Maxwell\'s equations

The Maxwell's equations , also called equations of Maxwell-Lorentz , are fundamental laws of the Physique. They constitute the postulates basic of the electromagnetism, with the expression of the electromagnetic Force of Lorentz.

These equations translate in form local different Théorème S (Gauss, Ampère, Faraday) which governed electromagnetism before Maxwell does not join together them in the form of integral equations. They thus give a mathematical framework precise to the fundamental concept of field introduces in physics by Faraday into the years 1830.

These equations show in particular that in stationary mode, the fields electric and magnetic are independent one of the other, whereas they are not it in variable mode. In the most general case, it is thus necessary to speak about the electromagnetic field, the dichotomy electric/magnetic being a fantastic notion. This aspect finds its formulation final in the formalism covariant presented in the second part of this article: the electromagnetic field is represented there by a single mathematical being of type Tenseur, the “tensor of Maxwell”, whose certain components are identified with those of the Electric field and others with those of the Magnetic field.

Historical aspects

The contribution of Maxwell

About 1865, Maxwell carried out a harmonious synthesis of the various experimental laws discovered by its predecessors (laws of the electrostatic , of the Magnétisme, the induction…), by expressing them in the form of a system of four partial derivative equations coupled. They were published in their final form in 1873 in the work Electricity and Magnetism .

But this synthesis was not possible that because Maxwell knew to exceed work of his precursors, by introducing into an equation a “missing link”, called the Courant of displacement, whose presence ensures the coherence of the unified building.

Heirs to Maxwell

The synthesis of Maxwell allowed later on the two larger projections of modern science:

the theory of the restricted Relativity (via the problem of the reference frame of hypothetical “the ether”). Indeed, the Maxwell's equations make it possible to predict the existence of a electromagnetic Onde, i.e. modification of one of the parameters (density of load, intensity of the current…) will have remote repercussions with a certain delay. However, the speed of these waves, C , calculated with the Maxwell's equations, are equal to the Speed of light which was measured in experiments. That made it possible to conclude that the light was an electromagnetic wave. The fact that C is the same one in all the directions and independent of the reference frame, conclusion which one draws from these equations, is one of the bases of the theory of relativity. In fact, one notices that if one changes reference frame, the traditional change of coordinates does not apply to the Maxwell's equations, it is necessary to use another transformation, the Transformation of Lorentz. Einstein tried to apply the transformations of Lorentz to traditional mechanics, and that led it to the restricted theory of relativity.

the Quantum physics. The study of the light and the electromagnetic waves, with in particular work of max Planck on the black Body and of Heinrich Hertz on the photoelectric Effet gave rise to the quantum theory in 1900.

Theory of Maxwell-Lorentz in the vacuum

One presents the microscopic theory below fundamental which gives the equations of Maxwell-Lorentz in the Vide in the presence of source S, which can be concentrated loadings and/or their microscopic electric currents associated if these loads are moving in the reference frame of study.

The macroscopic theory requiring the introduction of the fields D and H (and the associated Maxwell's equations) is discussed in details in the article: Electrodynamic of the continuous mediums.

One notes:

$\ rho (\ vec {R}, T)$ is the density of electric Charge local at the point $\ vec {R}$ at the moment $t$ .
$\ vec {J} (\ vec {R}, T)$ the vector Density of current.
$\ vec {E} (\ vec {R}, T)$ the vector Electric field.
$\ vec {B} (\ vec {R}, T)$ the magnetic pseudovector Induction.
$\ epsilon_0 \,$ the dielectric Permittivity of the vacuum.
$\ mu_0 \,$ the magnetic Permeability of the vacuum.

Equation of Maxwell-Gauss

The local equation of Maxwell

This local equation gives the divergence electric field according to the density of the electric charge. She is written:

This equation corresponds to a “ term of source ” : the density of charge electric is a source of the electric field. For example, for a concentrated loading $q \,$ fixed at the origin $O \,$ , the law of Coulomb giving the electrostatic field in a point $M \,$ of the space, not located by the vector position $\ vec {OM} = \ vec {R} = R \ \ vec {U} _r$ where $\ vec {U} _r \,$ is the radial unit vector, is written:

This electrostatic field checks the equation of Maxwell-Gauss for the static source:

where $\ delta^3 (\ vec {R}) \,$ is the Distribution of Dirac in space with three dimensions.

The theorem of Gauss

The equation of Maxwell-Gauss is inherited the Théorème of Gauss, which makes it possible to bind the flow electric field through a closed surface, to the interior load on this surface:

where $\ Sigma \,$ is an arbitrary closed surface, called surface of Gauss, and $Q_ {int} \,$ the interior total electric charge on this surface $\ Sigma \,$ .

Maxwell's equation of conservation of flow

The flow of the magnetic field through a surface $\ Sigma \,$ closed is always identically null:

The local equation of Maxwell

This local equation is with the magnetic field what the equation of Maxwell-Gauss is with the electric field, namely an equation with “term of source”, here identically no one:

It translates the following experimental fact: there does not exist magnetic monopoly . A magnetic monopoly would be a specific source of magnetic field, analog of the specific electric charge for the electric field. However, the basic object source of a magnetic field is the Aimant, which behaves like a magnetic Dipôle: a magnet indeed has a north pole and a south pole. The fundamental experiment consisting in trying to divide a magnet in two gives rise to two magnets , and not a north pole and a south pole separately.

Introduction of the vector potential

The vectorial Analysis watch which the divergence of rotational is always identically null:

Reciprocally, any function whose divergence is identically null can be expressed in the form of rotational.

The local equation of conservation of the magnetic flux thus makes it possible to define at least locally a vector potential $\ overrightarrow {has} \,$ such as:

The important problem of the unicity of the vector potential is discussed with the paragraph further: Invariance of gauge of the theory .

Equation of Maxwell-Faraday

This local equation translates the fundamental phenomenon of electromagnetic induction discovered by Faraday.

The local equation

It gives the rotational electric field according to the temporal derivative of the magnetic field:

That corresponds in the “variational term” : the variation of the magnetic field creates an electric field. Its integral form is the Faraday's law:

where $e \,$ is the electromotive force of induction in an electrical circuit and $\ Phi \,$ the magnetic flux through this circuit.

Introduction of the electric potential

The vectorial Analyze watch that the rotational one of a gradient is always identically null:

The equation of Maxwell-Faraday coupled with the local existence of a vector potential $\ vec {has} \,$ make it possible to define (at least locally) the electric potential $V \,$ (scalar) such as:

The important problem of the unicity of the electric potential is discussed with the paragraph further: Invariance of gauge of the theory .

Equation of Maxwell-Amp

The local equation of Maxwell

This equation is inherited the Théorème of Amp. In local form, she is written in terms of the vector density of current

\ overrightarrow {J} \,

Introduction of the displacement current

The preceding equation can be rewritten:

by introducing the Displacement current of Maxwell:

The integral form binds the circulation of the magnetic field on a contour $C$ closed, and the currents which cross surface being based on this contour:

Conservation equation of the load

Let us take the divergence of the equation of Maxwell-Amp:

One can write by permuting the derivative space and temporal, then by using the equation of Maxwell-Gauss:

One obtains finally the local equation of conservation of the electric charge:

The reader will have noted the essential presence of the term of Courant of displacement introduces by Maxwell for obtaining this equation.

Invariance of gauge of the theory

The vectorial Analysis watch which the divergence of rotational is always identically null:

The local equation of conservation of the magnetic flux thus makes it possible to define at least locally a vector potential $\ overrightarrow {has} \,$ such as:

The vectorial analysis also says to us that

Then the vector potential is not defined in a single way since the following transformation, with $f \,$ an unspecified function

\ overrightarrow {has} \ rightarrow \ overrightarrow {has} + \ overrightarrow \ nabla F

do not modify by the value of the field $\ overrightarrow {B} \,$ . This is an example of Transformation of gauge. It is thus necessary to impose conditions additional to determine $\ overrightarrow {has} \,$ in an not-ambiguous way. One calls that of the Conditions of gauge, for example the Jauge of Coulomb or the Jauge of Lorenz.

The reader will note that in traditional Physique, the vector potential seems to be only one convenient mathematical tool to analyze the solutions of the Maxwell's equations, but directly does not seem to be a physical size measurable . In 1959, within the framework of the Quantum physics, Aharonov and Bohm showed that the vector potential had an observable effect in quantum Mécanique: it is the Effet Bohm-Aharonov.

The vectorial Analyze watch that the rotational one of a gradient is always identically null:

The equation of Maxwell-Faraday coupled with the local existence of a vector potential $\ vec {has} \,$ make it possible to define (at least locally) the electric potential $V \,$ (scalar) such as:

The potential $V \,$ either is not defined to him in a single way but the transformation of associated gauge is related to that of $\ overrightarrow {has} \,$ is the following one (one points out that of $\ overrightarrow {has} \,$ by preoccupation with a clearness) and one has

\ left \ {
\begin{matrix}
V & \ rightarrow & V - \ partial_t F \ \
\ overrightarrow {has} & \ rightarrow & \ overrightarrow {have} + \ overrightarrow {\ nabla} F
\end{matrix}\right.
\,

These two equations give the invariance of complete gauge of the Maxwell's equations.

Solutions of the equations of the electromagnetic field.

To simplify, in accordance with the practice, we will allot these equations to Maxwell, by calling them “Maxwell's equations” (EM).

Mathematical solutions of the Maxwell's equations in the vacuum.

Let us solve the EM in the space possibly limited by conditions which keep the linearity.

Let us represent solutions by letters Q, R,… (whole of the formed 6-vectors of the six components of the field in any point of coordinates X, there, Z, T). By definition of the linearity, αQ + βR +…, where α, β… is real constants is a solution. Consequently, the solutions are represented by the points of a real vector space. In accordance with the definition introduced in acoustics, a mode is a ray of this space. A complete system of solutions constitutes a reference mark in this space named space of the solutions sometimes, sometimes space of the modes. A particular solution in a mode is obtained by multiplying a field of this mode posed like field of amplitude unit, by a real constant, the amplitude.

With a suitable system of units, energy W (Q) of a solution Q is the integral extended to all space, at a given moment, of Q ²; it is very often forgotten that this equation is not - linear, so that if one can add fields, corresponding energies are not added. By considering that W (Q) is the scalar square of Q, by processes of orthogonalization of Schmitt, one obtains complete systems of orthogonal solutions, or of the complete systems of orthogonal modes. In these systems, energies are independent.

Planck posed that energy in a monochromatic mode of frequency ν being propagated in a black body at the temperature T is W = hν/(exp (hν/kT) - 1) +K. The erroneous value of K given by Planck was corrected by Nernst in 1916; the value K = hν/2 is easily found because thermodynamics imposes that W tends towards kT when T tends towards the infinite one. This formula defines the temperature of a mode. However the interpretation of this formula is physically delicate because the definition of a pure frequency ν supposes an experiment of infinite duration.

Introduction of the electric charges

One can calculate the fields emitted by loads, for example the field emitted by an oscillating dipole. To be reduced to the preceding problem, one uses the “trick of Schwarzschild and Fokker”. The field emitted by a source is named “field delayed” Q^{R. Stripped source, this field is not not solution of the EM. To obtain an identical solution in the future, it is necessary to add a “advanced field” Q^{A to him. By this definition, Q^{A + Q^{R is solution of the EM. Thus, in substituent the advanced field with the source, one is brought back to the linear problem of a field in the vacuum and one can define modes.

Physical solutions of the Maxwell's equations.

The mathematician is free to be unaware of contingencies physics while making, in a more or less implicit way, the unrealistic assumptions; thus we supposed that there exist isolated electromagnetic systems in which it is permissible to introduce some selected fields; the establishment of the law of Planck is a remarkable example of this assumption. Let us show that this assumption is physically absurd.

Except perhaps in neutron stars, the matter consists of small particles compared to their distances. These particles are the sources of the electromagnetic fields; the field emitted by a particle decrease with the distance, it is thus much more intense in its vicinity than near other particles. However the absorption of a field is the addition of an opposite field; the generation of such a field with difficulty and is roughly obtained to obtain a absorption activates in acoustics, by means of high speakers, to reduce the intensity of a noise. The generation of opposed electromagnetic field created by a particle, intense in its vicinity requires the addition of the weak fields created by many other particles, which does nothing but complicate the problem which appears thus insoluble: there remains a stochastic residual field far from the sources.

The existence necessary of residual fields, known, seems it for a long time, is exploited, within the framework of electromagnetism, since the end of the nineteenth century by the charlatans who recognize themselves under names such as “radiesthesists”. In electromagnetism, the physicists encountered an impossibility of evaluating these “residual fields” until the determination of their median value hν/2 by monochromatic mode, with 0K, made by Planck and Nernst. The theory of the emission and absorption due to Einstein (1917) was supplemented by the interpretation of the spontaneous emission of light like an amplification of the residual field. The direction of energy exchange between a monochromatic source and the field depend on the interference of the field emitted with the field of external origin preexistent in the mode, therefore simply of the relative phases.

The residual field in a black body with 0K, usually named “field of item zero”, is often presented like a mysterious field of quantum origin, absurdity presentation since its median value was correctly evaluated by Nernst more than ten years before the birth of quantum mechanics; without its knowledge it would not have been possible to found the quantum electrodynamics by identifying the electromagnetic energy of a mode to that of a quantum harmonic oscillator.

It is necessary to take care not to think that in a mode there are two fields, a usual field and a field of item zero; such a design would be absurd since the field in a mode depends on only one real parameter, the amplitude of the field.

Thus, an absorption of a mode can reduce the amplitude of the field only up to one lower limit which corresponds, on average, with energy hν/2; an emission is an amplification of a preexistent field; it is known as spontaneous if the preexistent field corresponds to an energy close to hν/2, and induced if this energy is notably higher than hν/2.

Quantification in traditional electrodynamics.

A physical system has, in general, of the relative minima of energy. In nonevolutionary mode (stationary), the system, excited by an electromagnetic field about hν/2 in each mode which it is likely to emit (thus to absorb), remains in the vicinity of a minimum of energy; for each monochromatic mode, its excitation leads it to radiate a field in squaring with the field incidental, which does not produce any permanent energy exchange, but introduced a delay, the refraction. For a more intense field, in particular because of a favorable fluctuation of the field, the system can cross a collar of its diagram of energy and to absorb an energy hν this absorption can lead to a not very stable level from where the system can evolve/move quickly worms of other levels, in a more or less radiative cascade which brings it in a stationary state, stable.

In a classical theory, no paradox can be allowed, in particular the paradox of Einstein, Podolsky and Rosen does not exist: let us suppose that an atom loses a resonance energy hν, for example by the radiation of a dipole. The emission mode of this dipole is not orthogonal with the emission modes (thus of absorption) of other atoms whose amplitude can be increased; 0,1,2,… atoms can then absorb hν, even if, on average, only one atom is excited; the residual fields play the part of a thermodynamic bath.

Some usual errors

- It was written that the electron of a hydrogen atom following an orbit of Bohr emits a field, therefore rayon of energy and should fall on the core. The electron emits well a field, but, a very weak energy, because of the interference of the field emitted with the residual field; this energy falls to zero if the orbit is slightly corrected, so that the energy of the stationary state undergoes the displacement of Lamb.

- The study of the starting of a laser seems to indicate that the field of the item zero armature an emission twice more intense than a field of larger intensity. To take account of this result, one can introduce a “radiation of reaction”, ad hoc. The true explanation is very simple: an atom is excited by a field in the mode which it can emit, known as spherical; with the starting of the laser, there exists in this mode an amplitude corresponding to hν/2; the laser functions on a mode of wave planes of which it is necessary to take the spherical component to excite the atom, which divides energy by two.

- There does not exist isolated electromagnetic system; to forget that the minimum field is the field of the item zero conduit to errors when weak fields are detected.

Covariante formulation

NB This part follows traditional conventions of sign of MTW

This part also adopts the Convention of summation of Einstein.

Geometry of the space time of Minkowski

The space time of Minkowski (1908) is a differential Variété M punt provided with metric Lorentzian.

That is to say an unspecified frame of reference $x^ {\ driven}$ around an event (not) $P$ of the space time, and is ${\ mathbf E} _ {\ driven} (X)$ a local base of $T_xM$ , tangent space to the variety at the point $x \ in M$ . A tangent vector $\ mathbf W \ in T_xM$ is written then like the linear combination:

The $w^ {\ driven}$ are called the components contravariantes vector W . The metric Tenseur $\ mathbf \ eta$ is the symmetrical bilinear form:

In a orthonormée base of a inertial reference frame , its components covariantes $\ eta_ {\ driven \ naked}$ are:

Its components contravariantes $\ eta^ {\ driven \ naked}$ check:

One obtains explicitly:

Following usual conventions below will be used:

a Greek index varies from 0 to 3. It is associated with a size in the space time.
a Latin index varies from 1 to 3. It is associated with the space components of a size in the space time.

For example, the components contravariantes of the 4-vector position are written in an orthonormal frame of reference:

The metric tensor defines for each point $x \ in M$ of the space time a pseudo Produit scalar ( pseudo with the direction where the assumption of positivity is withdrawn) in tangent space $T_xM$ Euclidean in M in the point $x$ . If $\ mathbf u$ and $\ mathbf v$ is two vectors of $T_xM$ , their scalar product is written:

In particular, by taking two basic vectors, one obtains the components:

$w^ {\ driven}$ indicating the components contravariantes vector W , one can define in the same way his components covariantes by:

For example, the components covariantes of the 4-vector position are written in an orthonormal frame of reference:

Quadri-gradient

One introduces the differential operator quadri-gradient by his components covariantes:

Its components contravariantes are written:

The operator invariant of Alembertien is written for example:

Quadri-potential

One introduces the electromagnetic quadri-potential by his components contravariantes:

where $V$ is the electric potential scalar, and $\ vec {has}$ the magnetic vector potential. Its components covariantes are written:

The laws of transformation of gauge written previously are thus summarized in this notation in the form

$A^ \ driven \ rightarrow A^ \ driven + \ partial^ \ driven F
\,$

The condition of gauge of Lorenz is written for example way covariante:

Quadri-courant

One introduces the quadri-courant electromagnetic by his components contravariantes:

where $\ rho$ is the electric scalar density of load, and $\ vec {J}$ the vector density of current. Its components covariantes are written:

Tensor of Maxwell

The tensor of Maxwell is the antisymmetric tensor of definite row two starting from the quadri-potential by:

Its components covariantes are written explicitly:

One obtains his components contravariantes while writing:

The metric one being diagonal in an inertial reference frame, one then obtains the following formulas, without summation on the repeated indices :

$F^ {00} \ = \ \ eta^ {00} \ \ eta^ {00} \ F_ {00} \ = \ + \ F_ {00} \ = \ 0$
$F^ {0i} \ = \ \ eta^ {00} \ \ eta^ {II} \ F_ {0i} \ = \ - \ F_ {0i}$
$F^ {ij} \ = \ \ eta^ {II} \ \ eta^ {jj} \ F_ {ij} \ = \ + \ F_ {ij}$

that is to say explicitly:

Maxwell's equations in form covariante

The Maxwell's equations put in relativistic form covariante.

the two Maxwell's equations without terms of sources are written:

the two Maxwell's equations with terms of sources are written:

Since the tensor of Maxwell is antisymmetric, this last relation involves in particular that quadri-courant is preserved :

Equation of propagation for the quadri-potential out of gauge of Lorentz

By explicitly writing the tensor of Maxwell in terms of the quadri-potential in the equation covariante with term of sources, one obtains for the member of left:

In the gauge of Lorenz $\ partial_ {\ alpha} A^ {\ alpha} = 0$ , the second term disappears, and the Maxwell's equation with term of sources is reduced to an equation propagation for the quadri-potential:

The solution of this equation is written way a simple if one knows a Fonction of Green of the equation of propagation, i.e. function G (X) solution of the partial derivative equation:

where $\ delta (X)$ is the distribution of Dirac. One then obtains the quadri-potential in the form of a Produit convolution:

Example: delayed potentials

In traditional electrodynamics, one generally uses the Fonction of Green delayed which satisfies the assumption of Causalité:

This function of Green is written:

Equations of Maxwell-Lorentz in the material mediums

See also: Electrodynamic of the continuous mediums

Internal bonds

Vector of Poynting
Establishment of the equation of propagation starting from the Maxwell's equations
Electrodynamic of the continuous mediums
Invariance of Lorentz
Field of Maxwell

Virtual library

Ruth Durrer ; '' Electrodynamique II '' (PostScript): thorough course given by the author (Department of Theoretical physics, University of Geneva, Switzerland) to the students of second year of first cycle (123 pages).

Jean-Michel Raimond; '' Electromagnétisme & restricted relativity '' (pdf): thorough course (analytical mechanics, relativity & electromagnetism) given by the author (Kastler-Brossel laboratory, ENS Ulm, Paris) to the first-year students of Inter-University Magistère of Physics.

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