Introduction to Mathematical Physics/Electromagnetism/Electromagnetic field

Equations for the fields: Maxwell equations

Electromagnetic interaction is described by the means of Electromagnetic fields: $E$ field called electric field, $B$ field called magnetic field, $D$ field and $H$ field. Those fields are solution of Maxwell equations, \index{Maxwell equations}

{\mbox{ div }}D=\rho

{\mbox{ rot }}H=j+{\frac {\partial {D}}{\partial t}}

{\mbox{ div }}B=0

{\mbox{ rot }}E=-{\frac {\partial B}{\partial t}}

where $\rho$ is the charge density and $j$ is the current density. This system of equations has to be completed by additional relations called constitutive relations that bind $D$ to $E$ and $H$ to $B$ . In vacuum, those relations are:

D=\epsilon _{0}E

H={\frac {B}{\mu _{0}}}

In continuous material media, energetic hypotheses should be done (see chapter parenergint) .

Remark:

In harmonical regime\footnote{ That means that fields satisfy following relations:

E={\mathcal {E}}e^{j\omega t}

B={\mathcal {B}}e^{j\omega t}

} and when there are no sources and when constitutive relations are:

for $D$ field
$D(r,t)=\epsilon (r,t)*E(r,t)$
where $*$ represents temporal convolution\index{convolution} (value of $D(r,t)$ field at time $t$ depends on values of $E$ at preceding times) and:
for $B$ field:
$H={\frac {B}{\mu _{0}}},$

Maxwell equations imply Helmholtz equation:

\Delta {\mathcal {E}}+k^{2}{\mathcal {E}}=0.

Proof of this is the subject of exercise exoeqhelmoltz.

Remark:

Equations of optics are a limit case of Maxwell equations. Ikonal equation:

{\mbox{ grad }}^{2}L=n^{2}

where $L$ is the optical path and $n$ the optical index is obtained from the Helmholtz equation using WKB method (see section secWKB). Fermat principle can be deduced from ikonal equation {\it via} equation of light ray (see section secFermat). Diffraction's Huyghens principle can be deduced from Helmholtz equation by using integral methods (see section secHuyghens).

Conservation of charge

Local equation traducing conservation of electrical charge is:

eqconsdelacharge

{\frac {\partial \rho }{\partial t}}+{\mbox{ div }}{j}=0

secmodelcha

Modelization of charge

Charge density in Maxwell-Gauss equation in vacuum

{\mbox{ div }}E={\frac {\rho }{\epsilon _{0}}}

has to be taken in the sense of distributions, that is to say that $E$ and $\rho$ are distributions. In particular $\rho$ can be Dirac distribution, and $E$ can be discontinuous (see the appendix chapdistr about distributions). By definition:

a point charge $q$ located at $r=0$ is modelized by the distribution $q\delta (r)$ where $\delta (r)$ is the Dirac distribution.
a dipole\index{dipole} of dipolar momentum $P_{i}$ is modelized by distribution ${\mbox{ div }}(P_{i}\delta (r))$ .
a quadripole of quadripolar tensor\index{tensor} $Q_{i,j}$ is modelized by distribution $\partial _{x_{i}}\partial _{x_{j}}(Q_{i,j}\delta (r))$ .
in the same way, momenta of higher order can be defined.

Current density $j$ is also modelized by distributions:

the monopole doesn't exist! There is no equivalent of the point charge.
the magnetic dipole is ${\mbox{ rot }}A_{i}\delta (r)$

secpotelec

Electrostatic potential

Electrostatic potential is solution of Maxwell-Gauss equation:

\Delta V={\frac {\rho }{\epsilon _{0}}}

This equations can be solved by integral methods exposed at section chapmethint: once the Green solution of the problem is found (or the elementary solution for a translation invariant problem), solution for any other source can be written as a simple integral (or as a simple convolution for translation invariant problem). Electrical potential $V_{e}(r)$ created by a unity point charge in infinite space is the elementary solution of Maxwell-Gauss equation:

V_{e}(r)={\frac {1}{4\pi \epsilon _{0}r}}

Let us give an example of application of integral method of section chapmethint:

Example:

Potential created by an electric dipole, in infinite space:

V_{P_{i}}=\int V_{e}(r-r')\partial _{i}(P_{i}\delta (r'))

As potential is zero at infinity, using Green's formula:

V_{P_{i}}=-\int \partial _{i}(V_{e}(r-r'))(P_{i}\delta (r')).

From properties of $\delta$ distribution, it yields:

eqpotdipo

V_{P_{i}}=-\partial _{i}(V_{e}(r))P_{i}

seceqmaxcov

Covariant form of Maxwell equations

At previous chapter, we have seen that light speed $c$ invariance is the basis of special relativity. Maxwell equations should have a obviously invariant form. Let us introduce this form.

Current density four-vector

Charge conservation equation (continuity equation) is:

\nabla .j+{\frac {\partial \rho }{\partial t}}=0

Let us introduce the current density four-vector:

J=(j,ic\rho )

Continuity equation can now be written as:

\nabla J=0

which is covariant.

Potential four-vector

Lorentz gauge condition:\index{Lorentz gauge}

\nabla A-{\frac {\partial V}{\partial t}}=0

suggests that potential four-vector is:

A=(A,i{\frac {\phi }{c}})

Maxwell potential equations can thus written in the following covariant form:

\Box A_{\mu }=-\mu _{0}j_{\mu }

Electromagnetic field tensor

Special relativity provides the most elegant formalism to present electromagnetism: Maxwell potential equations can be written in a compact covariant form, but also, this is the object of this section, it gives new insights about nature of electromagnetic field. Let us show that $E$ field and $B$ field are only two aspects of a same physical being, the electromagnetic field tensor. For that, consider the equations expressing the potentials form the fields:

B=\nabla \wedge A

and

E=\nabla \phi -{\frac {\partial A}{\partial t}}.

Let us introduce the anti-symetrical tensor \index{tensor (electromagnetic field)} of second order $F$ defined by:

F_{\mu \nu }={\frac {\partial A_{\nu }}{\partial A_{\mu }}}-{\frac {\partial A_{\mu }}{\partial A_{\nu }}}.

Thus:

F_{\mu \nu }=\left({\begin{array}{cccc}0&B_{3}&-B_{2}&-{\frac {i}{c}}E_{1}\\-B_{3}&0&B_{1}&-{\frac {i}{c}}E_{2}\\B_{2}&-B_{1}&0&-{\frac {i}{c}}E_{3}\\{\frac {i}{c}}E_{1}&{\frac {i}{c}}E_{2}&{\frac {i}{c}}E_{3}&0\\\end{array}}\right)

Maxwell equations can be written as:

\partial _{\nu }F_{\mu \nu }=\mu _{0}j_{\mu }

This equation is obviously covariant. $E$ and $B$ field are just components of a same physical being^[1]

Footnote

↑ The electromagnetic interaction is an example of unification of interactions: before Maxwell's equations, electric and magnetic interactions were distinguished. Now, only one interaction, the electromagnetic interaction, needs to be considered. A unified theory unifies weak and electromagnetic interaction: the electroweak interaction ([#References|references]). The strong interaction (and the quantum chromodynamics) can be joined to the electroweak interaction {\it via} the standard model. One expects to describe one day all the interactions (the gravitational interaction included) in the frame of the great unification \index{unification}. }: the electromagnetic tensor. Expressing fields in various frames is now obvious using Lorentz transformation. For instance, it is clear why a point charge that has a uniform translation movement in a reference frame $R_{1}$ produces in this same reference frame a $B$ field.

[1] The electromagnetic interaction is an example of unification of interactions: before Maxwell's equations, electric and magnetic interactions were distinguished. Now, only one interaction, the electromagnetic interaction, needs to be considered. A unified theory unifies weak and electromagnetic interaction: the electroweak interaction ([#References|references]). The strong interaction (and the quantum chromodynamics) can be joined to the electroweak interaction {\it via} the standard model. One expects to describe one day all the interactions (the gravitational interaction included) in the frame of the great unification \index{unification}. }: the electromagnetic tensor. Expressing fields in various frames is now obvious using Lorentz transformation. For instance, it is clear why a point charge that has a uniform translation movement in a reference frame $R_{1}$ produces in this same reference frame a $B$ field.

[1]