-)	1	c−2	1	c−2	c−2	1	4π	4π
Electromagnetic[7] CGS (EMU, emu, or ab-)	c2	1	c−2	1	1	1	4π	4π
Gaussian[7] CGS	1	c−1	1	1	c−2	c−1	4π	4π
Heaviside–Lorentz[7] CGS	${\displaystyle {\frac {1}{4\pi }}}$	${\displaystyle {\frac {1}{4\pi c}}}$	1	1	${\displaystyle {\frac {1}{4\pi c^{2}}}}$	c−1	1	1
SI	${\displaystyle {\frac {1}{4\pi \epsilon _{0}}}}$	${\displaystyle {\frac {\mu _{0}}{4\pi }}}$	${\displaystyle \epsilon _{0}}$	${\displaystyle \mu _{0}}$	${\displaystyle {\frac {\mu _{0}}{4\pi }}}$	1	1	1

Also, note the following correspondence of the above constants to those in Jackson^[7] and Leung:^[12]

$${\displaystyle {\begin{aligned}k_{\rm {C}}&=k_{1}=k_{\rm {E}}\\\alpha _{\rm {B}}&=\alpha \cdot k_{2}=k_{\rm {B}}\\k_{\rm {A}}&=k_{2}=k_{\rm {E}}/c^{2}\\\alpha _{\rm {L}}&=k_{3}=k_{\rm {F}}\end{aligned}}}$$

Of these variants, only in Gaussian and Heaviside–Lorentz systems ${\displaystyle \alpha _{\rm {L}}}$ equals ${\displaystyle c^{-1}}$ rather than 1. As a result, vectors ${\displaystyle {\vec {E}}}$ and ${\displaystyle {\vec {B}}}$ of an electromagnetic wave propagating in vacuum have the same units and are equal in magnitude in these two variants of CGS.

In each of these systems the quantities called "charge" etc. may be a different quantity; they are distinguished here by a superscript. The corresponding quantities of each system are related through a proportionality constant.

Maxwell's equations can be written in each of these systems as:^[7][12]

System
CGS-ESU	${\displaystyle \nabla \cdot {\vec {E}}^{\text{ESU}}=4\pi \rho ^{\text{ESU}}}$	${\displaystyle \nabla \times {\vec {B}}^{\text{ESU}}-c^{-2}{\dot {\vec {E}}}^{\text{ESU}}=4\pi c^{-2}{\vec {J}}^{\text{ESU}}}$	${\displaystyle \nabla \cdot {\vec {B}}^{\text{ESU}}=0}$	${\displaystyle \nabla \times {\vec {E}}^{\text{ESU}}+{\dot {\vec {B}}}^{\text{ESU}}=0}$
CGS-EMU	${\displaystyle \nabla \cdot {\vec {E}}^{\text{EMU}}=4\pi c^{2}\rho ^{\text{EMU}}}$	${\displaystyle \nabla \times {\vec {B}}^{\text{EMU}}-c^{-2}{\dot {\vec {E}}}^{\text{EMU}}=4\pi {\vec {J}}^{\text{EMU}}}$	${\displaystyle \nabla \cdot {\vec {B}}^{\text{EMU}}=0}$	${\displaystyle \nabla \times {\vec {E}}^{\text{EMU}}+{\dot {\vec {B}}}^{\text{EMU}}=0}$
CGS-Gaussian	${\displaystyle \nabla \cdot {\vec {E}}^{\text{G}}=4\pi \rho ^{\text{G}}}$	${\displaystyle \nabla \times {\vec {B}}^{\text{G}}-c^{-1}{\dot {\vec {E}}}^{\text{G}}=4\pi c^{-1}{\vec {J}}^{\text{G}}}$	${\displaystyle \nabla \cdot {\vec {B}}^{\text{G}}=0}$	${\displaystyle \nabla \times {\vec {E}}^{\text{G}}+c^{-1}{\dot {\vec {B}}}^{\text{G}}=0}$
CGS-Heaviside–Lorentz	${\displaystyle \nabla \cdot {\vec {E}}^{\text{LH}}=\rho ^{\text{LH}}}$	${\displaystyle \nabla \times {\vec {B}}^{\text{LH}}-c^{-1}{\dot {\vec {E}}}^{\text{LH}}=c^{-1}{\vec {J}}^{\text{LH}}}$	${\displaystyle \nabla \cdot {\vec {B}}^{\text{LH}}=0}$	${\displaystyle \nabla \times {\vec {E}}^{\text{LH}}+c^{-1}{\dot {\vec {B}}}^{\text{LH}}=0}$
SI	${\displaystyle \nabla \cdot {\vec {E}}^{\text{SI}}=\rho ^{\text{SI}}/\epsilon _{0}}$	${\displaystyle \nabla \times {\vec {B}}^{\text{SI}}-\mu _{0}\epsilon _{0}{\dot {\vec {E}}}^{\text{SI}}=\mu _{0}{\vec {J}}^{\text{SI}}}$	${\displaystyle \nabla \cdot {\vec {B}}^{\text{SI}}=0}$	${\displaystyle \nabla \times {\vec {E}}^{\text{SI}}+{\dot {\vec {B}}}^{\text{SI}}=0}$

Electrostatic units (ESU)

In the electrostatic units variant of the CGS system, (CGS-ESU), charge is defined as the quantity that obeys a form of Coulomb's law without a multiplying constant (and current is then defined as charge per unit time):

${\displaystyle F={q_{1}^{\text{ESU}}q_{2}^{\text{ESU}} \over r^{2}}.}$

The ESU unit of charge, franklin (Fr), also known as statcoulomb or esu charge, is therefore defined as follows:^[13]

two equal point charges spaced 1 centimetre apart are said to be of 1 franklin each if the electrostatic force between them is 1 dyne.

Therefore, in CGS-ESU, a franklin is equal to a centimetre times square root of dyne:

${\displaystyle \mathrm {1\,Fr=1\,statcoulomb=1\,esu\;charge=1\,dyne^{1/2}{\cdot }cm=1\,g^{1/2}{\cdot }cm^{3/2}{\cdot }s^{-1}} .}$

The unit of current is defined as:

${\displaystyle \mathrm {1\,Fr/s=1\,statampere=1\,esu\;current=1\,dyne^{1/2}{\cdot }cm{\cdot }s^{-1}=1\,g^{1/2}{\cdot }cm^{3/2}{\cdot }s^{-2}} .}$

In the CGS-ESU system, charge q is therefore has the dimension to M^1/2L^3/2T⁻¹.

Other units in the CGS-ESU system include the statampere (1 statC/s) and statvolt (1 erg/statC).

In CGS-ESU, all electric and magnetic quantities are dimensionally expressible in terms of length, mass, and time, and none has an independent dimension. Such a system of units of electromagnetism, in which the dimensions of all electric and magnetic quantities are expressible in terms of the mechanical dimensions of mass, length, and time, is traditionally called an 'absolute system'.^[14]:3

Electromagnetic units (EMU)

In another variant of the CGS system, electromagnetic units (EMU), current is defined via the force existing between two thin, parallel, infinitely long wires carrying it, and charge is then defined as current multiplied by time. (This approach was eventually used to define the SI unit of ampere as well). In the EMU CGS subsystem, this is done by setting the Ampere force constant ${\displaystyle k_{\rm {A}}=1}$, so that Ampère's force law simply contains 2 as an explicit factor.

The EMU unit of current, biot (Bi), also known as abampere or emu current, is therefore defined as follows:^[13]

The biot is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one centimetre apart in vacuum, would produce between these conductors a force equal to two dynes per centimetre of length.

Therefore, in electromagnetic CGS units, a biot is equal to a square root of dyne:

${\displaystyle \mathrm {1\,Bi=1\,abampere=1\,emu\;current=1\,dyne^{1/2}=1\,g^{1/2}{\cdot }cm^{1/2}{\cdot }s^{-1}} }$.

The unit of charge in CGS EMU is:

${\displaystyle \mathrm {1\,Bi{\cdot }s=1\,abcoulomb=1\,emu\,charge=1\,dyne^{1/2}{\cdot }s=1\,g^{1/2}{\cdot }cm^{1/2}} }$.

Dimensionally in the CGS-EMU system, charge q is therefore equivalent to M^1/2L^1/2. Hence, neither charge nor current is an independent physical quantity in the CGS-EMU system.

Relations between ESU and EMU units

The ESU and EMU subsystems of CGS are connected by the fundamental relationship ${\displaystyle k_{\rm {C}}/k_{\rm {A}}=c^{2}}$ (see above), where c = 29979245800 ≈ 3×10¹⁰ is the speed of light in vacuum in centimetres per second. Therefore, the ratio of the corresponding "primary" electrical and magnetic units (e.g. current, charge, voltage, etc. – quantities proportional to those that enter directly into Coulomb's law or Ampère's force law) is equal either to c⁻¹ or c:^[13]

${\displaystyle \mathrm {\frac {1\,statcoulomb}{1\,abcoulomb}} =\mathrm {\frac {1\,statampere}{1\,abampere}} =c^{-1}}$

and

${\displaystyle \mathrm {\frac {1\,statvolt}{1\,abvolt}} =\mathrm {\frac {1\,stattesla}{1\,gauss}} =c}$.

Units derived from these may have ratios equal to higher powers of c, for example:

${\displaystyle \mathrm {\frac {1\,statohm}{1\,abohm}} =\mathrm {\frac {1\,statvolt}{1\,abvolt}} \times \mathrm {\frac {1\,abampere}{1\,statampere}} =c^{2}}$.

Practical CGS units

The practical CGS system is a hybrid system that uses the volt and the ampere as the units of voltage and current respectively. Doing this avoids the inconveniently large and small electrical units that arise in the esu and emu systems. This system was at one time widely used by electrical engineers because the volt and ampere had been adopted as international standard units by the International Electrical Congress of 1881.^[15] As well as the volt and ampere, the farad (capacitance), ohm (resistance), coulomb (electric charge), and henry (inductance) are consequently also used in the practical system and are the same as the SI units. The magnetic units are those of the emu system.^[16]

The electrical units, other than the volt and ampere, are determined by the requirement that any equation involving only electrical and kinematical quantities that is valid in SI should also be valid in the system. For example, since electric field strength is voltage per unit length, its unit is the volt per centimetre, which is one hundred times the SI unit.

The system is electrically rationalized and magnetically unrationalized; i.e., λ = 1 and λ′ = 4π, but the above formula for λ is invalid. A closely related system is the International System of Electric and Magnetic Units,^[17] which has a different unit of mass so that the formula for λ′ is invalid. The unit of mass was chosen to remove powers of ten from contexts in which they were considered to be objectionable (e.g., P = VI and F = qE). Inevitably, the powers of ten reappeared in other contexts, but the effect was to make the familiar joule and watt the units of work and power respectively.

The ampere-turn system is constructed in a similar way by considering magnetomotive force and magnetic field strength to be electrical quantities and rationalizing the system by dividing the units of magnetic pole strength and magnetization by 4π. The units of the first two quantities are the ampere and the ampere per centimetre respectively. The unit of magnetic permeability is that of the emu system, and the magnetic constitutive equations are B = (4π/10)μH and B = (4π/10)μ₀H + μ₀M. Magnetic reluctance is given a hybrid unit to ensure the validity of Ohm's law for magnetic circuits.

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.^[18] These include the Gaussian units and the Heaviside–Lorentz units.