Maxwell's equations

The Maxwell equations of James Clerk Maxwell describe the phenomena of electromagnetism. They are thus an important part of the modern physical world view.

The equations describe how electric and magnetic fields with each other and related to electrical charges and electricity under given boundary conditions. Together with the Lorentz force them to explain all the phenomena of classical electrodynamics. They therefore also form the theoretical basis of the optical and electrical engineering. The equations are named after the Scottish physicist James Clerk Maxwell, who has it worked from 1861 to 1864. He combined this the Ampere's law and the Gaussian law with the law of induction and resulted in addition in order not to violate the continuity equation, the named after him also a displacement current.

The Maxwell equations are a special system of linear partial differential equations of first order. They can also be presented in an integral form in differential geometric shape and in covariant form.

3.1 Overview
3.2 Explanations 3.2.1 Electricity
3.2.2 Electric field
3.2.3 Magnetic field

4.1 The three-dimensional approach 4.1.1 The inhomogeneous Maxwell equations
4.1.2 The homogeneous Maxwell equations
4.1.3 The material equations

4.2.1 The homogeneous Maxwell equations
4.2.2 The material equations
4.2.3 The entire Maxwell equations with only two differential forms
4.2.4 Abstract integral formulation and interpretation

5.1 Maxwell's equations for a constant frequency in complex notation
5.2 covariate formulation of the Maxwell equations
5.3 Maxwell's equations taking into account hypothetical magnetic monopoles
5.4 Maxwell's equations and photon mass

7.1 Systematic transformation behavior (SI ↔ cgs )

Maxwell equations in the flux line picture

The electric and the magnetic field can be represented by field lines. The electric field is represented by the fields of the electric field strength and the electric flux density, while the magnetic field is represented by the fields of the magnetic field strength and the magnetic flux density.

The electric field strength and the magnetic flux density can be illustrated by the principle of force to charge. The relationships are described in the article about the Lorentz force accurately. In the case of electric field the profile of the electric field strength indicates the direction of force exerted by the field strength (the force acting in the direction of the tangent to the field line at any place ), the field line density (the vicinity of the field lines to one another ), the field strength in this area dar. In the case of the magnetic field, the force acting perpendicular to the direction of the magnetic flux density, and perpendicular to the direction of movement of the charge.

The following figure shows the field pattern is illustrated using a positive and a negative charge. The electric field is strongest to the carriers, and increases with greater distance from:

In source fields, the field lines are characterized by a beginning and an end (or vanish at infinity ). In eddy fields, the field lines are closed curves.

The Gauss's law for electric fields states that electric charges are sources and sinks of the field of the electric flux density, that is the beginning and end of the associated field lines represent. Electric fields without sources and sinks, so-called vortex fields, however, occur in induction processes.

Gauss's law for magnetism indicates that the field of the magnetic flux density having no sources. The magnetic flux density has therefore only closed field lines.

Induction Faraday's Law: Temporal changes of the magnetic flux cause electric vortex field.

Advanced Ampere's law, also called Maxwell Ampèresches Act: Electric currents - including a temporal change of the electric flux density - lead to a magnetic vortex field.

Equations

In a narrower sense, the Maxwell equations are the mathematical description of these laws. Directly analogous to the laws they can be described by four coupled differential equations, but there are also equivalent formulations.

Notation

Denotes the nabla operator. The differential operators mean:

Denotes the divergence of
Denotes the rotation of

Microscopic Maxwell equations

Microscopic Maxwell equations combine the electric field strength and the magnetic flux density of the charge density (charge per volume) and the electric current density ( current per -carrying surface ).

Macroscopic Maxwell equations

In the presence of matter, the microscopic Maxwell equations are unwieldy on the one hand, because after all, everyone charge carriers must be considered in every atom of the medium. On the other hand, the magnetic properties can be ( for example by a permanent magnet ) is derived in principle without additional physical findings of the quantum mechanics of the microscopic Maxwell equations.

The macroscopic Maxwell equations take into account the properties of matter in the form of material parameters, the empty space will be assigned to the parameters of permittivity and permeability. Maxwell himself was not of an empty room, but - as usual in his time - from the filled with a so-called "ether" of space. The term " macroscopic" comes from the fact that the properties of matter ultimately characterize locally averaged properties of matter. With respect to the charges is made between the free charge carriers (such as the conduction electrons in the electrical conductor ) and bonded charge carriers (such as shell electrons ), it is discriminated and it is assumed that the bound charge carriers through microscopic processes on a macroscopic polarization or magnetization.

The presence of matter requires that the electric and magnetic fields are each described by two additional vector fields, the electric flux density and the magnetic field strength.

And it is, for example,

Differential and integral formulation

The following sections and tables indexing of charge and current is used with respect to a semantically equivalent convention: Namely or written without index and referred to as "true charges " or " real flows", while conversely the microscopic to the ' equations ' occurring non-indexed variables are written as rms quantities or. In vacuum, " microscopic equations " and apply to the indexing can be avoided. The following equations apply, however, in matter, and one has to rely on a uniform notation, mostly the used below, although here too different conventions are not excluded.

Survey

Here, the Maxwell equations are given in SI units, eg. Formulations in other unit systems are listed at the end and are illustrated by the comments in the text.

The symbols given below in the right column in the center of single or double integrals emphasize that one has to do with closed curves or surfaces.

Any change in the field leads to a mating electrical field. The vortex of the electric field depends on the temporal change of the magnetic induction.

The vortex of the magnetic field depends on the electrical line and of the current density of the electric flux density. The time change of displacement is referred to as current density and is the sum of the conduction current density, the total current density:

Notes

It should be noted that all variables must be measured from an arbitrary but the same for all sizes inertial system. If using the above equations, for example, the voltage induced in a moving conductor loop are considered, it is advantageous to convert the variables in the moving parts of the system using the Lorentz transformation to the rest frame.

In this context it should be mentioned that some textbooks write the following approximation instead of the law of induction:

The equation measures the field strength at the respective speed of the line element and all other quantities in the rest frame. This equation is not compatible with the special theory of relativity and valid only for small velocities.

Set in the formulation of the law of induction, the magnetic flux are used, instead, lends itself to the following formula:

The equation is due ( no magnetic monopoles, which is one of Maxwell's statements ) is equivalent to the induction law in differential form, and thus with the special theory of relativity tolerated.

Electricity

The electric current density can be purely formal summarizes both the normal conduction current density according to the flow of electrical charge carriers and the displacement current ( the temporal change of the electric field ), which played an important role in the discovery of Maxwell's equations by Maxwell. Usually, however, the displacement current is listed separately. The electric current density is linked via the constitutive equations of electrodynamics and the resultant electrical conductivity with the electric field strength.

Electric field

Is the electric flux density, historic and somewhat confusing also called electric displacement density or electrical excitation. It is the density of the electric flux that emanates from electrical charges. The electric flux density is linked via the constitutive equations of electrodynamics and the thereby occurring dielectric conductivity with the electric field strength. More generally applies to the electric polarization, the electric dipole moment per unit volume.

Magnetic field

Is the magnetic flux density, historically referred to as induction. It is the density of the magnetic flux which is caused by moving electrical charges, or by permanent magnets. The magnetic flux density is linked via the constitutive equations of electrodynamics and the thereby occurring magnetic conductivity with the magnetic field strength. More generally applies to the magnetic polarization, the magnetic dipole moment per unit volume (as magnetization is equivalent to the hereafter designated size ).

The magnetic polarization should not (or more precisely to the line current density) with the current density to be confused. Rather, the following applies:

Explanation of the Maxwell equations with matter

The material equations occurring in all three areas are not counted directly to the Maxwell equations, but the three sets of equations:

Maxwell equations
Constitutive equations of electrodynamics
Continuity equations of electrodynamics

Put together and complement each other and the foundation of the electrodynamic field theory represents the material equations are valid in the general form for both the empty space as well as filled with matter space areas.

For historical reasons, and sometimes certain processes represent specific calculation, the equations and the matter occurring in three conductivities to be cleaved in each of the proportion of the empty space, respectively, and the proportion of the conductivity, which is caused by the matter, and.

The electric field is obtained by the splitting of the permittivity the possibility of introduction of an additional vector field, the electric polarization (actually dielectric polarization, but also referred to as electric polarization, since the electric field assigned).

Similarly, the magnetic polarization of the loosened describes the characteristics of the empty space in the material conditions of the magnetic field. From the magnetic polarization results in the magnetization. ( In the cgs system, the situation is confusing: where they are labeled the same as cgs magnetization, and differ only by a factor depending on whether or meant is. )

Principle can be dispensed without loss of the introduction of vector fields of the electrical polarization, and the magnetic polarization (or the equivalent thereto of magnetization ). Instead, the dependencies in the material equations and the corresponding conductivities generally taken in the form of tensors of higher order are considered. Furthermore, the conductivities can also represent functions to capture non-linear properties of matter can. These can even depend on the pre-treatment, so be explicitly time-dependent. This procedure is also recommended for a systematic approach when this is done via the SI unit system. For historical reasons, but also in specific sub- fields of physics, but is sometimes very intense from the - and - ( or - ) vector fields made use of, which is why this subject is dealt with in more detail below.

In matter is generally

As well as

Which still results in the special case of linearity with isotropy or cubic systems, the following simplification:

And

In homogeneous isotropic materials (ie, the sizes and are scalar and constant) is obtained for the Maxwell equations

In anisotropic non- linear cubic matter are the scalars and tensors second stage, the relationships remain valid. In nonlinear materials, the conductivities depend on their instantaneous values of the field strengths or the most general case of the entire history (see hysteresis). The - and - fields, called electric or magnetic polarization, vanish outside of matter, which is equivalent to the statement in the above special cases, that is.

The dielectric polarization is then with the electrical susceptibility, and the relative permittivity and the vacuum permittivity ( dielectric constant ) linked as follows ( in SI system, i.e. in the unit ):

With

For the magnetic polarization or the magnetization of a corresponding equation with the magnetic susceptibility or the relative permeability and the vacuum permeability ( magnetic constant ) with the unit:

With

(Caution: are in the cgs system and multiplied by! )

Next there is the definition of the refractive index with

And the relationship between the speed of light and electric and magnetic field constant

This causes the propagation of light in the material constants of the medium in combination. Thus, the phase velocity in the medium is

Without the dispersion equal to the group velocity.

Summary

Explanation: The last specified, brackets relations are only valid for a linear relationship. The stated before that definitions of and on the other hand generally.

Traditionally, the two last- mentioned so-called material laws and Ohm's law will be (here is the specific electrical conductance ) usually not involved in the Maxwell equations. The continuity equation as a description of the charge conservation follows from the Maxwell equations.

The electric field strength and the magnetic flux densities are interpreted as physically existing force fields. Even Maxwell linked these force fields to the electrical potential field and the vector potential:

The relationship between field strengths and potentials, although defined only up to gauge transformations, the potentials comes but in the quantum theory of fundamental importance to ..

Maxwell's equations in differential forms (differential geometric formulation)

The description of the vector analysis has the great disadvantage that they

Is on the flat or limited
Principle "contaminated metric " is because either the Euclidean or the Minkowski metric is installed in the operators, although the Maxwell equations are metric- free defined
The choice of a map of the underlying manifold is unphysical, since laws of nature must be properly independent of the chosen coordinate.

It is therefore better to write the equations with alternating differential forms and thus to consistently use the methods of differential geometry.

The three-dimensional approach

In this three-dimensional approach, the time is treated as an external parameter, as usual, from classical mechanics.

The inhomogeneous Maxwell equations

Let be a differential form on the arbitrary smooth manifold of dimension 3 and the Cartan'sche exterior derivative. So Then

Because there can not be different from 0 differential form of degree 4 on a three-dimensional manifold. On a star-shaped region secures the lemma of Poincaré that a potential form exists, so that

Furthermore, it is postulated that the time derivative of the charge from a variety of power is opposed by the boundary (read: everything wants from the " volume " out, must pass through the boundary surface flow ).

This statement thus corresponds to the conservation law associated with the continuity equation for the total charge ( the arbitrariness of the manifold ensures analogous to the law of Gauss, that this also applies without integrals). is called the current density (two dimensionally ). So:

This mathematical statement implies but according to the lemma of Poincaré that on a star -shaped region, a differential form of degree 1 exists, so

It should be noted that the law of Gauss follows purely from the geometry of the problem, and ultimately has no physical meaning: the only physical input is the existence of electric charges and the continuity equation, which opens in the Maxwell - Ampère 's law. The inhomogeneous equations are thus a consequence of the charge conservation. Not affected is basically just the so-called spin magnetism, that is, those magnetic phenomena, which do not originate from here exclusively treated Ampère circuit currents ( the vertebrae of j ) (see Mathematical structure of quantum mechanics, especially the section about the spin, and the article about the so-called gyromagnetic ratio). This applies to the dominant part of the permanent magnetism. But this is basically just that classical electrodynamics is not complete in itself, although it mathematically and theoretically - physically it seems.

Maxwell's equations, the homogeneous

Similarly, the continuity equation, the law of induction is postulated. The time change of the magnetic flux through an area associated with the induction of a voltage at its opposite annular edge. This is completely analogous to the continuity equation, only one dimension lower.

Here, the magnetic flux density (two dimensionally ), and the electric field. The randomness of the surface ensures that the law of induction can be written without integral:

Thus, it can be seen that the manifold may be dependent only on the (room) components, but not on the time. However, the expression on the left of the equal sign does not depend on the choice of coordinates. Must therefore be f (x, y, z) disappear. In addition, the equation can only be Lorentz invariant. It follows the source of freedom of the magnetic flux density ( two dimensionally ) (ie, the non-existence of magnetic charges, see above):

Again, only a postulate is a, the induction law; the source of freedom is then a purely mathematical consequence.

The constitutive equations

Because the one-forms E and H are not compatible with the two forms of D and B, one needs to establish a relationship between them. This is done with the Hodge operator, which is on a three-dimensional manifold is an isomorphism between one-forms and two- forms.

Here it is obvious why H and B or E and D can be easily identified already for mathematical reasons (up to a factor ). H is indeed a one-form and is integrated over a curve B is a two- form and needs a (2- dimensional ) surface of integration. ( In addition, the associated vector fields are also physically much different in polarizable media. ) It can thus already consist of mathematics not establish proportionality between these variables, as suggested by the description of the vector analysis. The same applies to E and D: the first variable describes a differential form of degree 1, that the integration takes a curve as in a motor integral; the second variable is a two- form, so needs a face like a river integral. This difference seems pedantic, but is fundamental.

It should be noted that only the metrics play a role in the equations with the Hodgeoperator. Maxwell's equations without the constitutive equations are independent of the choice of the metric, and even independent of the nature of the manifold, as long as three-dimensional. Only the effect of the constitutive equations would change.

The four-dimensional approach

Is a smooth manifold of dimension 4 and a smooth submanifold of dimension 3 ( from the 3- dimensional approach) and the metric tensor with coefficients representation.

( There are many equivalent forms which can be obtained by multiplying a number by the amount of 1, for example. )

The metric only needs to be set, so that one can explicitly write down the ensuing four-potential ( physics: " contravariant quantities" ), without going through the coefficients of a vector field ( physics: " covariant quantities" ) to go with

The definition of the Minkowski space, which can among other things, needed to " spacelike " and " timelike " vector or tensor components to be distinguished, or in the definition of the duality operation ( see below), that is not necessary here. You could choose the metric also free, then see the components of the one-form

Only different, because

Be so from here the multiplicity of flat Minkowski space, ie without loss of generality. Then the vector potential is given by

Maxwell's equations, the homogeneous

Now let the exterior derivative of A is given by, ie by the so-called field strength tensor ( Faradayzweiform ):

Impressive is the fact that the outer derivative of F always will disappear irrespective of the A looks. This yields the so-called gauge freedom and justifies why the restriction not to hurt the Minkowski space the general public. However, since the equations do not require any physical input, it directly follows that the homogeneous Maxwell equations are merely a consequence of the geometry of space and the formalism used ( the same is true indeed for the relationship: a closed differential form is still largely free, namely to the exterior derivative of a lower to a degree form. ).

The constitutive equations

The Faradayzweiform can be written also in the sizes already known:

The dual to F two- form G is called Maxwell two- form and is given by the already known variables, namely:

In physical theories corresponds to the field strength tensor F and G its dual tensor ( see below).

The entire Maxwell equations with only two differential forms

We define now a three- form, we obtain the exterior derivative

This corresponds to the conservation law for the total charge already mentioned.

Now, while the two homogeneous Maxwell equations ( Maxwell I and II) can be summarized by the statement that the electric and magnetic fields and are represented by a single differential equation of the second closed position (), applies to the remaining non-homogeneous Maxwell equations III and IV, the statement that the outer discharge of the dual form with the current waveform is the same. so

Thus, the totality of all four Maxwell equations is in mathematical shorthand by only two differential forms, and expressed. ( In particular, it follows from the last equation also immediately Kontuitätsgleichung because the two-time exterior derivative always results in zero. )

Again, the metric plays no direct role ( indirectly it is very important for example in the definition of duality, which is required in the calculation of charges and currents from the fields and the disclosure of the explicit form of the Lorentz invariance ). The manifold is optional, as long as it has dimension 4. Ultimately, however, even here the physical metric significantly, not only in the just-mentioned duality. But also here it comes not only to the four-dimensionality of the manifold, but also on the distinction between space and time coordinates (or between so-called space-like and time-like vectors, tensor and field components), which indeed expressed by means of the metric tensor. This is not so given by eg orally by Dh one does not have one but, as I said, to do with a manifold. The distinction of " spacelike " and " timelike " sizes in the metric is also related to the difference between electric and magnetic fields. Although the field components of these variables can be transformed into each other by the Lorentz relations ( six in total), is the characterization of a field as essentially " electric" and " magnetic" an invariant of the theory, because the Lagrangian, one from F *, F and J composite invariant function, from which the equations of motion (ie, the Maxwell equations ) can be calculated, is the cgs - system is substantially equal to B2- E2. ( Note: A Minkowski vector is space-like or time-like or light-like, depending on whether is positive or negative or zero analog is an electromagnetic field substantially magnetically or electrically or wave depending on whether the Lagrangian for. is positive or negative or zero. )

Abstract integral formulation and interpretation

This abstract differential formulation of the Maxwell equations using the theory of so-called alternating differential forms, in particular the so-called outer differential. The corresponding abstract integral formulation is obtained by application of the generalized Stoke 's theorem from this mathematical theory: People are focusing to within the specified three -manifold V with Minkowski metric ( eg, embedded in the space ) particularly on the edge of a closed two- manifold, and obtain:

For all V, and ( with ):

Here, the really interesting part is behind the clip and it is emphasized by the character in the sense of physics that the domain of integration is a closed manifold. The first of the two given equations containing the Faraday induction law and the law of the non-existence of magnetic charges. In the last equation, the Maxwell - Ampère law and the law of Gauss is included. Both laws of a pair So each belong together. Gauss's law e.g. states in the abstract formulation given here: The flow of electromagnetic form by the edge of the manifold V is equal to the total contained in V " charge ", as it results from the current form.

The specified gauge freedom arises geometrically from the fact that you can find a pre-specified margin many different manifolds V, the " fit " in it.

Special formulations and special cases

Maxwell's equations for a constant frequency in complex notation

The field vectors occurring in the Maxwell equations are not only functions of location, but also the time, for example in general. In the partial differential equations occurs in addition to the local variables on the time variable. To simplify the solution of the differential equations is limited in practice often be harmonic ( sinusoidal ) operations. This representation is for practical field calculation, for example in the calculation of electromagnetic shields or the antenna technique is essential.

Using the complex notation can be avoided when the time dependence of harmonic processes, as lifts out of the complex time factor here. The field quantities appearing in the Maxwell equations are then complex amplitudes and only features of the site. Instead of the partial differentiation with respect to time, the multiplication occurs with the imaginary factor. The factor is referred to as angular frequency.

As usual in electrical engineering, the imaginary unit is denoted with (it should not be confused with the variable commonly used for current density ) - in mathematics and theoretical physics it is usually written.

In complex form - complex quantities are underlined to distinguish - are the Maxwell equations in differential form:

Covariant formulation of the Maxwell equations

The electrodynamics, as it is described by the Maxwell equations, in contrast to Newtonian mechanics compatible with special relativity. This implies that the Maxwell equations are valid in all inertial systems, without changing their shape when changing the reference system. The historically played in the development of the theory of relativity by Albert Einstein an important role ..

Technical Maxwell's equations are formulated relativistically covariant or invariant, that is, that they do not change their shape under Lorentz transformations.

This property is not to be considered the Maxwell equations in the form described above without further notice. It may therefore be useful to work out through a reformulation of the theory of the invariance, in other words: " covariant manifest" the theory of writing.

For this purpose, it is expedient that occur above sizes, etc. expressed by variables which have a clearly defined, simple transformation properties under Lorentz transformations, ie by Lorentz scalars, four-vectors and four - tensors of higher levels.

Starting point for the reformulation form the electromagnetic potentials ( scalar potential ) and ( vector potential ), from which one the electric and magnetic fields by

Is obtained ( see also electrodynamics). These variables can be combined into a four-vector, the four-potential

Summarized. Similarly, one can consist of charge density and current density, the four-current density, with

The electrodynamic field tensor is derived from the four-potential, just the components that are up to sign and constant prefactors that depend on the unit system, the electric and magnetic fields. It has the shape

It now defines the Vierergradienten, the relativistic form of the derivation, as

With these variables, you can see the two inhomogeneous Maxwell equations in vacuum by the covariant equation

Replace. The Einstein summation convention is, as usual, used, that is, over repeated indices in products (here) is summed. Furthermore, as usual, carried the upward and downward pulling of indices with the metric tensor

Note that due to the anti- symmetry of the Feldstärketensors the continuity equation (disappearance of the four - divergence) follows

The two homogeneous Maxwell equations obtained in vacuum, the manifest covariant form

This is also often written with the Levi- Civita symbol compact than

With the dual field strength tensor

The components of which can also make contact by the vectors is replaced by and by. so

Differential forms provide a particularly clear presentation of the Maxwell equations, which is also automatically covariant. This four-potential and four-current density can be represented by the 1-forms and the field strength tensor by the 2 - form and its dual by the 2 - form (the symbol stands for differential forms on the Cartan derivative). The Maxwell equations in vacuum can then be written

And

Maxwell equations taking into account hypothetical magnetic monopoles

Magnetic monopoles occur in some GUT theories as possible or necessary components. With them, the quantization of electric charge could explain how Paul Dirac in 1931 recognized. So far, magnetic monopoles were observed only as quasiparticles. Real particles as monopolies have not been found. Therefore, it is also assumed in Maxwell's equations mentioned above, that no magnetic monopoles ( magnetic charges ) exist.

If in the future still be found such magnetic charges, so these can be easily taken into account in the Maxwell equations.

If, for the monopole charge density on the current density and the velocity of the moving magnetic monopole charges so change only two of the four above equations in differential form to

Interpretation: The field lines of the magnetic flux density in the start and end of a magnetic charge.

Interpretation: Explore time-varying magnetic flux density or the presence of magnetic current densities lead to electrical eddy fields.

The two other equations remain unchanged while but of course for the two new differential ( i.e. local ) equations also new integral (ie global) representations arise, but which are readily calculated with the integral sets of Gauss and Stokes can.

The case of the vanishing monopolies leads back to the known equations given above.

Maxwell equations and photon mass

The photon mass vanishes according to the Maxwell equations. These equations are the limiting case of the more general Maxwell - Proca equations with a non-negative photon mass. Instead of the Coulomb potential causes in the Maxwell - Proca theory an electric point charge the Yukawa potential and only has a range of some of the Compton wavelength belongs to.

Historical Remarks

Maxwell published his equations 1865. In 1873, Maxwell brought his equations in a quaternionic representation. In the course of Maxwell and the magnetic field potential and the magnetic composition into its inserted equations and inserted these field variables in the equation of the electromagnetic force. However, Maxwell did not count right in this notation, but treated the scalar part and the vector part separately.

The now common notation were formulated later by Oliver Heaviside and independently Josiah Willard Gibbs and Heinrich Hertz on the basis of the original Maxwell equations of 1865. These are easier to read and easier to apply in most cases, which is why they are common even today.

Maxwell equations in cgs systems

In a distributed version of Gaussian cgs system are the Maxwell equations:

Thus, the Maxwell equations are ( besides the one nor the SI system used), for example, in the famous textbook by Jackson wrote. There are also versions of the Gaussian Cgs system, which use a different definition of the current, and in which the Ampere's law is ( eg in the popular textbook by Panofsky and Phillips):

For the potential in the Cgs system is set:

Furthermore, applies

In the formulation of the Maxwell equations in the Cgs system of Heaviside - Lorentz factors omitted in the above equations prior to Jackson and the pre-factor is omitted. In the formulation of the so-called " natural units " must be set to 1 and the factors that are omitted in the above equations c.

Systematic transformation behavior (SI ↔ cgs )

One can in a few lines describe the transformation behavior between SI and Cgs systems systematically, although the transformations already are therefore not entirely trivial because the latter system three basic sizes ( " length", "Earth", "time" ), the former system but four of them has ( in addition, the " electric current "). In fact, in the cgs system practice two like-charged point masses, the distance r is successive from the Coulomb force, while in the SI system is the same force.

It is therefore necessary, first: After a quite analogous law itself also transforms the electric moment p and the electric polarization ( electric moment per unit volume ) P, and the electric current density j = The electric field strength, however, transforms complementary to q, because the product " charge times field strength " must be invariant. So:
Second, the following applies:
Thirdly ( because vacuum but is ).

For the corresponding magnetic quantities (: first, the magnetic moment m and the magnetic polarization J = μ0 M ( connection: m = JΔV = μ0MΔV ), secondly, the magnetic field strength H, thirdly, the magnetic induction B) are similar laws in which takes the place of.

However, both the Ampere's law and Faraday's law of induction coupled electric and magnetic quantities. At this point, the speed of light c comes into play, by the fundamental relationship

If, for example, the Ampere's law considered, which reads in the SI system as follows: we obtain the first of the specified precisely in the equations in Table Cgs system.

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