Gaussian units

Major differences between Gaussian and SI systems

Rationalized unit systems

One difference between the Gaussian and SI systems is in the factor 4π in various formulas that relate the quantities that they define. With SI electromagnetic units, called rationalized, Maxwell's equations have no explicit factors of 4π in the formulae, whereas the inverse-square force laws – Coulomb's law and the Biot–Savart law – do have a factor of 4π attached to the r². With Gaussian units, called unrationalized (and unlike Heaviside–Lorentz 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, which reflects the geometry of the configuration. 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 the Gaussian system and the ISQ is in the respective definitions of the quantity charge. In the ISQ, a separate base dimension, electric current, with the associated SI unit, the ampere, is associated with electromagnetic phenomena, with the consequence that a unit of electrical charge (1 coulomb = 1 ampere × 1 second) is a physical quantity that cannot be expressed purely in terms of the mechanical units (kilogram, metre, second). On the other hand, in the Gaussian system, the unit of electric charge (the statcoulomb, statC) can be written entirely as a dimensional combination of the non-electrical base units (gram, centimetre, second), as:

1 statC = 1 g^1/2⋅cm^3/2⋅s⁻¹.

For example, Coulomb's law in Gaussian units has no constant: $${\displaystyle F=\frac{Q_1^{{}_{\mathrm{G}}}{Q}_2^{{}_{\mathrm{G}}}}{r^2},}$$ where F is the repulsive force between two electrical charges, Q^G
₁ and Q^G
₂ are the two charges in question, and r is the distance separating them. If Q^G
₁ and Q^G
₂ are expressed in statC and r in centimetres, then the unit of F that is coherent with these units is the dyne.

The same law in the ISQ is: $${\displaystyle F=\frac{1}{4\pi {\epsilon }_0}\frac{Q_1^{{}_{\mathrm{I}}}{Q}_2^{{}_{\mathrm{I}}}}{r^2}}$$ where ε₀ is the vacuum permittivity, a quantity that is not dimensionless: it has dimension (charge)² (time)² (mass)⁻¹ (length)⁻³. Without ε₀, the equation would be dimensionally inconsistent with the quantities as defined in the ISQ, whereas the quantity ε₀ does not appear in Gaussian equations. This is an example of how some dimensional physical constants can be eliminated from the expressions of physical law by the choice of definition of quantities. In the ISQ, 1/ε₀ converts or scales electric flux density, D, to the corresponding electric field, E (the latter has dimension of force per charge), while in the Gaussian system, electric flux density is the same quantity as electric field strength in free space aside from a dimensionless constant factor.

In the Gaussian system, the speed of light c appears directly in electromagnetic formulas like Maxwell's equations (see below), whereas in the ISQ it appears via the product μ₀ε₀ = 1/c².

Units for magnetism

In the Gaussian system, unlike the ISQ, the electric field E^G and the magnetic field B^G have the same dimension. This amounts to a factor of c 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 the magnetic field, H, and magnetization, M.) For example, in a planar light wave in vacuum, |E^G(r, t)| = |B^G(r, t)| in Gaussian units, while |E^I(r, t)| = c |B^I(r, t)| in the ISQ.

Polarization, magnetization

There are further differences between Gaussian system and the ISQ in how quantities related to polarization and magnetization are defined. For one thing, in the Gaussian system, all of the following quantities have the same dimension: E^G, D^G, P^G, B^G, H^G, and M^G. A further point is that the electric and magnetic susceptibility of a material is dimensionless in both the Gaussian system and the ISQ, 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 the Gaussian system and the International System of Quantities (ISQ). Most symbol names are not given; for complete explanations and definitions, please click to the appropriate dedicated article for each equation. A simple conversion scheme for use when tables are not available may be found in Garg (2012). All formulas except otherwise noted are from Ref.

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.

Other basic laws

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.

where

• E and D are the electric field and the displacement field, respectively;
• P is the polarization density;
• ${\displaystyle \epsilon }$ is the permittivity;
• ${\displaystyle {\epsilon }_0}$ is the permittivity of vacuum (used in the SI system, but meaningless in Gaussian units); and
• ${\displaystyle {\chi }_{\mathrm{e}}}$ is the electric susceptibility.

The quantities ${\displaystyle {\epsilon }^{{}_{\mathrm{G}}}}$ and ${\displaystyle {\epsilon }^{{}_{\mathrm{I}}}/{\epsilon }_0}$ are both dimensionless, and they have the same numeric value. By contrast, the electric susceptibility ${\displaystyle {\chi }_{\mathrm{e}}^{{}_{\mathrm{G}}}}$ and ${\displaystyle {\chi }_{\mathrm{e}}^{{}_{\mathrm{I}}}}$ are both unitless, but have different numeric values for the same material: $${\displaystyle 4\pi {\chi }_{\mathrm{e}}^{{}_{\mathrm{G}}}={\chi }_{\mathrm{e}}^{{}_{\mathrm{I}}}.}$$

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.

where

• B and H are the magnetic fields;
• M is the magnetization;
• ${\displaystyle \mu }$ is the magnetic permeability;
• ${\displaystyle {\mu }_0}$ is the permeability of vacuum (used in the SI system, but meaningless in Gaussian units); and
• ${\displaystyle {\chi }_{\mathrm{m}}}$ is the magnetic susceptibility.

The quantities ${\displaystyle {\mu }^{{}_{\mathrm{G}}}}$ and ${\displaystyle {\mu }^{{}_{\mathrm{I}}}/{\mu }_0}$ are both dimensionless, and they have the same numeric value. By contrast, the magnetic susceptibility ${\displaystyle {\chi }_{\mathrm{m}}^{{}_{\mathrm{G}}}}$ and ${\displaystyle {\chi }_{\mathrm{m}}^{{}_{\mathrm{I}}}}$ are both unitless, but has different numeric values in the two systems for the same material: $${\displaystyle 4\pi {\chi }_{\mathrm{m}}^{{}_{\mathrm{G}}}={\chi }_{\mathrm{m}}^{{}_{\mathrm{I}}}}$$

Vector and scalar potentials

The electric and magnetic fields can be written in terms of a vector potential A and a scalar potential ϕ:

Electrical circuit

where

• Q is the electric charge
• I is the electric current
• V is the electric potential
• Φ is the magnetic flux
• R is the electrical resistance
• C is the capacitance
• L is the inductance

Fundamental constants

Electromagnetic unit names

Note: All the equations are in SI, and the quantities ${\displaystyle {\epsilon }_0}$ and ${\displaystyle {\mu }_0}$ satisfy ⁠${\displaystyle {\epsilon }_0{\mu }_0=1/c^2}$⁠.

The conversion factors are written both symbolically and numerically. The numerical conversion factors can be derived from the symbolic conversion factors by substitution of equals for equals. For example, the top row says ${\displaystyle 1/\sqrt{4\pi {\epsilon }_0}\approx 2.998\times {10}^9\mathrm{F}\mathrm{r}/1\mathrm{C}}$, a relation which can be verified by substituting for ${\displaystyle {\epsilon }_0}$ and Fr their values as given in the permittivity row and Gaussian unit column, respectively, and expressing them and C in SI base units.

The equations of the table, including those in the Gaussian unit column and the note but excluding those containing a numerical conversion factor, constitute an interpretation (or model) in the SI of the Gaussian system, both its quantity and unit equations. An interpreted unit equation, however, is not a conversion, which makes use of the excluded equations. For example, the equation q^G = 1 Fr is interpreted to mean q^I/(4πε₀)^1/2 = 10^−9/2 m^3/2⋅kg^1/2⋅s⁻¹ but is converted to q^I = 3.3356 × 10⁻¹⁰ C.

A popular interpretation of the SI in the Gaussian system uses the equations q^I = q^G and 4πε₀ = 1. The equation q^I = 1 C is rewritten as q^I/(4πε₀)^1/2 = 1 C/(4πε₀)^1/2 = 2.9979 × 10^9/2 m^3/2⋅kg^1/2⋅s⁻¹, which is interpreted to mean q^G = 2.9979 × 10⁹ cm^3/2⋅g^1/2⋅s⁻¹ = 2.9979 × 10⁹ Fr. This model can be derived from Table 1 by substituting ε₀ = 1/(4π) and μ₀ = 4π/c².

Coulomb's law in the unrationalized MKSA system (U) may be modeled in the SI by either of the following triples of equations: (1) q^U = q^I, C^U = C^I, and ε^U = 4πε^I; or (2) q^U = q^I/(4π)^1/2, C^U = C^I/(4π)^1/2, and ε^U = ε^I. Each triple may be extended to a complete interpretation of the unrationalized MKSA system in the SI. The first interpretation is preferred, and interpretations like it (i.e., in which q is invariant, as in the preceding example) are used to model the SI in electrostatic and electromagnetic units.

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 t seconds, the half-life of the discharge is ~0.05 t 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 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:

General rules to translate a formula

Any formula can be converted between Gaussian and SI units by using the symbolic conversion factors from Table 1 above.

For example, the electric field of a stationary point charge has the ISQ formula $${\displaystyle {\mathbf{E}}^{{}_{\mathrm{I}}}=\frac{q^{{}_{\mathrm{I}}}}{4\pi {\epsilon }_0{r}^2}\hat{\mathbf{r}},}$$ where r is distance, and the "I" superscript indicates that the electric field and charge are defined as in the ISQ. If we want the formula to instead use the Gaussian definitions of electric field and charge, we look up how these are related using Table 1, which says: $${\displaystyle \begin{array}{cc}\frac{{\mathbf{E}}^{{}_{\mathrm{G}}}}{{\mathbf{E}}^{{}_{\mathrm{I}}}}& =\sqrt{4\pi {\epsilon }_0},\\ \frac{q^{{}_{\mathrm{G}}}}{q^{{}_{\mathrm{I}}}}& =\frac{1}{\sqrt{4\pi {\epsilon }_0}}.\end{array}}$$

Therefore, after substituting and simplifying, we get the Gaussian-system formula: $${\displaystyle {\mathbf{E}}^{{}_{\mathrm{G}}}=\frac{q^{{}_{\mathrm{G}}}}{r^2}\hat{\mathbf{r}},}$$ which is the correct Gaussian-system formula, as mentioned in a previous section.

For convenience, the table below has a compilation of the symbolic conversion factors from Table 1. To convert any formula from the Gaussian system to the ISQ using this table, replace each symbol in the Gaussian column by the corresponding expression in the SI column (vice versa to convert the other way). Replace ${\displaystyle 1/c^2}$ by ${\displaystyle {\epsilon }_0{\mu }_0}$ (or vice versa). 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.

After the rules of the table have been applied and the resulting formula has been simplified, replace all combinations ${\displaystyle {\epsilon }_0{\mu }_0}$ by ${\displaystyle 1/c^2}$.