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}

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:

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

Conservation of charge
Local equation traducing conservation of electrical charge is:

Modelization of charge
Charge density in Maxwell-Gauss equation in vacuum

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: Current density $$j$$ is also modelized by distributions:
 * 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.
 * the monopole doesn't exist! There is no equivalent of the point charge.
 * the magnetic dipole is $$\mbox{ rot } A_i\delta(r)$$

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:

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

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:

Let us introduce the current density four-vector:

Continuity equation can now be written as:

which is covariant.

Potential four-vector
Lorentz gauge condition:\index{Lorentz gauge}

suggests that potential four-vector is:

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

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:

and

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

Thus:

Maxwell equations can be written as:

This equation is obviously covariant. $$E$$ and $$B$$ field are just components of a same physical being


 * Footnote