# 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:

$D=\epsilon_0E$

$H=\frac{B}{\mu_0}$

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:

$\Box A_\mu=-\mu_0j_\mu$

### 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:

$\partial_{\nu}F_{\mu\nu}=\mu_0j_{\mu}$

This equation is obviously covariant. $E$ and $B$ 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 $R_1$ produces in this same reference frame a $B$ field.