Are Maxwell's equations used in engineering?
The four Maxwell's equations describe the generation of electrical and magnetic fields by charges and currents, as well as the interaction between these two fields, which occurs in time-dependent fields. They are the basis of electrodynamics and theoretical electrical engineering and were developed by James Clerk Maxwell between 1861 and 1864. In essence, Maxwell summarized the laws discovered up to this point in time
in a unified theory and added to it
to get consistency with the continuity equation. Thus, Maxwell's equations are a standard example of a unified theory that can explain various phenomena, here magnetic and electrical, in a closed form. Maxwell's equations were developed from experience and experiments, therefore Maxwell's theory is also a macroscopic theory.
A basic understanding of vector analysis is required to understand the following equations. Maxwell's equations can be represented in differential and integral form. The equivalence of both formulations is based on the Stokes theorem and the Gauss theorem. There is also an elegant four-dimensional formulation, the so-called covariant form (see below). B. is used in relativity and quantum electrodynamics.
See also: Differential and integrated form
The symbol ρ stands for the space charge density and represents the electrical charge distributed over space. The distribution of the charge in space is a scalar field.
The vector fields introduced above can be divided into three areas, which are described by their respective set of equations. These three areas are the electric flow field, the electric field and the magnetic field.
Electric flow field
The electric current density As an essential component of the flow field, it indicates how much current flows per area in which direction. This includes both the conduction current density, which is usually caused by the flow of electrical charge carriers in electrical conductors, and the displacement current. The electrical current density is based on the material equations of electrodynamics and the electrical conductivity σ that occurs with the electrical field strength connected.
is the electrical flux density, also known as historical and somewhat confusing electrical displacement density or as electrical excitation designated. This is the density of the electrical flow that emanates from electrical charges. The electrical flux density is based on the material equations of electrodynamics and the resulting dielectric conductivity with the electric field strength connected. It is even more general , with the electrical polarization , the electric dipole moment per volume.
is the magnetic flux density, also historically called induction designated. This is the density of the magnetic flux which is caused by moving electrical charges or permanent magnets (e.g. "spin magnetism!"). The magnetic flux density is based on the material equations of electrodynamics and the resulting magnetic conductivityμ with the magnetic field strength connected. It is even more general , with the magnetic polarization, the magnetic dipole moment per volume (as magnetization will the i.W. to equivalent size designated).
The magnetic polarization should not match the current density be confused.
The material equations occurring in all three areas are not counted directly to Maxwell's equations, but rather the three sets of equations:
- Maxwell's equations
- Material equations of electrodynamics
- Continuity equations in electrodynamics
represent the foundation of electrodynamic field theory together and with mutual supplementation. The material equations apply in the general form both to empty space and to areas of space filled with matter.
For historical reasons and sometimes also to specifically represent certain calculation processes, the equations of matter and the three conductivities that occur in them are each in the proportion of empty space or μ0 and the part of the conductivity which is caused by the matter and μr split up. The electrical flow field does not exist in empty space and is always bound to matter, which means that with the electrical conductivity σ as a material constant, this splitting cannot take place.
For the electric field, the splitting of the dielectric conductivity results in the possibility of introducing a further vector field, the electric polarization. Note: In technical literature, dielectric polarization is sometimes referred to as electrical polarization because it is about the electric field. Since the flow field has no polarization in this context, this polarization is always clearly assigned to the electric field. describes the changed conditions in matter for the electric field separately from the properties of empty space.
Magnetic polarization describes this analogously the relationships in matter for the magnetic field, 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 there as cgs magnetization, and only differ by a factor of 4π, depending on whether or is meant.)
In principle, the vector fields of electrical polarization can be introduced without loss and magnetic polarization (or the equivalent magnetization ) can be waived. 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 pretreatment, 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 P and J (or M) vector fields are sometimes used very intensively, which is why this fact is presented in more detail below.
Maxwell's equations in matter
The following applies in general to matter:
where in the special case of linearity with isotropy or with cubic systems the following simplification results
- and .
In anisotropic non-cubic linear matter the scalars become and μr to tensors 2nd level where the relationships remain valid. In non-linear materials, the conductivities of which depend, for example, on the respective instantaneous values of the field strengths and μr to be understood as functions of frequency and / or wave number. 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.
The dielectric polarization is then with the electrical susceptibilityχe, or the relative permittivity and the vacuum permittivity (dielectric constant) linked as follows (in the SI system, i.e. in As / Vm):
- , With
For magnetic polarization or the magnetization applies accordingly, with the magnetic susceptibilityχm, or the relative permeability μr and the vacuum permeability (magnetic field constant) μ0 in Vs / Am:
- , with μr = 1 + χm
(Attention: in the cgs system there are χe and χm to be multiplied by 4π!)
The definition of the refractive index also results
- and the speed of light in a vacuum
which connects the speed of light in the material with the corresponding constants. So is the phase velocity in the medium
which, with a frequency-independent refractive index (without dispersion), is equal to the group velocity in the medium.
Maxwell's 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 essential for the practical field calculation, for example when calculating electromagnetic screens or for antenna technology.
With the help of the complex notation, the time dependency in harmonic processes can be avoided, since the complex time factor ejωt highlights. The field sizes appearing in Maxwell's equations are then complex amplitudes and only functions of the location. Instead of the partial differentiation according to time, there is multiplication with the imaginary factor jω. ω is also referred to as the angular frequency.
In complex form, complex quantities are underlined to distinguish, the Maxwell equations are in differential form:
Covariant formulation of Maxwell's equations
- In this paragraph, as in the rest of the article, the SI system of units is used. This and the associated factors μ0, etc., many theorists perceive the covariant formulation of electrodynamics as unnatural 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.
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 changing their form when the reference system is changed. Historically, this played an important role in the development of the theory of relativity by Albert Einstein.
The Maxwell equations are formulated more technically relativistic covariant or form invariant, that is, they do not change their shape under Lorentz transformations.
However, this property is not to be taken directly into 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 use the sizes above , etc. express by 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.
The starting point for this reformulation are the electromagnetic potentials φ (scalar potential) and (Vector potential), from which one can get the electric and magnetic fields through
receives (see also electrodynamics). These quantities can be converted into a four-vector, the Four potential
sum up. Likewise one can get from the charge density ρ and the current density the Four-current density put together, with
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
The four-fold gradient, the relativistic form of the derivative, is now defined as
- , so, as well as the differentials dxα = (cdt, dx, dy, dz), which are required when dealing with Maxwell's equations in the article Differential forms, which is also recommended at this point.
With these sizes you can do the two inhomogeneous Maxwell's equations in a vacuum by the covariant equation
replace. As usual, Einstein's summation convention is used, that is, summation is carried out using double indices in products (here α). Furthermore, as usual, the indexes are pulled up and down with the tensor = diag (+ 1, -1, -1, -1).
Note that because of the antisymmetry of the field strength tensor, the continuity equation (disappearance of the four-fold divergence) also follows
The two homogeneous Maxwell equations receive the manifest covariant form in a vacuum
This is also often spelled more compactly than with the Levi-Civita symbol
with the dual field strength tensor
whose components can also be derived from those of F.αβ can be obtained by looking at the vectors by and by replaced. So
Differential forms enable a particularly clear representation of Maxwell's equations, which is also automatically covariant if one does not work in Euclidean space from the beginning, but in Minkowski space. The four potential and four current density are given by the 1-forms and represented, the field strength tensor by the 2-form and its dual through the 2-form (In differential forms, the symbol d stands for a formal derivation and not for an infinitely small differential). The Maxwell equations in a vacuum are then (in Heaviside-Lorentz units) and .
Maxwell's equations taking into account hypothetical magnetic monopoles
Magnetic monopoles appear as possible or necessary components in some GU theories. They could be used to explain the quantification of the electrical charge, as Paul Dirac recognized as early as 1931. So far, however, no magnetic monopole has been observed. 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 Maxwell's equations.
If we put ρm for the monopoly charge density, for the current density and for the speed of the moving magnetic monopoly charges, only two of the four equations mentioned above change in differential form to
Interpretation: The field lines of the magnetic flux density begin and end in a magnetic charge.
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 the vanishing monopoles ρm = 0 leads back to the well-known equations given above.
Maxwell's equations taking into account a hypothetical photon rest mass
The rest mass of photons is 0. As an academic exercise, Maxwell's equations can be formulated taking into account a tiny photon rest mass different from 0. These are then called Maxwell-Proca equations. According to Proca (1930, 1946) this case can be described with a modified Lagrange function. The potentials get a special meaning through the photon rest mass: For example, the static electrical potential of a point charge becomes a spherically symmetrical Yukawa potential. This massively limits the range of electrical forces (power transmission by photons).
- Adolf J. Schwab: Conceptual world of field theory. Springer-Verlag Berlin 2002, ISBN 3-540-42018-5
- U. Krey, A. Owen: Basic Theoretical Physics - A Concise Overwiew, Springer-Verlag Berlin 2007, ISBN 978-3-540-36804-5
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