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} Template:IMP/eq Template:IMP/eq Template:IMP/eq Template:IMP/eq 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: Template:IMP/eq Template:IMP/eq In continuous material media, energetic hypotheses should be done (see chapter parenergint) . Template:IMP/rem Template:IMP/rem

Conservation of charge

Local equation traducing conservation of electrical charge is: Template:IMP/label Template:IMP/eq

Modelization of charge

Charge density in Maxwell-Gauss equation in vacuum Template:IMP/eq 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 $\text{ div }(P_i\delta (r))$.
• 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 $\text{ rot }{A}_i\delta (r)$

Electrostatic potential

Electrostatic potential is solution of Maxwell-Gauss equation:

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: Template:IMP/eq Let us give an example of application of integral method of section chapmethint: Template:IMP/exmp

Template:IMP/label

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: Template:IMP/eq Let us introduce the current density four-vector: Template:IMP/eq Continuity equation can now be written as: Template:IMP/eq which is covariant.

Potential four-vector

Lorentz gauge condition:\index{Lorentz gauge} Template:IMP/eq suggests that potential four-vector is: Template:IMP/eq Maxwell potential equations can thus written in the following covariant form: Template:IMP/eq

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: Template:IMP/eq and Template:IMP/eq Let us introduce the anti-symetrical tensor \index{tensor (electromagnetic field)} of second order $F$ defined by: Template:IMP/eq Thus: Template:IMP/eq Maxwell equations can be written as: Template:IMP/eq This equation is obviously covariant. $E$ and $B$ field are just components of a same physical being

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