Introduction to Mathematical Physics/Electromagnetism/Electromagnetic field

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Equations for the fields: Maxwell equations[edit]

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[edit]

Local equation traducing conservation of electrical charge is:

eqconsdelacharge


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

secmodelcha

Modelization of charge[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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.