# Introduction to Mathematical Physics/Electromagnetism/Electromagnetic field

## Equations for the fields: Maxwell equations

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

${\displaystyle {\mbox{ div }}D=\rho }$
${\displaystyle {\mbox{ rot }}H=j+{\frac {\partial {D}}{\partial t}}}$
${\displaystyle {\mbox{ div }}B=0}$
${\displaystyle {\mbox{ rot }}E=-{\frac {\partial B}{\partial t}}}$

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

${\displaystyle D=\epsilon _{0}E}$
${\displaystyle H={\frac {B}{\mu _{0}}}}$

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

Remark:

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

${\displaystyle E={\mathcal {E}}e^{j\omega t}}$
${\displaystyle B={\mathcal {B}}e^{j\omega t}}$

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

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

Maxwell equations imply Helmholtz equation:

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

Proof of this is the subject of exercise exoeqhelmoltz.

Remark:

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

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

where ${\displaystyle L}$ is the optical path and ${\displaystyle n}$ 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

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

secmodelcha

## Modelization of charge

Charge density in Maxwell-Gauss equation in vacuum

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

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

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

Current density ${\displaystyle j}$ is also modelized by distributions:

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

secpotelec

## Electrostatic potential

Electrostatic potential is solution of Maxwell-Gauss equation:

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

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 ${\displaystyle V_{e}(r)}$ created by a unity point charge in infinite space is the elementary solution of Maxwell-Gauss equation:

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

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

Example:

Potential created by an electric dipole, in infinite space:

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

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

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

From properties of ${\displaystyle \delta }$ distribution, it yields:

eqpotdipo

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

seceqmaxcov

## Covariant form of Maxwell equations

At previous chapter, we have seen that light speed ${\displaystyle c}$ 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:

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

Let us introduce the current density four-vector:

${\displaystyle J=(j,ic\rho )}$

Continuity equation can now be written as:

${\displaystyle \nabla J=0}$

which is covariant.

### Potential four-vector

Lorentz gauge condition:\index{Lorentz gauge}

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

suggests that potential four-vector is:

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

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

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

### 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 ${\displaystyle E}$ field and ${\displaystyle B}$ 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:

${\displaystyle B=\nabla \wedge A}$

and

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

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

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

Thus:

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

Maxwell equations can be written as:

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

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

Footnote
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 ${\displaystyle R_{1}}$ produces in this same reference frame a ${\displaystyle B}$ field.