# 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

Footnote

<references>

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.