Maxwell's equations

The Maxwell equations by James Clerk Maxwell (1831–1879) describe the phenomena of electromagnetism . They are therefore an important part of the modern physical worldview.

The equations describe how electric and magnetic fields are related to each other and to electric charges and electric current under given boundary conditions . Together with the Lorentz force , they explain all the phenomena of classical electrodynamics . They therefore also form the theoretical basis of optics and electrical engineering . The equations are named after the Scottish physicist James Clerk Maxwell, who worked them out from 1861 to 1864. He combined the law of flux and Gauss's law with the law of induction and, in order not to violate the continuity equation , also introduced the displacement current, which is also named after him .

James Clerk Maxwell

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

Maxwell equations in the field line image

The field lines of the electric field run between the positive and negative charges.

The field lines of the magnetic flux density form closed paths or are infinitely long. ${\displaystyle {\vec {B}}}$

The electric and magnetic fields 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 . ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {D}}}$ ${\displaystyle {\vec {H}}}$ ${\displaystyle {\vec {B}}}$

The electric field strength and the magnetic flux density can in principle be illustrated by the exertion of force on charges. The relationships are described in more detail in the article on the Lorentz force . In the case of the electric field, the course of the electric field strength shows the direction of the force exerted by the field (the force acts in the direction of the tangent to the field line at the respective location), the field line density (the proximity of the field lines to each other) represents the field strength in this area In the case of the magnetic field, the force acts normal to the direction of the magnetic flux density and normal to the direction of movement of the charge. ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {B}}}$

In the following figure, the field lines are illustrated using a positive and a negative charge. The electric field is strongest at the charge carriers and decreases with greater distance:

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

• Gauss's law for electric fields states that electric charges are sources and sinks of the field of electric flux density , i.e. they represent the beginning and end of the associated field lines. Electric fields without sources and sinks, so-called eddy fields, on the other hand, occur during induction processes. ${\displaystyle {\vec {D}}}$
• Gauss's law for magnetism states that the field of magnetic flux density has no sources. The magnetic flux density therefore only has field lines that have no end. A magnetic field line is therefore either infinitely long or leads back to itself in a closed path. ${\displaystyle {\vec {B}}}$
• Faraday's law of induction : Changes in the magnetic flux over time lead to an electrical vortex field.
• Extended Ampère's law , also called flow-through or Maxwell-Ampère's law: Electric currents - including a change in the electrical flux density over time - 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 with four coupled differential equations, but there are also other equivalent formulations.

notation

The methods of vector analysis (and the associated surface integral , curve integral ) are used. denotes the Nabla operator . The differential operators mean: ${\displaystyle {\vec {\nabla }}}$

• ${\displaystyle {\vec {\nabla }}\cdot {\vec {E}}\equiv \operatorname {div} {\vec {E}}}$denotes the divergence of ${\displaystyle {\vec {E}}}$
• ${\displaystyle {\vec {\nabla }}\times {\vec {E}}\equiv \operatorname {rot} {\vec {E}}}$denotes the rotation of ${\displaystyle {\vec {E}}}$

Microscopic Maxwell equations

The microscopic Maxwell equations link the electric field strength and the magnetic flux density with the charge density (charge per volume) and the electric current density (current per flow area). ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle \,\rho }$ ${\displaystyle {\vec {j}}}$

Surname	SI	Physical content
Gaussian law	${\displaystyle {\vec {\nabla }}\cdot {\vec {E}}={\frac {\rho }{\varepsilon _{0}}}}$	Electric field lines diverge from one another in the presence of electric charge; the charge is the source of the electric field.
Gaussian law for magnetic fields	${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$	Magnetic field lines do not diverge , the field of the magnetic flux density is source-free; there are no magnetic monopoles .
Induction law	${\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}}$	Changes in the magnetic flux density lead to an electrical vortex field . The minus sign is reflected in Lenz's rule .
Extended law of flooding	${\displaystyle {\vec {\nabla }}\times {\vec {B}}=\mu _{0}{\vec {j}}+\mu _{0}\varepsilon _{0}{\frac {\partial {\vec {E}}}{\partial t}}}$	Electric currents - including the displacement current - lead to a magnetic vortex field.

This can also be used. ${\displaystyle \mu _{0}\varepsilon _{0}={\frac {1}{c^{2}}}}$

Macroscopic Maxwell's equations

Microscopic electrical dipoles and circular currents as well as the spin dipoles (not shown) (relativistic quantum theory) lead to macroscopic polarizations or . ${\displaystyle {\vec {P}}}$ ${\displaystyle {\vec {M}}}$

In the presence of matter, the microscopic Maxwell equations are, on the one hand, unwieldy because, after all, every charge carrier in every atom of the medium must be taken into account. On the other hand, the magnetic properties (for example of a permanent magnet) cannot in principle be derived from the microscopic Maxwell equations without additional physical knowledge of quantum mechanics.

The macroscopic Maxwell equations take into account the properties of matter in the form of material parameters, whereby the parameters permittivity and permeability are assigned to the empty space . Maxwell himself did not start from an empty space, but - as was customary in his time - from the space filled with a so-called “ether” . The term "macroscopic" comes about because the properties of the matter ultimately characterize locally averaged properties of the matter. With regard to the charges, a distinction is made between free charge carriers (e.g. conduction electrons in the electrical conductor) and bound charge carriers (e.g. shell electrons), and it is assumed that the bound charge carriers lead to macroscopic polarization or magnetization through microscopic processes . ${\displaystyle \varepsilon _{0}}$ ${\displaystyle \mu _{0}}$ ${\displaystyle {\vec {P}}}$ ${\displaystyle {\vec {M}}}$

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  . ${\displaystyle {\vec {D}}:={\varepsilon _{0}}{\vec {E}}+{\vec {P}}}$ ${\displaystyle {\vec {H}}:={\frac {1}{\mu _{0}}}{\vec {B}}-{\vec {M}}}$

Surname	SI	Physical content
Gaussian law	${\displaystyle {\vec {\nabla }}\cdot {\vec {D}}=\rho _{\text{frei}}}$	The charge is the source of the electric field.
Gaussian law for magnetic fields	${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$	The field of the magnetic flux density is source-free; there are no magnetic monopoles.
Induction law	${\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}}$	Changes in the magnetic field lead to an electrical vortex field.
Extended law of flooding	${\displaystyle {\vec {\nabla }}\times {\vec {H}}={\vec {j}}_{\text{frei}}+{\frac {\partial {\vec {D}}}{\partial t}}}$	Electric currents - including the displacement current - lead to a magnetic vortex field.

and it is z. B. ${\displaystyle \rho _{\mathrm {frei} }=\rho -\rho _{\mathrm {pol} }=\rho +\operatorname {div} {\vec {P}}\,.}$

Differential and integral formulation

In the following sections and tables, a semantically equivalent convention is used with regard to the indexing of charge and current: namely, written or without an index and referred to as "true charges" or "true currents", while conversely those in the 'microscopic Equations' occurring non-indexed quantities are written as effective quantities or respectively . In a vacuum, the “microscopic equations” apply and indexing can be dispensed with. The following equations, on the other hand, apply in matter, and one has to rely on a uniform notation, mostly the one used below, although here too different conventions are not excluded. ${\displaystyle \rho _{\mathrm {frei} }}$ ${\displaystyle {\vec {j}}_{\mathrm {frei} }}$ ${\displaystyle \rho _{E}}$ ${\displaystyle {\vec {j}}_{B}}$

Overview

Here u. a. the Maxwell equations are given in SI units. Formulations in other systems of units are listed at the end or are explained by comments in the text.

The symbols given in the following in the right column in the center of single or double integrals emphasize that one is dealing with closed   curves or surfaces.

Maxwell's equations in SI units

differential shape	linking integral theorem			Integral form
Physical Gaussian law : The field is a source field. The charge (charge density ) is the source of the electric field. ${\displaystyle {\vec {D}}}$ ${\displaystyle \rho }$		Gauss	The (electrical) flow through the closed surface of a volume is directly proportional to the electrical charge inside. ${\displaystyle \partial V}$ ${\displaystyle V}$
${\displaystyle \operatorname {div} {\vec {D}}={\vec {\nabla }}\cdot {\vec {D}}=\rho }$		${\displaystyle \Longleftrightarrow }$	${\displaystyle \iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset {\vec {D}}\cdot \mathrm {d} {\vec {A}}=\iiint _{V}\rho \ \mathrm {d} V=Q(V)}$
No source of the B-field : The -field is source-free. There are no magnetic monopoles. ${\displaystyle {\vec {B}}}$		Gauss	The magnetic flux through the closed surface of a volume is equal to the magnetic charge in its interior, namely zero, since there are no magnetic monopoles.
${\displaystyle \operatorname {div} {\vec {B}}={\vec {\nabla }}\cdot {\vec {B}}=0}$		${\displaystyle \Longleftrightarrow }$	${\displaystyle \iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\subset \!\supset {\vec {B}}\cdot \mathrm {d} {\vec {A}}=0}$
Induction law : Every change in thefield leads to an opposing electric field. The eddies of the electric field depend on the change in the magnetic flux density over time. ${\displaystyle {\vec {B}}}$		Stokes	The (electrical) circulation over the edge curve of a surface is equal to the negative temporal change of the magnetic flux through the surface. ${\displaystyle \partial A}$ ${\displaystyle A}$
${\displaystyle \operatorname {rot} {\vec {E}}={\vec {\nabla }}\times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}}$		${\displaystyle \Longleftrightarrow }$	${\displaystyle \oint _{\!\!\!\partial A}\!\!\!{\vec {E}}\cdot \mathrm {d} {\vec {s}}=-\!\!\iint _{A}\!{\frac {\partial {\vec {B}}}{\partial t}}\cdot \mathrm {d} {\vec {A}}}$
Flooding Act : The vortices of the magnetic field depend on the conduction current density and the electrical flux density . The time change in is also referred to as the displacement current density and, as the sum with the conduction current density, gives the total current density ${\displaystyle \textstyle {\vec {j}}_{\mathrm {l} }}$ ${\displaystyle \textstyle {\vec {D}}}$ ${\displaystyle {\vec {D}}}$ ${\displaystyle {\vec {j}}_{\mathrm {v} }}$ ${\displaystyle {\vec {j}}_{\,{\text{total}}}={\vec {j}}_{\mathrm {l} }+{\vec {j}}_{\mathrm {v} }}$		Stokes	The magnetic circulation over the edge curve of a surface is equal to the sum of the conduction current and the change in the electrical flow through the surface over time. ${\displaystyle \partial A}$ ${\displaystyle A}$
${\displaystyle \operatorname {rot} {\vec {H}}={\vec {\nabla }}\times {\vec {H}}={\vec {j}}_{\mathrm {l} }\;+\;{\frac {\partial {\vec {D}}}{\partial t}}}$		${\displaystyle \Longleftrightarrow }$	${\displaystyle \oint _{\!\!\!\partial A}\!\!\!{\vec {H}}\cdot \mathrm {d} {\vec {s}}=\iint _{A}\!{\vec {j}}_{\mathrm {l} }\cdot \mathrm {d} {\vec {A}}\;\;+\;\;\iint _{A}\!{\frac {\partial {\vec {D}}}{\partial t}}\cdot \mathrm {d} {\vec {A}}}$

Explanations

It should be noted that all quantities must be measured from any inertial system that is the same for all quantities . Should using the above Equations, for example, the induced voltage in a moving conductor loop are considered, it is beneficial to convert the variables in the moving parts of the system into the rest system using the Lorentz transformation .

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

${\displaystyle \oint _{\partial A}{\vec {E}}'\cdot {\text{d}}{\vec {s}}=-\iint _{A}{\frac {\partial {\vec {B}}}{\partial t}}\cdot {\text{d}}{\vec {A}}+\oint _{\partial A}{\vec {v}}\times {\vec {B}}\cdot {\text{d}}{\vec {s}}\,\,\left(=-{\frac {\text{d}}{{\text{d}}t}}\iint _{A}{\vec {B}}\cdot {\text{d}}{\vec {A}}\,\right),}$

The field strength is measured in a reference system in which the line element is at rest. This equation only applies to speeds that are small compared to the speed of light, ${\displaystyle {\vec {E}}'}$ ${\displaystyle \mathrm {d} {\vec {s}}}$ ${\displaystyle |{\vec {v}}|\ll c.}$

Electrical current

In purely formal terms, both the usual conduction current density corresponding to the flow of electrical charge carriers and the displacement current density (the change in electrical flux density over time) can be summarized in the electrical current density , which played an important role in Maxwell's discovery of the Maxwell equations. Usually, however, the displacement current is listed separately. The electrical current density is linked to the electrical field strength via the material equations of electrodynamics and the electrical conductivity that occurs. ${\displaystyle {\vec {j}}}$ ${\displaystyle \sigma }$ ${\displaystyle {\vec {E}}}$

Electric field

${\displaystyle {\vec {D}}}$is the electrical flux density, historically and somewhat confusingly also known as electrical displacement density or electrical excitation . It is the density of the electrical flow that emanates from electrical charges. The electrical flux density is linked to the electrical field strength via the material equations of electrodynamics and the resulting dielectric conductivity . With the electrical polarization , the electrical dipole moment per volume is even more general . ${\displaystyle \varepsilon }$ ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {D}}=\varepsilon _{0}\,{\vec {E}}+{\vec {P}}}$ ${\displaystyle {\vec {P}}}$

Magnetic field

${\displaystyle {\vec {B}}}$is the magnetic flux density, also known historically as induction . This is the density of the magnetic flux caused by moving electrical charges or permanent magnets. The magnetic flux density is linked to the magnetic field strength via the material equations of electrodynamics and the resulting magnetic conductivity . With the magnetic polarization , the magnetic dipole moment per volume applies even more generally (the quantity that is to be equivalent in the following is referred to as magnetization ). ${\displaystyle \mu }$ ${\displaystyle {\vec {H}}}$ ${\displaystyle {\vec {B}}=\mu _{0}{\vec {H}}+{\vec {J}}}$ ${\displaystyle {\vec {J}}}$ ${\displaystyle {\vec {J}}}$ ${\displaystyle {\vec {M}}={\frac {\vec {J}}{\mu _{0}}}}$

The magnetic polarization should not be confused with the current density (more precisely: with the conduction current density ). Rather, the following applies: ${\displaystyle {\vec {J}}}$ ${\displaystyle {\vec {j}}}$

${\displaystyle \operatorname {rot} {\frac {{\vec {B}}-{\vec {J}}}{\mu _{0}}}={\vec {j}}+{\frac {\partial {\vec {D}}}{\partial t}}}$

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's equations
• Material equations of electrodynamics
• Continuity equations in electrodynamics

represent the foundation of the electrodynamic field theory together and with mutual supplementation. The material equations apply in their general form to empty space as well as to areas of space filled with matter.

For historical reasons, and sometimes specific to represent specific calculation processes, the material and the equations developed therein three conductivities are in each case the proportion of the empty space or the fraction of the conductivity, which is caused by the matter, and split. ${\displaystyle \varepsilon _{0}}$ ${\displaystyle \mu _{0}}$ ${\displaystyle \varepsilon _{\mathrm {r} }}$ ${\displaystyle \mu _{\mathrm {r} }}$

For the electric field, the splitting of the dielectric conductivity results in the possibility of introducing a further vector field, the electric polarization (actually dielectric polarization, which is also referred to as electric polarization because it is assigned to the electric field). ${\displaystyle {\vec {P}}}$

Analogous to this, the magnetic polarization describes the relationships in matter for the magnetic field, which are detached from the properties of empty space. The magnetization results from the magnetic polarization . (In the cgs system, the relationships are more confusing: and are referred to in the same way as cgs magnetization , and only differ by one factor , depending on whether or is meant.) ${\displaystyle {\vec {J}}}$ ${\displaystyle {\vec {M}}={\tfrac {\vec {J}}{\mu _{0}}}}$ ${\displaystyle {\vec {J}}}$ ${\displaystyle {\vec {M}}}$ ${\displaystyle 4\pi }$ ${\displaystyle {\vec {B}}}$ ${\displaystyle {\vec {H}}}$

In principle, the introduction of the vector fields of the electrical polarization and the magnetic polarization (or the magnetization equivalent thereto ) can be dispensed with without loss . Instead, the dependencies in the material equations and the corresponding general conductivities are taken into account in the form of higher-order tensors. Furthermore, the conductivities can also represent functions in order to be able to record non-linear properties of the matter. These can even depend on the pre-treatment, i.e. they can be explicitly time-dependent. This procedure is also recommended for systematic access if this is done using the SI system of units . For historical reasons, but also in certain sub-areas of physics, the - and - (or -) vector fields are sometimes used very intensively , which is why this fact is presented in more detail below. ${\displaystyle {\vec {P}}}$ ${\displaystyle {\vec {J}}}$ ${\displaystyle {\vec {M}}}$ ${\displaystyle {\vec {P}}}$ ${\displaystyle {\vec {J}}}$ ${\displaystyle {\vec {M}}}$

In matter it applies generally

${\displaystyle {\vec {D}}=\varepsilon _{0}{\vec {E}}+{\vec {P}}}$

such as

${\displaystyle {\vec {H}}={\frac {1}{\mu _{0}}}{\vec {B}}-{\vec {M}}}$

or.

${\displaystyle {\vec {B}}=\mu _{0}({\vec {H}}+{\vec {M}})=\mu _{0}{\vec {H}}+{\vec {J}}\,,}$with the "magnetic polarization" introduced above , ${\displaystyle {\vec {J}}=\mu _{0}{\vec {M}}}$

where in the special case of linearity with isotropy or with cubic systems the following simplification results:

${\displaystyle {\vec {D}}=\varepsilon _{0}\varepsilon _{\mathrm {r} }\,{\vec {E}}}$

and

${\displaystyle {\vec {B}}=\mu _{0}\mu _{\mathrm {r} }\,{\vec {H}}}$.

In homogeneous isotropic materials (i.e. the quantities and are scalar and constant) one obtains for the Maxwell equations ${\displaystyle \varepsilon \,\,(=\varepsilon _{0}\varepsilon _{\mathrm {r} })\,}$ ${\displaystyle \mu \,\,(=\mu _{0}\mu _{\mathrm {r} })}$

${\displaystyle {\vec {\nabla }}\times {\vec {H}}=\varepsilon \,{\frac {\partial {\vec {E}}}{\partial t}}+\sigma \,{\vec {E}}}$

${\displaystyle -{\vec {\nabla }}\times {\vec {E}}=\mu \,{\frac {\partial {\vec {H}}}{\partial t}}}$

${\displaystyle \varepsilon \,{\vec {\nabla }}\cdot {\vec {E}}=\rho }$

${\displaystyle \mu \,{\vec {\nabla }}\cdot {\vec {H}}=0}$.

In anisotropic non-cubic linear matter, the scalars and become tensors of the 2nd order, whereby the relationships remain valid. In non-linear materials, the conductivities depend on the respective instantaneous values ​​of the field strengths or, in the most general case, on their entire history (see hysteresis ). The - and - fields, called electrical or magnetic polarization, disappear outside the matter, which in the special cases mentioned is equivalent to the statement that becomes. ${\displaystyle \varepsilon _{\mathrm {r} }}$ ${\displaystyle \mu _{\mathrm {r} }}$ ${\displaystyle {\vec {P}}}$ ${\displaystyle {\vec {J}}}$ ${\displaystyle \varepsilon _{\mathrm {r} }=\mu _{\mathrm {r} }=1}$

The dielectric polarization is then linked to the electrical susceptibility or the relative permittivity and the vacuum permittivity ( dielectric constant ) as follows (in SI, i.e. in the unit ): ${\displaystyle \chi _{e}}$ ${\displaystyle \varepsilon _{r}}$ ${\displaystyle \varepsilon _{0}}$ ${\displaystyle \mathrm {\tfrac {A\,s}{V\,m}} }$

${\displaystyle {\vec {P}}=\varepsilon _{0}\chi _{e}{\vec {E}}=\varepsilon _{0}\cdot ({\varepsilon _{\mathrm {r} }}-1){\vec {E}}}$,

With

${\displaystyle \varepsilon _{\mathrm {r} }=1+\chi _{e}}$.

For the magnetic polarization or the magnetization , a corresponding equation applies with the magnetic susceptibility or the relative permeability and the vacuum permeability ( magnetic field constant ) with the unit : ${\displaystyle {\vec {J}}}$ ${\displaystyle {\vec {M}}={\tfrac {\vec {J}}{\mu _{0}}}}$ ${\displaystyle \chi _{m}}$ ${\displaystyle \mu _{\mathrm {r} }}$ ${\displaystyle \mu _{0}}$ ${\displaystyle \mathrm {\tfrac {V\,s}{A\,m}} }$

${\displaystyle {\vec {J}}=\mu _{0}\,\chi _{m}\,{\vec {H}}=\mu _{0}\cdot \left(\mu _{\mathrm {r} }-1\right)\,{\vec {H}}}$,

With

${\displaystyle \mu _{\mathrm {r} }=1+\chi _{m}}$.

(Caution: are cgs system and to multiply!) ${\displaystyle \chi _{e}}$ ${\displaystyle \chi _{m}}$ ${\displaystyle 4\pi }$

Furthermore, the definition of the resulting refractive index with

${\displaystyle n:={\sqrt {\varepsilon _{\mathrm {r} }\,\mu _{\mathrm {r} }}}}$

and the relationship between the speed of light and the electrical and magnetic field constant

${\displaystyle c={\frac {1}{\sqrt {\varepsilon _{0}\,\mu _{0}}}}}$.

This connects the propagation of light in matter with the constants of the medium. So is the phase velocity in the medium

${\displaystyle c_{p}={\frac {c}{n}}={\frac {1}{\sqrt {\varepsilon _{0}\,\mu _{0}\,\varepsilon _{\mathrm {r} }\,\mu _{\mathrm {r} }}}}}$,

which is equal to the group velocity without dispersion .

Summary

Maxwell equations in matter in SI units

Flooding Act	${\displaystyle {\vec {\nabla }}\times {\vec {H}}={\frac {\partial {\vec {D}}}{\partial t}}+{\vec {j}}}$
Induction law	${\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}}$
Gaussian law	${\displaystyle {\vec {\nabla }}\cdot {\vec {D}}=\rho }$
Gaussian law of magnetism	${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$
see explanation	${\displaystyle {\vec {D}}=\varepsilon _{0}\,{\vec {E}}+{\vec {P}}\quad }$(or also ) ${\displaystyle {\vec {D}}=\varepsilon \,{\vec {E}}}$
see explanation	${\displaystyle {\vec {B}}=\mu _{0}\,({\vec {H}}+{\vec {M}})\quad }$ (or also ) ${\displaystyle {\vec {B}}=\mu \,{\vec {H}}}$

Explanation:
The last specified, parenthesized relationships only apply to a linear relationship. The definitions of and given above are general. ${\displaystyle {\vec {P}}}$ ${\displaystyle {\vec {M}}}$

Traditionally, the last two so-called material laws and Ohm's law ( here is the specific electrical conductance ) are usually not included in the Maxwell equations. The continuity equation as a description of the conservation of charge follows from the Maxwell equations. ${\displaystyle {\vec {j}}=\sigma \,{\vec {E}}}$ ${\displaystyle \sigma }$ ${\displaystyle {\tfrac {\partial \rho }{\partial t}}+\operatorname {div} \,{\vec {j}}=0\,}$

The electric field strengths and the magnetic flux densities are interpreted as physically existing force fields. Maxwell already connected these force fields with the electric potential field and the vector potential : ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle \,\phi }$ ${\displaystyle {\vec {A}}}$

${\displaystyle {\vec {E}}=-{\vec {\nabla }}\,\phi -{\frac {\partial {\vec {A}}}{\partial t}}\,,}$

${\displaystyle {\vec {B}}={\vec {\nabla }}\times {\vec {A}}\,.}$

The relationship between field strengths and potentials is only defined with the exception of gauge transformations , but potentials are of fundamental importance in quantum theory.

Maxwell equations with differential forms (differential geometric formulation)

The description by vector analysis has the big disadvantage that it

• on the flat or is limited ${\displaystyle \mathbb {R} ^{3}}$ ${\displaystyle \mathbb {R} ^{4}}$
• is in principle “metrically contaminated”, since either the Euclidean or the Minkowski metric is built into the operators , although the Maxwell equations are defined without metrics
• the choice of a map of the underlying manifold is unphysical, since natural laws must be correct regardless of the coordinates chosen .

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

The three-dimensional approach

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

The inhomogeneous Maxwell equations

Let be a differential form on the arbitrary smooth manifold of dimension 3 and be Cartan's outer derivative . So then applies ${\displaystyle \rho \in \Lambda ^{3}({\mathcal {M}})}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \mathrm {d} }$

${\displaystyle \mathrm {d} \rho =0\quad \Leftrightarrow \quad \rho {\text{ ist eine geschlossene Differentialform}}\,,}$

because there cannot be a differential form of degree 4 other than 0 on a three-dimensional manifold. On a star-shaped area , Poincaré's lemma ensures that a potential form exists such that ${\displaystyle D\in \Lambda ^{2}({\mathcal {M}})}$

${\displaystyle \mathrm {d} D=\rho \quad \Leftrightarrow \quad \int _{\mathcal {M}}\mathrm {d} D=\int _{\mathcal {M}}\rho \quad {\stackrel {\text{Stokes}}{\Longleftrightarrow }}\quad \oint _{\!\!\!\partial {\mathcal {M}}}D=\int _{\mathcal {M}}\,\,\rho \quad }$ (Gauss law).

Furthermore, it is postulated that the time derivative of the charge from a manifold is opposite to a current through the boundary (that is, everything that wants to get out of the “volume” must flow through the boundary surface ). ${\displaystyle \partial _{t}Q}$ ${\displaystyle {\mathcal {M}}}$ ${\displaystyle \partial {\mathcal {M}}}$

${\displaystyle \partial _{t}Q=-I\quad \Leftrightarrow \quad \partial _{t}\int _{\mathcal {M}}\rho +\oint _{\!\!\!\partial {\mathcal {M}}}j=0\quad {\stackrel {\text{Stokes}}{\Longleftrightarrow }}\quad \int _{\mathcal {M}}\underbrace {\left(\partial _{t}\rho +\mathrm {d} j\right)} _{\text{Kontinuitätsgleichung}}=0\,.}$

This statement corresponds to the conservation law for the total charge belonging to the continuity equation (the arbitrariness of the manifold ensures, analogous to Gauss's law, that this also applies without integrals). is called current density (two-form). So: ${\displaystyle {\mathcal {M}}}$ ${\displaystyle j\in \Lambda ^{2}({\mathcal {M}})}$

${\displaystyle \partial _{t}\rho +\mathrm {d} j=0\quad \Leftrightarrow \quad \mathrm {d} (\partial _{t}D+j)=0\quad \Leftrightarrow \quad (\partial _{t}D+j){\text{ ist eine geschlossene Differentialform}}\,.}$

However, according to Poincaré's lemma, this mathematical statement implies that a differential form of degree 1 exists in a star-shaped area , so that ${\displaystyle H\in \Lambda ^{1}({\mathcal {M}})}$

${\displaystyle \mathrm {d} H=\partial _{t}D+j\quad \Leftrightarrow \quad \int _{\mathcal {S}}\mathrm {d} H=\int _{\mathcal {S}}\left(\partial _{t}D+j\right)\quad {\stackrel {\text{Stokes}}{\Longleftrightarrow }}\quad \oint _{\partial {\mathcal {S}}}H=\int _{\mathcal {S}}\left(\partial _{t}D+j\right)\quad }$(Maxwell-Ampère law).

It should be noted that Gauss’s law follows purely from the geometry of the problem, i.e. ultimately has no physical meaning: The only physical input is the existence of electrical charges or the continuity equation, which leads to Maxwell-Ampère’s law. The inhomogeneous equations are therefore a consequence of the conservation of charge. Basically only the so-called spin magnetism is not affected, i.e. H. of those magnetic phenomena that do not  originate from the Ampère's circular currents (the eddies of j ) , which are exclusively dealt with here (see Mathematical Structure of Quantum Mechanics , especially the section on spin , as well as the article on the so-called gyromagnetic ratio ). This affects the dominant part of permanent magnetism . Basically, however, this only shows that classical electrodynamics is not self-contained, although mathematically and theoretically-physically it seems so.

The homogeneous Maxwell equations

Similar to the continuity equation, the law of induction is postulated. The change in the magnetic flux through a surface over time is accompanied by the induction of an opposite ring voltage on its edge . This is completely analogous to the continuity equation, only one dimension deeper. ${\displaystyle {\mathcal {S}}}$ ${\displaystyle \partial S}$

${\displaystyle U=-\partial _{t}\Phi _{\text{mag}}\quad \Rightarrow \quad \oint _{\!\!\!\partial {\mathcal {S}}}E=-\int _{\mathcal {S}}\partial _{t}B\quad {\stackrel {\text{Stokes}}{\Longleftrightarrow }}\quad \int _{\mathcal {S}}\left(\mathrm {d} E+\partial _{t}B\right)=0\quad {\text{(Induktionsgesetz)}}}$

Thereby the magnetic flux density (two-form) and the electric field. The arbitrariness of the surface ensures that the law of induction can also be written without an integral: ${\displaystyle B\in \Lambda ^{2}({\mathcal {M}})}$ ${\displaystyle E\in \Lambda ^{1}({\mathcal {M}})}$ ${\displaystyle {\mathcal {S}}}$

${\displaystyle \mathrm {d} E+\partial _{t}B=0\quad {\stackrel {\mathrm {d} }{\Rightarrow }}\quad \underbrace {d^{2}} _{=0}E+\partial _{t}\mathrm {d} B=0\quad \Rightarrow \mathrm {d} B=f(x,y,z)}$

So one recognizes that it can only depend on the (space) -components of the manifold , but not on the time. However, the expression to the left of the equal sign does not depend on the choice of coordinates. So f (x, y, z) must vanish. In addition, the equation can only then be Lorentz invariant. It follows that the magnetic flux density is swell free (two-form) (i.e. the non-existence of magnetic charges, see above): ${\displaystyle \mathrm {d} B}$ ${\displaystyle {\mathcal {M}}}$

${\displaystyle \mathrm {d} B=0\quad \Leftrightarrow \quad \int _{\mathcal {M}}\mathrm {d} B=0\quad {\stackrel {\text{Stokes}}{\Longleftrightarrow }}\quad \oint _{\!\!\!\partial {\mathcal {M}}}B=0\quad {\text{(Quellfreiheit)}}}$

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

The material equations

Because the one-forms and are not compatible with the two-forms and , one has to establish a relationship between them. This happens with the Hodge star operator , which is an isomorphism between one-form and two-form on a three-dimensional manifold. ${\displaystyle E}$ ${\displaystyle H}$ ${\displaystyle D}$ ${\displaystyle B}$ ${\displaystyle *}$

${\displaystyle D=\varepsilon _{0}*E\quad {\text{und}}\quad B=\mu _{0}*H\quad {\text{(Materialgleichungen)}}}$

Here it becomes obvious why and or and cannot simply be identified (apart from one factor) for mathematical reasons. is a one-form and is integrated via a curve, is a two-form and needs a (2-dimensional) surface for integration. (In addition, the associated vector fields are also physically significantly different in polarizable media.) So, mathematically speaking, there cannot be any proportionality between these quantities, as the description suggests through vector analysis. The same applies to and : The first quantity describes a differential form of degree 1, so it needs a curve for integration, like a force integral; the second quantity is a two-form, so it needs an area like a flux integral. This difference seems pedantic, but it is fundamental. ${\displaystyle H}$ ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle D}$ ${\displaystyle H}$ ${\displaystyle B}$ ${\displaystyle E}$ ${\displaystyle D}$

It should be noted that the metric only plays a role in the equations with the Hodge operator. The Maxwell equations without the material equations are independent of the choice of metric and even independent of the nature of the manifold as long as is three-dimensional. Only the effect of in the material equations would change. ${\displaystyle {\mathcal {M}}}$ ${\displaystyle *}$

The four-dimensional approach

${\displaystyle {\mathcal {N}}}$Let be a smooth manifold of dimension 4 and a smooth submanifold of dimension 3 (from the 3-dimensional approach) and the metric tensor with coefficient representation. ${\displaystyle {\mathcal {M}}\subset {\mathcal {N}}}$ ${\displaystyle g\in {\mathcal {T}}_{2}^{0}({\mathcal {N}})}$

${\displaystyle g_{\mu \nu }=\left(\operatorname {diag} (-1,1,1,1)\right)_{\mu \nu }\quad {\text{(Minkowski-Metrik)}}}$

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

The metric only needs to be set in order to the ensuing four-potential can write down explicitly (physics: "contravariant sizes"), without going through the coefficients of a vector field (physics: "covariant quantities") to go with ${\displaystyle A\in \Lambda ^{1}({\mathcal {N}})}$ ${\displaystyle {\mathcal {A}}\in {\mathcal {X}}({\mathcal {N}})}$

${\displaystyle A=g({\mathcal {A}},\cdot )\quad {\text{wobei}}\quad {\mathcal {A}}=a^{\mu }\partial _{\mu }}$.

The commitment to the Minkowski room, which u. a. required to differentiate between “space-like” and “time-like” vector or tensor components, or for the definition of the duality operation (see below), so it is not required here. One could also freely choose the metric, then see the components of the one-form

${\displaystyle A=a_{\mu }dx^{\mu }\quad {\text{(Viererpotential)}}}$

just different, because

${\displaystyle a_{\mu }=g_{\mu \nu }a^{\nu }\quad {\text{(Transformation zwischen Vektorfeld und Differentialform)}}}$.

So from here on let the manifold be the flat Minkowski space, that is, o. B. d. A. . Then the vector potential is given by ${\displaystyle {\mathcal {N}}=M^{4}=(\mathbb {R} ^{4},g)}$

${\displaystyle A=-{\frac {\phi }{c}}\ \mathrm {d} t+a_{1}\ \mathrm {d} x+a_{2}\ \mathrm {d} y+a_{3}\ \mathrm {d} z\quad }$for the vector field . ${\displaystyle \quad {\mathcal {A}}=+{\frac {\phi }{c}}\ \partial _{t}+a^{1}\ \partial _{x}+a^{2}\ \partial _{y}+a^{3}\ \partial _{z}}$

The homogeneous Maxwell equations

Let the external derivative of be given by , i.e. by the so-called field strength tensor (Faraday two-form): ${\displaystyle A}$ ${\displaystyle F\in \Lambda ^{2}({\mathcal {N}})}$

${\displaystyle F=\mathrm {d} A\quad {\stackrel {\mathrm {d} }{\Rightarrow }}\quad \mathrm {d} F=\underbrace {\mathrm {d} ^{2}} _{=0}A=0\quad {\text{(homogene Maxwell-Gleichungen)}}}$.

What is impressive is the fact that the outer derivative of always disappears, regardless of what it looks like. This results in the so-called freedom from calibration and also explains why the restriction to the Minkowski area does not harm the general public. However, since the equations manage without any physical input, it immediately follows that the homogeneous Maxwell equations are merely a consequence of the geometry of the space and the formalism used (the same applies to the relationship : a closed differential form is still largely free, namely up to on the outer differential of a shape lower by one degree.). ${\displaystyle F}$ ${\displaystyle A}$ ${\displaystyle \mathrm {d} ^{2}\equiv 0\,}$

The material equations

The Faraday two-form can also be written in the already known sizes:

${\displaystyle F=E\wedge \mathrm {d} t+B\quad {\stackrel {*}{\Rightarrow }}*F=-{\sqrt {\frac {\mu _{0}}{\varepsilon _{0}}}}G\quad {\text{(Materialgleichungen)}}}$.

The two-form G that is dual to F is called Maxwell's two-form and is given by quantities that are already known, namely:

${\displaystyle G=D-H\wedge \mathrm {d} t}$ .

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

The entire Maxwell equations with only two differential forms

If one now defines a three-form  , its external derivative results ${\displaystyle J=\rho -j\wedge \mathrm {d} t\in \Lambda ^{3}({\mathcal {N}})}$

${\displaystyle \mathrm {d} J=0\quad {\text{(Kontinuitätsgleichung)}}\,.}$

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

While the two homogeneous Maxwell equations (Maxwell I and II) can be summarized by the statement that the electric or magnetic fields or are represented by a single closed differential form of the second level ( ), the following applies to the remaining inhomogeneous Maxwell equations III and IV the statement that the outer derivative of the dual form is identical to the current form . So ${\displaystyle E}$ ${\displaystyle B}$ ${\displaystyle \,F}$ ${\displaystyle \mathrm {d} F=0}$ ${\displaystyle G\,:=\,*F}$ ${\displaystyle J}$

${\displaystyle \mathrm {d} (*F)=J\quad {\text{(inhomogene Maxwell-Gleichungen, III und IV)}}\,.}$.

Thus the totality of all four Maxwell equations is expressed in mathematical short form by only two differential forms, and ,. (In particular, the continuity equation immediately follows from the last equation, because the two-fold outer derivative always results in zero.) ${\displaystyle F}$ ${\displaystyle J}$

Again, the metric does not play a direct role (indirectly it is very important, e.g. when defining the duality, which is required when calculating the charges and currents from the fields and when specifying the explicit form of the Lorentz invariance). The manifold is also arbitrary as long as it has dimension 4. Ultimately, however, the metric is physically essential here too, not only with the duality just mentioned. Here, too, it is not just the four-dimensionality of the manifold that matters, but also the distinction between space and time coordinates (or between so-called space-like and time-like vectors, tensor and field components), which are expressed with the help of the metric tensor. This is not given by but z. B. by One is not dealing with a  - but, as already said, with a -  manifold. The distinction between “space-like” and “time-like” quantities in metrics is also related to the difference between electric and magnetic fields. Although the (a total of six) field components of these quantities can be transformed into one another by the Lorentz relations , the characterization of a field as essentially "electrical" or "magnetic" is an invariant of the theory because the Lagrange function , one of * F , F and J composite invariant function, from which the equations of motion (i.e. the Maxwell equations) can be calculated, is essentially equal to B ² -E ² in the cgs system . (Note: A Minkowski vector is space-like or time-like or light-like , depending on whether positive or negative, or is zero analog is an electromagnetic field substantially. Magnetically or electrically or wave depending on whether the Lagrangian for  , is positive or negative or zero.) ${\displaystyle {\mathcal {N}}}$ ${\displaystyle \mathrm {d} s^{2}=+c^{2}\mathrm {d} t^{2}+\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}\,,}$ ${\displaystyle \mathrm {d} s^{2}=-c^{2}\mathrm {d} t^{2}+\mathrm {d} x^{2}+\mathrm {d} y^{2}+\mathrm {d} z^{2}\,.}$ ${\displaystyle \mathbb {R} ^{4}}$ ${\displaystyle \mathbb {M} ^{4}}$ ${\displaystyle {\vec {v}}}$ ${\displaystyle (\mathrm {d} s)^{2}[{\vec {v}}]}$ ${\displaystyle {\vec {J}}=0}$

Abstract integral formulation and interpretation

This abstract differential formulation of Maxwell's equations uses the theory of the so-called alternating differential forms, in particular the so-called external differential. The corresponding abstract integral formulation results from the application of the generalized Stokes' theorem from this mathematical theory: For this purpose, one concentrates in the specified three-manifold with Minkowski metric (e.g. embedded in the space ), especially on the edge of a closed two-manifold , and receives: ${\displaystyle V}$ ${\displaystyle \mathbb {M} ^{4}\,}$ ${\displaystyle \partial V\,,}$

${\displaystyle \left(\iiint _{V}\mathrm {d} F\equiv \right)\,\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset F=0}$

for everyone , as well as (with ): ${\displaystyle V}$ ${\displaystyle G=*F\,}$

${\displaystyle \left(\iiint _{V}\mathrm {d} G\equiv \right)\,\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset G=\iiint _{V}J\,.}$

The part that is actually of interest is behind the brackets and the symbol in the sense of physics emphasizes that the area of ​​integration is a closed manifold. The first of the two equations given contains Faraday's law of induction and the law of the non-existence of magnetic charges. The last equation contains Maxwell-Ampère's law and Gauss's law. Both laws of a couple therefore belong together. The Gaussian law z. B. says in the abstract formulation given here: The flow of the electromagnetic form through the edge of the manifold V is equal to the total “charge” contained in V , as it results from the current form . ${\displaystyle \;\;\iint _{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset \!\supset }$ ${\displaystyle \partial V}$ ${\displaystyle cG}$ ${\displaystyle J}$

The specified freedom from calibration results geometrically from the fact that many different manifolds can be found at a given edge that “fit into” it. ${\displaystyle \Gamma =\partial V}$ ${\displaystyle V}$

Special formulations and special cases

Maxwell equations for constant frequencies ω in complex notation

The field vectors appearing in Maxwell's equations are generally not only functions of location but also of time, for example . In the partial differential equations, in addition to the position variables, the time variable also appears. To simplify the solution of these differential equations, one often limits oneself in practice to harmonic (sinusoidal) processes. This representation is of essential importance for the practical field calculation, for example when calculating electromagnetic screens or for antenna technology. ${\displaystyle {\vec {H}}(x,\,y,\,z,\,t)}$

With the help of the complex notation, the time dependency in harmonic processes can be avoided, since the complex time factor is emphasized and the Maxwell equations thus become a Helmholtz equation . The field sizes appearing in the Maxwell equations are then complex amplitudes and only functions of the location. Instead of the partial differentiation according to time, there is multiplication by the imaginary factor . The factor is also known as the angular frequency . ${\displaystyle e^{\mathrm {j} \,\omega \,t}}$ ${\displaystyle \mathrm {j} \,\omega }$ ${\displaystyle \omega }$

As is customary in electrical engineering, the imaginary unit is denoted by (it should not be confused with the variable often used for current density ) - it is mostly written in mathematics and theoretical physics . ${\displaystyle \mathrm {j} }$ ${\displaystyle {\vec {j}}}$ ${\displaystyle \mathrm {i} }$

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

${\displaystyle {\vec {\nabla }}\cdot {\underline {\vec {D}}}=\rho }$

${\displaystyle {\vec {\nabla }}\cdot {\underline {\vec {B}}}=0}$

${\displaystyle {\vec {\nabla }}\times {\underline {\vec {E}}}=-\mathrm {j} \,\omega \,{\underline {\vec {B}}}}$

${\displaystyle {\vec {\nabla }}\times {\underline {\vec {H}}}={\underline {\vec {j}}}+\mathrm {j} \,\omega {\underline {\vec {D}}}=\left(\sigma +\mathrm {j} \,\omega \,\varepsilon \right)\,{\underline {\vec {E}}}}$

Covariant formulation of Maxwell's equations

In this paragraph, as in the rest of the article, the SI system of units is used. Many theorists find this and the associated factors , etc., to be unnatural, especially with the covariant formulation of electrodynamics, and use other systems, such as Gauss units or Heaviside-Lorentz units , in which the basic quantities of electrodynamics are defined differently. In the literature, therefore, preliminary factors can be omitted, added or moved to other places compared to this representation. ${\displaystyle \,\mu _{0}}$ ${\displaystyle \varepsilon _{0}}$

In contrast to Newtonian mechanics, electrodynamics, as described by Maxwell's equations, is compatible with the special theory of relativity . This includes that the Maxwell equations are valid in every inertial system without their form changing when the reference system is changed. Historically, this played an important role in the development of the theory of relativity by Albert Einstein .

In more technical terms, the Maxwell equations are relativistically covariant or form invariant , which means that they do not change their shape under Lorentz transformations .

However, this property is not straightforward to consider in the Maxwell equations in the form described above. It can therefore be useful to work out the form invariance by reformulating the theory, in other words: to write the theory “manifestly covariant”.

For this purpose, it is useful to express the quantities appearing above , etc., in terms of quantities that have a clearly defined, simple transformation behavior under Lorentz transformations, i.e. by Lorentz scalars , four vectors and four tensors of higher levels. ${\displaystyle {\vec {E}}}$ ${\displaystyle {\vec {B}}}$

The starting point for this reformulation are the electromagnetic potentials (scalar potential ) and ( vector potential ), from which the electric and magnetic fields can be drawn through ${\displaystyle \,\phi }$ ${\displaystyle {\vec {A}}}$

• ${\displaystyle {\vec {E}}=-{\vec {\nabla }}\phi -\partial _{t}{\vec {A}}}$
• ${\displaystyle {\vec {B}}={\vec {\nabla }}\times {\vec {A}}}$

receives (see also electrodynamics ). These quantities can be converted into a four-vector, the four-potential

${\displaystyle A^{\mu }=\left({\frac {\phi }{c}},{\vec {A}}\right)}$

sum up. You can also compose the four-current density from the charge density and current density , with ${\displaystyle \,\rho }$ ${\displaystyle {\vec {j}}}$

${\displaystyle j^{\mu }=(c\rho ,{\vec {j}})}$.

The electrodynamic field strength tensor is derived from the four potential, the components of which are precisely those of the electric and magnetic fields, apart from the sign and constant prefactors that depend on the system of units. He has the shape

${\displaystyle F^{\alpha \beta }=\partial ^{\alpha }A^{\beta }-\partial ^{\beta }A^{\alpha }={\begin{pmatrix}0&-{\frac {E_{x}}{c}}&-{\frac {E_{y}}{c}}&-{\frac {E_{z}}{c}}\\{\frac {E_{x}}{c}}&0&-B_{z}&B_{y}\\{\frac {E_{y}}{c}}&B_{z}&0&-B_{x}\\{\frac {E_{z}}{c}}&-B_{y}&B_{x}&0\\\end{pmatrix}}}$.

The four-fold gradient , the relativistic form of the derivative, is now defined as

${\displaystyle \partial ^{\alpha }=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},-{\vec {\nabla }}\right)}$, that is   , as well as the differentials that are required when dealing with the Maxwell equations in the article Differential Forms, which is also recommended at this point. ${\displaystyle \partial _{\alpha }=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},+{\vec {\nabla }}\right)}$ ${\displaystyle \,\mathrm {d} x^{\alpha }=(c\mathrm {d} t,\mathrm {d} x,\mathrm {d} y,\mathrm {d} z)}$

With these quantities, the two inhomogeneous Maxwell equations can be found in a vacuum using the covariant equation

${\displaystyle \,\partial _{\alpha }F^{\alpha \beta }=\mu _{0}j^{\beta }}$

replace. As usual, Einstein's summation convention is used, which means that double indices in products (here ) are summed up. Furthermore, as usual, the indexes are pulled up and down with the metric tensor ${\displaystyle \alpha }$

${\displaystyle {\overset {\leftrightarrow }{\eta }}={\begin{pmatrix}+1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\\\end{pmatrix}}}$.

Note that because of the antisymmetry of the field strength tensor, the continuity equation (disappearance of the four-fold divergence ) also follows

${\displaystyle \mu _{0}(\partial _{t}\rho +\operatorname {div} {\vec {j}})=\mu _{0}\partial _{\beta }j^{\beta }=\partial _{\alpha }\partial _{\beta }F^{\alpha \beta }=-\partial _{\alpha }\partial _{\beta }F^{\beta \alpha }=-\partial _{\beta }\partial _{\alpha }F^{\beta \alpha }=0}$.

The two homogeneous Maxwell equations are given the manifestly covariant form in a vacuum

${\displaystyle \,\partial _{\alpha }F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }+\partial _{\gamma }F_{\alpha \beta }=0}$

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

${\displaystyle \varepsilon ^{\alpha \beta \gamma \delta }\partial _{\alpha }F_{\gamma \delta }=0}$

or

${\displaystyle \partial _{\alpha }{\tilde {F}}^{\alpha \beta }=0}$

with the dual field strength tensor

${\displaystyle {\tilde {F}}^{\alpha \beta }:={\frac {1}{2}}\varepsilon ^{\alpha \beta \gamma \delta }F_{\gamma \delta },}$

whose components can also be obtained from those of by replacing the vectors with and by . So ${\displaystyle \,F^{\alpha \beta }}$ ${\displaystyle {\vec {E}}/c}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle {\vec {B}}}$ ${\displaystyle -{\vec {E}}/c}$

${\displaystyle {\tilde {F}}^{\alpha \beta }={\begin{pmatrix}0&-B_{x}&-B_{y}&-B_{z}\\B_{x}&0&{\frac {E_{z}}{c}}&-{\frac {E_{y}}{c}}\\B_{y}&-{\frac {E_{z}}{c}}&0&{\frac {E_{x}}{c}}\\B_{z}&{\frac {E_{y}}{c}}&-{\frac {E_{x}}{c}}&0\\\end{pmatrix}}}$.

Differential forms enable a particularly clear representation of Maxwell's equations, which is also automatically covariant. The four-potential and four-current density are 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 the Cartan derivative in differential forms ). The Maxwell equations in vacuum then read ${\displaystyle {\vec {A}}}$ ${\displaystyle {\vec {j}}}$ ${\displaystyle {\vec {F}}=\mathrm {d} {\vec {A}}}$ ${\displaystyle *{\vec {F}}}$ ${\displaystyle \mathrm {d} }$

${\displaystyle *\mathrm {d} *{\vec {F}}=\mu _{0}{\vec {j}}}$

and

${\displaystyle \mathrm {d} {\vec {F}}=0}$.

Maxwell's equations taking hypothetical magnetic monopoles into account

Magnetic monopoles appear as possible or necessary components in some GUT theories. They could be used to explain the quantification of the electric charge, as Paul Dirac recognized as early as 1931. So far, magnetic monopoles have only been observed as quasiparticles . Real particles as monopoles have not yet been found. Therefore, in the Maxwell equations mentioned above, it is also assumed that no magnetic monopoles (magnetic charges) exist.

Should such magnetic charges be found in the future, they can easily be taken into account in the Maxwell equations.

If one sets for the monopole charge density , for the current density and for the speed of the moving magnetic monopole charges, only two of the four equations mentioned above change in differential form ${\displaystyle \rho _{m}}$ ${\displaystyle {\vec {j}}_{m}=\rho _{m}{\vec {v}}_{m}}$ ${\displaystyle {\vec {v}}_{m}}$

${\displaystyle \operatorname {div} {\vec {B}}=\rho _{m}\,\,.}$

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

${\displaystyle \operatorname {rot} {\vec {E}}=-\left({\frac {\partial }{\partial t}}{\vec {B}}+{\vec {j}}_{m}\right)\,.}$

Interpretation: Magnetic flux densities that change over time or the presence of magnetic current densities lead to electrical vortex fields.

The other two equations remain unchanged, while of course new integral (i.e. global) representations also result for the two new differential (i.e. local) equations, which can, however, easily be calculated with the integral theorems of Gauss and Stokes.

The case of vanishing monopoles leads back to the well-known equations given above. ${\displaystyle \rho _{m}=0}$

Maxwell's equations and photon mass

The photon mass disappears according to Maxwell's equations. These equations are the limiting case of the more general Maxwell-Proca equations with a non-negative mass of the exchange particles (in the electromagnetic case of the photons). Instead of the Coulomb potential effects in the Maxwell Proca theory a point charge the Yukawa potential , and has only a range of about the Compton wavelength . ${\displaystyle m=0}$ ${\displaystyle m}$ ${\displaystyle {\tfrac {q}{r}}}$ ${\displaystyle q}$ ${\displaystyle {\tfrac {q}{r}}\,\mathrm {e} ^{-m\,c\,r/\hbar }}$ ${\displaystyle \lambda _{m}=h/{m\,c}}$

Historical remarks

Maxwell published his equations in 1865 ( A dynamic theory of the electromagnetic field ). However, this system of originally twenty equations also contained those that contained definitions and equations that are no longer included in the actual Maxwell equations today (such as the continuity equation due to the conservation of charge and preforms of the Lorentz force). The twenty equations also included the three components that are now combined in a vector equation. In 1873, Maxwell's A Treatise on Electricity and Magnetism in Volume 2 (Part 4, Chapter 9) contains a somewhat modified list, which, however, largely corresponds to the list from 1865. In addition, Maxwell brought his equations into a quaternionic representation, an alternative to the vector calculus that was particularly popular in Great Britain at the time. In the course of this, Maxwell also introduced the magnetic potential field and the magnetic mass into his equations and inserted these field variables into the equation for the electromagnetic force . Maxwell did not calculate directly in this quaternionic notation, but treated the scalar part and the vector part separately. ${\displaystyle {\vec {\Omega }}}$ ${\displaystyle m}$ ${\displaystyle {\vec {F}}}$

The vector notations commonly used today were formulated later by Oliver Heaviside and independently Josiah Willard Gibbs and Heinrich Hertz on the basis of the original Maxwell equations from 1865. They also restricted the original system (in vector notation) to four equations. These are easier to read and in most cases easier to use, which is why they are still common today.

Maxwell's equations in natural unit systems

In natural systems of units , the natural constants do not apply.

(See also: Electromagnetic Units → Important Units with the formulation of Maxwell's equations in five different systems of units.)

Gaussian system of units

Since the Gaussian system of units is based on the CGS system , not all fundamental constants can be reduced.

In a common version of the Gaussian CGS system, the Maxwell equations are:

Maxwell's equations in the Gaussian cgs system

Flooding Act	${\displaystyle {\vec {\nabla }}\times {\vec {H}}=4\pi {\frac {\vec {j}}{c}}+{\frac {1}{c}}\,{\frac {\partial {\vec {D}}}{\partial t}}}$
Induction law	${\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\frac {1}{c}}{\frac {\partial {\vec {B}}}{\partial t}}}$
Gaussian law	${\displaystyle {\vec {\nabla }}\cdot {\vec {D}}=4\pi \rho }$
Gaussian law of magnetism	${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$

For example, the Maxwell equations are written in the famous Jackson textbook (which also uses the International System of Units (SI)). There are also versions of the Gaussian cgs system, which use a different definition of the current intensity and in which the law of flow reads (e.g. in the widely used textbook by Panofsky and Phillips :)

${\displaystyle {\vec {\nabla }}\times {\vec {H}}=4\pi {\vec {j}}+{\frac {1}{c}}\,{\frac {\partial {\vec {D}}}{\partial t}}}$

For the potentials, the following is set in the cgs system:

${\displaystyle {\vec {E}}=-{\vec {\nabla }}\,\phi -{\frac {1}{c}}\,{\frac {\partial {\vec {A}}}{\partial t}}}$ such as ${\displaystyle {\vec {B}}={\vec {\nabla }}\times {\vec {A}}\,.}$

Furthermore applies

${\displaystyle {\vec {D}}={\vec {E}}+4\pi {\vec {P}}}$ and ${\displaystyle {\vec {B}}={\vec {H}}+4\pi {\vec {M}}\,.}$

Systematic transformation behavior (SI ↔ cgs)

The transformation behavior between SI and cgs systems can be described systematically in a few lines, although the transformations are not entirely trivial because the latter system has three basic variables ("length", "mass", "time"), the former system but has four of them (plus the "electrical current"). In the cgs system, two equally charged point masses whose distance is equal to each other exert the Coulomb force , while in the SI the same force is. ${\displaystyle r}$ ${\displaystyle q_{\mathrm {cgs} }^{2}/r^{2}}$ ${\displaystyle q_{\mathrm {SI} }^{2}/(4\pi \varepsilon _{0}r^{2})}$

• First of all, according to a very analogous law, the electrical moment or the electrical polarization (electrical moment per volume) and the electrical current density are also transformed. The electrical field strength, on the other hand, transforms in a complementary way because the product “charge times field strength” is invariant got to. ${\displaystyle q_{\mathrm {cgs} }=q_{\mathrm {SI} }/{\sqrt {4\pi \varepsilon _{0}}}\,.}$ ${\displaystyle p}$ ${\displaystyle P}$ ${\displaystyle j=\rho v.}$ ${\displaystyle q}$
• Second:    ${\displaystyle \textstyle E_{\mathrm {cgs} }=E_{\mathrm {SI} }\cdot {\sqrt {4\pi \varepsilon _{0}}}\,.}$
• Thirdly:  (because in a vacuum but is.) ${\displaystyle \textstyle D_{\mathrm {cgs} }=D_{\mathrm {SI} }\cdot {\sqrt {\frac {4\pi }{\varepsilon _{0}}}}\,}$ ${\displaystyle D_{\mathrm {SI} }=\varepsilon _{0}\,E_{\mathrm {SI} }\,,}$ ${\displaystyle D_{\mathrm {cgs} }=E_{\mathrm {cgs} }}$

For the corresponding magnetic quantities (first: the magnetic moment or the magnetic polarization (connection:) , second: the magnetic field strength , third: the magnetic induction ), similar laws apply, in which takes the place of . ${\displaystyle m}$ ${\displaystyle J=\mu _{0}M}$ ${\displaystyle m=J\Delta V=\mu _{0}M\Delta V}$ ${\displaystyle H}$ ${\displaystyle B}$ ${\displaystyle \mu _{0}}$ ${\displaystyle \varepsilon _{0}}$

However, both the law of flux and Faraday's law of induction couple electrical and magnetic quantities. This is where the speed of light comes into play, through the fundamental relationship ${\displaystyle c}$ ${\displaystyle \varepsilon _{0}\mu _{0}\equiv 1/c^{2}\,.}$

If you z. For example, consider the law of flow, which in the SI reads as follows: in the cgs system, one obtains the first of the equations just given in the table. ${\displaystyle {\operatorname {rot} }{\vec {H}}={\vec {j}}+{\frac {\partial {\vec {D}}}{\partial t}}\,,}$

Heaviside-Lorentz unit system

Since the Heaviside-Lorentz system of units is rationalized, the factors are omitted. Combined with the Planck system of units , the Maxwell equations do not contain any constants: ${\displaystyle 4\pi }$

	HLE	combined with Planck units
Flooding Act	${\displaystyle {\vec {\nabla }}\times {\vec {H}}={\frac {\vec {j}}{c}}+{\frac {1}{c}}\,{\frac {\partial {\vec {D}}}{\partial t}}}$	${\displaystyle {\vec {\nabla }}\times {\vec {H}}={\vec {j}}+{\frac {\partial {\vec {D}}}{\partial t}}}$
Induction law	${\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\frac {1}{c}}{\frac {\partial {\vec {B}}}{\partial t}}}$	${\displaystyle {\vec {\nabla }}\times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}}$
Gaussian law	${\displaystyle {\vec {\nabla }}\cdot {\vec {D}}=\rho }$	${\displaystyle {\vec {\nabla }}\cdot {\vec {D}}=\rho }$
Gaussian law of magnetism	${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$	${\displaystyle {\vec {\nabla }}\cdot {\vec {B}}=0}$

literature

• Richard Becker , Fritz Sauter : Theory of Electricity. Volume 1 (introduction to Maxwell's theory, electron theory, relativity theory). Teubner, Stuttgart 1969.
• Richard P. Feynman , Robert B. Leighton , Matthew Sands : The Feynman Lectures on Physics. Volume 2. Definitive edition. Pearson Addison-Wesley, San Francisco CA u. a. 2006, ISBN 0-8053-9047-2 .
• John David Jackson : Classical Electrodynamics. John Wiley, New York NY 1962 (3rd edition. Ibid 1999, ISBN 0-471-30932-X ; German: 4th revised edition. De Gruyter, Berlin et al. 2006, ISBN 3-11-018970-4 ).
• Uwe Krey, Anthony Owen: Basic Theoretical Physics. A Concise Overview . Springer, Berlin a. a. 2007, ISBN 978-3-540-36804-5 .
• Lew Landau , Jewgeni Lifschitz : Theoretical Physics. Volume 2: Classical Field Theory. 12th revised edition. German, Thun u. a. 1997, ISBN 3-8171-1327-7 .
• Günther Lehner: Electromagnetic field theory for engineers and physicists . 5th edition. Springer-Verlag, Berlin a. a. 2006, ISBN 3-540-26550-3 .
• Wolfgang Panofsky , Melba Phillips : Classical Electricity and Magnetism. Addison-Wesley, Reading MA 1955 (2nd edition. Dover, Mineola NY 2005, ISBN 0-486-43924-0 ).
• Károly Simonyi : Theoretical electrical engineering. 10th edition. Barth, Leipzig a. a. 1993, ISBN 3-335-00375-6 .

Web links

Commons : Maxwell equations  - collection of images, videos, and audio files

Wikiversity: Maxwell Introduction  - Course Materials

• Literature on the Maxwell equations in the catalog of the German National Library
• Very extensive site about the Maxwell equations
• Compact, clear presentation of Maxwell's equations (PDF)
• Clear explanation of Maxwell's equations, almost free of formulas

References and comments

1. ↑ Steffen Paul, Reinhold Paul: Fundamentals of electrical engineering and electronics 2: Electromagnetic fields and their applications . Springer-Verlag, 2012, ISBN 3-642-24157-3 , p. 200 ( limited preview in Google Book Search).
2. ↑ Wolfgang Nolting: Basic Course Theoretical Physics 3, Chap. 4.1.3, books.google.de ; it speaks of four equations
3. ↑ These microscopic processes are generally described by quantum mechanics, whereby in the case of spin magnetism even the relativistic form of quantum mechanics, the so-called Dirac equation , has to be used.
4. ↑ ^{a b} The bracketed double integral is zero if the magnetic or electrical induction remains constant. In this case, too, there is an electromotive effect if there is a change in the integration area in the time under consideration , which leads to a Lorentz force . See the second of the equations given in the immediately following section. ${\displaystyle dt}$ ${\displaystyle {\vec {A}}}$
   
5. ↑ In the physics literature, and if clearly recognizable from the context, the conduction current density is usually referred to as . The term is common in electrical engineering . ${\displaystyle {\vec {j_{\mathrm {l} }}}}$ ${\displaystyle {\vec {j}}}$ ${\displaystyle {\vec {J}}}$
6. ↑ Klaus W. Kark: Antennas and radiation fields - electromagnetic waves on lines in free space and their radiation . 3. Edition. Vieweg + Teubner, Wiesbaden 2010, chap. 3.7.1, p. 46 f.
7. ^ ^{A b} James Clerk Maxwell: A Dynamical Theory of the Electromagnetic Field . (PDF) In: Royal Society Transactions , 155, 1865, pp. 459-512, submitted 1864 .
8. ^ Yakir Aharonov, David Bohm: Significance of Electromagnetic Potentials in the Quantum Theory . In: Physical Review , 115/3, 1959.
9. ↑ The representation of Maxwell's equations in the differential form calculus is presented (similar to this one), for example, in the lectures by Martin R. Zirnbauer (University of Cologne), which are to be published in book form by Springer Verlag, or, for example, in Friedrich W. Hehl , Yuri Oboukhov: Foundations of Classical Electrodynamics: Charge, Flux and Metric . Birkhäuser 2003. Hehl, Oboukhov, Rubilar: Classical Electrodynamics - a tutorial on its foundations . 1999, arxiv : physics / 9907046 . Hehl, Oboukhov: A gentle introduction to the foundations of classical electrodynamics . 2000, arxiv : physics / 0005084 .
10. ↑ The duality operation swaps u. a. covariant and contravariant vector components, so it depends on the metric tensor.
11. ↑ At this point it is accepted that "magnetic polarization" can be confused with the variable named in the same way (see above) ${\displaystyle J}$ ${\displaystyle ,\equiv \mu _{0}M\,,}$
12. ↑ Albert Einstein: On the electrodynamics of moving bodies . In: Annalen der Physik und Chemie , June 17, 30, 1905, pp. 891-921.
13. ↑ with the convention it becomes ${\displaystyle \hbar =c=1}$ ${\displaystyle {\tfrac {q}{r}}\,\mathrm {e} ^{-m\,r}}$
14. ↑ with the reduced Compton wavelength the Yukawa potential is simplified to ${\displaystyle {\lambda \!\!\!^{-}}=\hbar /{m\,c}}$ ${\displaystyle {\tfrac {q}{r}}\,\mathrm {e} ^{-r/{\lambda \!\!\!^{-}}}}$
15. ↑ James Clerk Maxwell: A Treatise on Electricity & Magnetism . Dover Publications, New York 1873, ISBN 0-486-60636-8 and ISBN 0-486-60637-6 .
16. ↑ Oliver Heaviside: On the Forces, Stresses and Fluxes of Energy in the Electromagnetic Field . In: Philosophical Transactions of the Royal Society , 183A, 1892, p. 423
17. ^ EB Wilson: Vector Analysis of Josiah Willard Gibbs - The History of a Great Mind . Charles Scribner's Sons, New York 1901.
18. ↑ On the development of the notation at Maxwell: Gerhard Bruhn: The Maxwell equations - from the original to the modern notation . TU Darmstadt.
19. ^ Jackson: Classical Electrodynamics
20. ↑ z. B. Panofsky, Phillips, 2nd edition 1978, p. 466. There are also explanations of the units of measurement in the appendix. For the ambiguities of the Gaussian cgs systems used, see also footnote in Jackson, p. 817.
21. ↑ The connection between SI and cgs systems is shown particularly clearly in a special chapter of the third and subsequent editions of "Jackson" (see above).
22. ↑ The given transformation equations are not only valid in a vacuum.